Semi-Hyperbolicity and Bi-Shadowing

P. Diamond, P. Kloeden, V. Kozyakin, A. Pokrovskii z

Semi-Hyperbolicity and Bi-Shadowing December 6, 2010

Springer

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Preface

Hyperbolicity is a very important concept in dynamical systems theory. It has been extensively investigated over the past half century along with its associated concepts of robustness of dynamical behaviour [11,13,18,19,24–26, 37, 61, 62, 87,92, 94, 102]. Many deep results have been obtained and there are now numerous monographs and textbooks on the subject [1, 14, 20, 109, 113]. Most of the work on hyperbolicity, however, concerns abstract differentiable dynamical systems and it is often very difficult to show that the results apply to systems generated by specific differential equations such as the Lorenz equations. In addition, in recent years, there have been many interesting and useful generalizations and extensions of dynamical systems ideas to what were previously ‘non-mainstream’ applications arising, for example, in nonsmooth systems, numerically generated systems and systems with hysteresis, to name just a few. The existing theory of hyperbolicity does not apply directly to these situations, but many of the associated results on robustness and shadowing are nevertheless very important and useful for them too. Since we were not able to not find any suitable ‘working’ tools in the literature to handle such real, numerical or physical applications, during the 1990s, we developed a practial approach, first for noninvertible differentiable mappings and later for Lipschitz mappings, to investigate ‘robustness’ issues for ‘real-world’ systems. We called this concept semi-hyperbolicity. It arose indirectly in the context of our research on the effects of spatial discretization on the behaviour of a dynamical system, in particular by using finite machine arithmetic in computer representations of dynamical systems. Subsequently, this idea rapidly broadened into a series of papers in which differing aspects and applications of the concept were explored. These and some recent papers form the basis of this monograph, the aim of which is to present a more complete and systematic development of the concept of semi-hyperbolicity, as well as to illustrate its usefulness. The concept of bi-shadowing was also developed in the above papers. Shadowing is a well known consequence of hyperbolicity, and also holds in weaker

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situations of semi-hyperbolicity. Essentially it says that there is always a true solution near a pseudo-solution, which could be arbitrary or, more interestingly, a solution of some approximating system. Bi-shadowing includes the converse effect and is definitely nontrivial when the pseudo-solutions that are near a given true solution come from a specified class or approximating solution. We would like to stress that semi-hyperbolicity has, essentially, very little in common with ‘classical’ hyperbolicity — no invariant splitting, no ‘real’ smoothness of a system, no invertibility, and so on. Of course, when all of these properties are present in a system, then semi-hyperbolicity is often a sufficient condition for hyperbolicity of the system. For this reason, in this monograph we cite only very general facts about hyperbolic systems that we need here, rather than going into very deep problems of hyperbolicity that are usually considered in monographs on the subject. We reiterate that while the connection with the theory of hyperbolic systems is important and cannot be ignored, much of our motivation comes from our interest and background in applications of dynamical systems and this has naturally influenced the types of questions asked and investigated here. The theory of semi-hyperbolicity and bi-shadowing is developed systematically in this monograph in nine chapters. There are also two appendices, one by Marcin Mazur, Jacek Tabor and Piotr Ko´scielniak on the the relationship between hyperbolicity and semi-hyperbolicity in linear systems and one by Janosch Rieger on semi-hyperbolicity and bi-shadowing in set-valued systems. Brisbane–Frankfurt–Moscow–Cork 1992–2010

Phil Diamond Peter Kloeden Victor Kozyakin Alexei Pokrovskii

Dedication This book is dedicated to the memory of our coauthor and friend Alexei Pokrovskii, who passed away, much too soon, shortly after the manuscript was completed. Alexei was the principal source of many of the ideas in this book. December 2010

Phil Diamond Peter Kloeden Victor Kozyakin

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Modeling Dynamical Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Semi-Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Outline of Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 4

2

Smooth Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Hyperbolic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Hyperbolic Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Anosov Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Hyperbolic Diffeomorphisms . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Expansivity and Shadowing . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Conjugate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Hyperbolic Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 7 9 10 12 13 13 14 16 17 18

3

Semi-Hyperbolic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Differentiable Mappings on Compact Manifolds . . . . . . 3.1.2 Differentiable Mappings on Compact Sets . . . . . . . . . . . 3.1.3 Lipschitz Mappings on Compact Sets . . . . . . . . . . . . . . . 3.1.4 Continuous Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Elementary Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Anosov Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . .

21 21 23 24 24 25 28 28 29

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Contents

4

Semi-Hyperbolic Sequences of Matrices . . . . . . . . . . . . . . . . . . . 4.1 The Split Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 A Perturbation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Semi-Hyperbolicity Implies Hyperbolicity . . . . . . . . . . . . . . . . . . 4.2.1 Sequence Operators R and L . . . . . . . . . . . . . . . . . . . . . 4.2.2 An Equivariant Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Hyperbolicity of Sequences of Matrices . . . . . . . . . . . . . . 4.3 A More Abstract Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Semi-Hyperbolic Linear Operators . . . . . . . . . . . . . . . . . . 4.3.2 Equivalent Operators in Sequence Spaces . . . . . . . . . . . . 4.3.3 An Inversion Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Hyperbolicity of the Linear Operator A . . . . . . . . . . . . .

35 35 37 39 40 44 51 53 53 57 58 63

5

Semi-Hyperbolicity and Hyperbolicity . . . . . . . . . . . . . . . . . . . . 5.1 Perturbation of Anosov Endomorphisms . . . . . . . . . . . . . . . . . . . 5.2 Converse Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Alternative Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 72 73

6

Expansivity and Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Expansivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Lipschitz Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Direct Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Inverse Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Bi-Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Bi-Shadowing of Finite Trajectories . . . . . . . . . . . . . . . . . 6.3.2 Bi-Shadowing of Infinite Trajectories . . . . . . . . . . . . . . . . 6.3.3 Cyclic Bi-Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 75 79 83 84 85 87 94 95 102 104

7

Structural Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Topological Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Structural Stability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Weak Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Weak Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Definition of Chaotic Behaviour . . . . . . . . . . . . . . . . . . . . 7.3.2 Perturbations of Bi-Shadowing Systems . . . . . . . . . . . . . 7.3.3 Semi-Hyperbolic Lipschitz Mappings . . . . . . . . . . . . . . . . 7.3.4 Perturbations on Sets of Chain Recurrent Points . . . . .

107 107 109 111 115 117 117 119 123 124

Contents

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8

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Semi-Hyperbolic Mappings in Banach Spaces . . . . . . . . . . . . . . 8.1.1 Bi-Shadowing: Completely Continuous Perturbations . 8.1.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Nonautonomous Difference Equations . . . . . . . . . . . . . . . . . . . . . 8.2.1 Semi-Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Semi-Hyperbolicity Implies Bi-Shadowing . . . . . . . . . . . 8.2.3 Twisted Horseshoe Mapping . . . . . . . . . . . . . . . . . . . . . . . 8.3 Chaotic Attractors under Discretization . . . . . . . . . . . . . . . . . . . 8.4 Delay Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Systems with Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Transducer Stop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Ishlinskii and Besseling Systems . . . . . . . . . . . . . . . . . . . .

129 129 129 131 131 137 139 141 142 144 148 151 152 153

9

Extensions and Further Applications . . . . . . . . . . . . . . . . . . . . . 9.1 Split-Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Split-Hyperbolicity in Product-Spaces . . . . . . . . . . . . . . . 9.1.2 An Application to Hysteresis . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Split-Hyperbolicity and Strong Compatibility . . . . . . . . 9.2 Topological Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Semiconductor Lasers Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Truncated Lang–Kobayashi equations . . . . . . . . . . . . . . . 9.3.2 Poincaré Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Avian Influenza in a Seabird Colony . . . . . . . . . . . . . . . . . . . . . 9.4.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Numerical Evidence of Chaotic Trajectories . . . . . . . . . . 9.4.3 Main Assertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Existence of Chaotic Behaviour . . . . . . . . . . . . . . . . . . . . 9.5 Chaotic Canard-Type Trajectories . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Kaldor Model of the Business Cycle . . . . . . . . . . . . . . . . 9.6.2 Periodically Forced KdVB Equations . . . . . . . . . . . . . . . 9.6.3 Piecewise Linear Oscillator . . . . . . . . . . . . . . . . . . . . . . . . 9.6.4 Oscillations of a Ferromagnetic Pendulum . . . . . . . . . . .

155 155 155 157 159 160 163 164 165 167 168 169 171 172 174 175 179 179 180 181 183

A

Semi-Hyperbolicity: Estimations . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Functional Calculus and its Consequences . . . . . . . . . . . . . . . . . A.3 General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 185 190 192

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B

Contents

Set-Valued Semi-Hyperbolic Mappings . . . . . . . . . . . . . . . . . . . B.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.1 Strong semi-hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . B.1.2 Weak semi-hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Expansivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Bi-Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197 198 200 202 205

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Chapter 1

Introduction

In this chapter the appeal of introducing ideas of semi-hyperbolicity and bishadowing is explained from the point of view of mathematical modeling of real processes.

1.1 Modeling Dynamical Processes Real world dynamical processes can be extremely complicated, yet very many quite simple idealized mathematical models often appear to capture the essence of what is happening and allow useful and testable predictions to be made. This fact, which is to some extent justifiable mathematically, has been central to the resounding success of rationalist scientific thinking on the physical world over the past four hundred years. Moreover, a large amount of mathematical analysis owes its origin to the development of concepts and techniques needed to formulate and investigate such idealized mathematical models, which, though relatively simple to state, are by no means trivial to analyze and develop. Traditionally mathematical models have involved differential equations, both ordinary and partial, and thus represent continuous time dynamical systems on appropriately chosen state spaces. In contrast, much of the modern theory of dynamical systems has focussed on discrete time dynamical systems generated by iteration of a mapping f of a given state space X into itself, that is on difference equations xn+1 = f (xn ),

n = 0, 1, 2, . . . .

(1.1)

Though seemingly less complicated, the dynamical behaviour can often be richer without the restricting constraint of continuous time, but in any case many results can often be transferred to and from the differential context by means of suspensions, time-1 maps or Poincaré return maps. Indeed, mo-

1

2

1 Introduction

tivated by the properties of solutions of smooth differential equations, the mapping f in (1.1) is often assumed to be a diffeomorphism and the state space X to be a compact manifold. Investigations of long term or asymptotic behaviour have dominated the development of the theory of dynamical systems. Not only has the behaviour of individual systems been of interest, especially the existence of attractors, but also the classification of classes of systems that share common behavioural characteristics or retain them under transformations and perturbations. Stability is the crucial concept here, both within a given system and across a class of systems, and respectively described as dynamical stability or structural stability. Both types of stability here have connotations of robustness of dynamical behaviour. An idealized mathematical model f is typically taken as given and as the starting point of a mathematical investigation, for which it remains a fixed reference point. Approximation methods may be required, but an approximating system fh , for example the difference equation (1.1) implemented in the finite arithmetic field of a computer, and its behaviour is then then usually referred back to that of the original mathematical model. In practice, a mathematical model f is precisely that, a conveniently selected idealization of some nearby, but only imprecisely known system f˜. In applications it is thus desirable that the properties of an approximation fh to a modeled system f also relate to those of possible underlying systems f˜ on which the model is based. Hence, robustness of dynamical behaviour is a particulary important issue in mathematical modeling.

1.1.1 Hyperbolicity The concept of hyperbolicity in dynamical systems provides an elegant means of addressing one the the fundamental problems raised in mathematical modeling, namely that all systems close to an idealized model should share its essential dynamical properties. In a sense it combines and subsumes the idea of total stability, where all systems close to one with an asymptotically stable steady state have a small attracting set about this steady state, along with the analogous idea of total instability. It is based on the elementary observation that a linear mapping with a saddle point, hence with both contracting stable and expanding unstable directions, is robust under small parameter changes of the linear mapping itself as well as under small nonlinear perturbations. This extends easily to a hyperbolic cycle with the proviso that the corresponding stable and unstable linear manifolds are successively mapped onto their counterparts under the linearized mapping. The deep insight of Anosov [6–8] was to generalize this idea to an arbitrary compact invariant subset, in his case a manifold, for each point of which there is a splitting of the tangent space into stable and unstable linear manifolds.

1.1 Modeling Dynamical Processes

3

Hyperbolicity and its implications have been intensively investigated since the 1960s led by Smale and his coworkers, with interest being heightened by the presence of chaos generating mechanisms within noncyclic hyperbolic sets (see, e.g. [60, 65, 112] and bibliography therein). Identifying and classifying such sets have been fundamental tasks, as also has been establishing their structural stability, where all systems in some neighbourhood of a system with such a hyperbolic set are homeomorphic to it. This last generalizes the robustness proved in the Hartman–Grobman theorem [65, 123, 124] for a system with a simple saddle point. Symbolic dynamics [65, 131] has been a key tool here, in showing the existence of a homeomorphism between a system restricted to a hyperbolic set and the shift operator on a space of symbol sequences. This conjugacy of these systems allows the more transparent complicated behaviour of the latter to be transferred back to the original system. Shadowing [16, 29–32, 35, 82, 103, 104, 110, 111, 133], which asserts the existence of a true trajectory close to any approximate, pseudo-trajectory, is another useful property, suggesting that computer simulations do indeed reflect the dynamics of chaotic hyperbolic systems. The importance of hyperbolicity in the theory of dynamical systems cannot be understated, yet there remains a very wide gap between the deep theoretical understanding that it provides and applying these mathematical results to specific, practical dynamical systems. There are two reasons for this, one involving the conditions placed upon the systems, and the other concerning the nature of the mathematical theorems themselves and their manner of proof. Much of the theory of hyperbolic systems deals with diffeomorphisms on compact manifolds, yet many interesting systems lack the invertibility, as do many difference equations in population modeling, or the smoothness of diffeomorphisms, for example in hysteresis or control switching. Of course, there have been attempts to generalize hyperbolicity to homeomorphisms or to noninvertible maps, as in Ruelle’s pre-hyperbolicity, but then the second issue of practical application becomes paramount. Hyperbolicity is an extremely difficult property to verify for specific systems generating diffeomorphisms and even more so for its generalizations to homeomorphisms and noninvertible maps. Indeed, nearly all mathematically confirmed examples of hyperbolic systems are artificial constructs. The nature of many theorems on hyperbolic systems and their proofs is another obstacle to their applicability to specific systems, particularly as many proofs are nonconstructive or lack clear, tight estimates. Moreover, many assertions in theorems hold only generically, that is for systems belonging to some residual subset of systems. Although typical in such a sense, it says little about any specific model to hand.

4

1 Introduction

1.1.2 Semi-Hyperbolicity Although in part restrictive, as argued above, hyperbolicity is nevertheless an ubiquitous and important property of dynamical systems, at least in the sense that many dynamical systems satisfy some if not all of its defining properties and enjoy many of the dynamical consequences. In particular, smoothness and invertibility of the system are not essential, nor are continuity and equivariance of the tangent space splitting at each point of a hyperbolic set, nor indeed is the invariance of the hyperbolic set itself. As mentioned above, various generalizations of the definition of hyperbolicity have been proposed, but their actual applicability to specific systems is perhaps questionable. What is urgently required is a pragmatic reformulation of the concept of hyperbolicity that is both widely and effectively applicable and at the same time directly addresses both the practical and theoretical issues raised by the use and analysis of mathematical models. The authors have previously introduceded such a reformulation, first for differentiable mappings and then for Lipschitz mappings, called semi-hyperbolicity. This development arose indirectly in the context of research on the effect of spatial discretization on the behaviour of a dynamical system, in particular that resulting from computer representations, in finite machine arithmetic, of a dynamical system, and appeared as a series of papers in which differing aspects and applications of the concept were explored. These papers [2–5, 41–51, 74, 75, 80] form the basis of this monograph, the aim of which is to present a more coherent and systematic development of the concept of semi-hyperbolicity, and to illustrate its practicality in applications. While the relationship with the theory of hyperbolic systems is important and will be discussed, much of the motivation of this work stems from the authors’ interests and background in applications of dynamical systems. This has naturally influenced the types of questions asked and investigated.

1.2 Outline of Book The book consists of nine Chapters, two Appendices, and a List of Symbols, Index and Bibliography. In Chapter 1 the appeal of introducing ideas of semi-hyperbolicity and bi-shadowing is explained from the point of view of mathematical modeling of real processes. Chapter 2 briefly reviews background material on dynamical systems, particularly that involving differentiable hyperbolic mappings, and introduces terminology and results that will be required in later chapters. In Chapter 3 the concept of semi-hyperbolicity is introduced and some examples are considered. Here, modified definitions of semi-hyperbolicity are given to apply hyperbolicity to other contexts, such as differentiable mappings

1.2 Outline of Book

5

which need not be invertible, differentiable mappings on a general compact subset of Rd and for Lipschitz mappings. In the last two cases the subset on which the mapping is considered is not necessarily invariant. Further generalizations of the concept of semi-hyperbolicity to mappings in Banach spaces are given in Chaps. 8 and 9, and Appendix A. The usefulness of first Liapunov approximation methods to investigate the properties of semi-hyperbolic mappings and their trajectories naturally leads to considering linear operators in spaces of sequences generated in one way or another by derivatives of semi-hyperbolic mappings. Some elementary properties of such linear operators that will be needed throughout the book are considered in Chap. 4. Chapter 4 also shows that a semi-hyperbolic sequence of matrices is hyperbolic, that is in the linear case semi-hyperbolicity implies hyperbolicity. This is generally not true for nonlinear mappings, as an example in Sect. 5.1 demonstrates that a semi-hyperbolic mapping may not possess an invariant splitting and so cannot be a hyperbolic. Nevertheless, Sect. 5.2 shows that a semi-hyperbolic mapping which is smooth and invertible in a neighbourhood of a compact invariant set is hyperbolic on that set. The proofs depend substantially on background material and results in Chap. 4. In Sect. 2.2 it was mentioned that hyperbolic systems possess some rather strong and useful properties such as expansivity (see Definition 2.8 and Theorem 2.9) and shadowing (see Theorem 2.10). Chapter 6 shows that these and further properties are also true for semi-hyperbolic systems. Moreover, explicit values or sharp estimates of relevant parameters and parameter regions in which semi-hyperbolicity holds are found. Chapter 7 considers properties of semi-hyperbolic mappings which can be considered analogous to structural stability results. More precisely, it is shown that topological entropy can only increase following to continuous perturbation of a Lipschitz semi-hyperbolic mapping. Then problems of conjugation and factorization of semi-hyperbolic mappings are be studied. and chaotic behaviour of semi-hyperbolic mappings is discussed. In Chaps. 8 and 9 we briefly consider several applications of semi-hyperbolicity and its consequences, in particular to delay differential equations, systems with hysteresis and the numerical approximation of chaotic attractors. The first Appendix A is contributed by Marcin Mazur, Jacek Tabor and Piotr Ko´scielniak of the Jagiellonian University, Institute of Mathematics, Krakow, Poland. It contains recent work on some relationships between semihyperbolicity and hyperbolicity, and parameter estimations suitable for numerical verification of hyperbolicity using the semi-hyperbolic parameters. The second Appendix B by Janosch Rieger of the Goethe Universität, Frankfurt am Main, considers extensions of the concepts of semi-hyperbolicity and bi-shadowing to set-valued dynamics generated by a set-valued mapping.

Chapter 2

Smooth Dynamical Systems

This chapter briefly reviews background material on dynamical systems, particularly that involving differentiable hyperbolic mappings, and introduces terminology and recalls results that are required in later chapters.

2.1 Hyperbolic Mappings Major theoretical advances in the theory of dynamical systems by Anosov in the early 1960s to extended the concept of a hyperbolic set from a finite cyclic set to a general compact invariant subset K [6], opening the way for major advances in the theory. In particular, Smale used this generalization of hyperbolicity used to describe highly non-trivial recurrent behaviour in dynamical systems as well as the robustness of such behaviour [132]. Their work and that of many others was in the context of diffeomorphisms on smooth compact manifolds, motivated in part by the fact that time-one mappings and return mappings of smooth differential equations are diffeomorphisms, and developed differentiable dynamics. A principal objective of this monograph is to show that complicated dynamical behaviour is also possible under far less stringent assumptions. In what follows k · k will denote a fixed, but otherwise arbitrary norm on Rd .

2.1.1 Hyperbolic Cycles Let X be an open subset of Rd and f : X → X a differentiable mapping. This mapping generates a discrete time dynamical system by successive iteration, xn+1 = f (xn ),

n = 0, 1, 2, . . . ,

7

8

2 Smooth Dynamical Systems

determining the trajectory of the system starting at the point x0 ∈ X. A trajectory starting at x0 is called a cycle of f with period p ≥ 1, or p-cycle, if the successive points x0 , x1 , . . . , xp−1 are distinct and xp = x0 . A point xk of a p-cycle is called a p-periodic point of f , and a hyperbolic (k) periodic point 1 , if all eigenvalues λj of the derivative Dfxk satisfy the nonunit modulus condition: (k) |λj | = 6 1. The eigenvalues of Dfxk are actually the same at each of the points {x0 , x1 , . . . , xp−1 } of the cycle, so the superindex k is be omitted in what follows. Suppose that the eigenvalues, after possible rearrangement, satisfy |λj | < 1,

0 ≤ j ≤ ns ,

|λj | > 1,

ns + 1 ≤ j ≤ d

(2.1)

for some 0 ≤ ns ≤ d. Then the linear subspaces Exsk and Exuk spanned by the eigenvectors, and generalized eigenvectors if necessary, of Dfxk corresponding to the eigenvalues with modulus less than 1 and modulus greater than 1, respectively, form a splitting or decomposition of Rd , that is with Rd = Exsk ⊕ Exuk . with dimensions dim Exsk = ns ,

dim Exuk = nu := d − ns

(2.2)

and projection operators Pxsk : Rd → Exsk ,

Pxuk : Rd → Exuk .

Note that an equivalent norm k · kxk , which may depend on xk , can be found such that Dfxk |Exs is contractive and Dfxk |Exs is expansive in the sense that k k there is a constant λ > 1 such that kDfxk ukf (xk ) ≤ λ−1 kukxk ,

kDfxk vkf (xk ) ≥ λkvkxk

for any u ∈ Exsk and v ∈ Exuk . The linear subspaces Exsk and Exuk are called the stable and unstable subspaces at xk , respectively. Condition (2.1) is satisfied at each point of the cycle with the same ns , so the corresponding stable and unstable spaces for each point of the cycle have the same dimension (2.2). They are related by the equivariance property: Dfxk (Exsk ) = Efs (xk ) ,

Dfxk (Exuk ) = Efu(xk )

for each k. It is a trivial, yet nevertheless important observation that the set 1

A hyperbolic periodic point of period 1 is called a hyperbolic fixed point.

2.1 Hyperbolic Mappings

9

K = {x0 , . . . , xp−1 } of points of a p-cycle of f is invariant under f , or f -invariant, that is f (K) = K. In addition, the behaviour of the dynamical system generated by the mapping f is robust with respect to small perturbations in the vicinity of any cyclic set K if its points are hyperbolic. This robustness manifests itself globally when the mapping f has only a finite number of cycles each of which is hyperbolic.

2.1.2 Anosov Systems Anosov began the investigation of non-cyclic hyperbolic sets with what is now called an Anosov system, for a diffeomorphism f on an m-dimensional closed differentiable manifold M in Rd . Recall that a diffeomorphism f : M → M is an invertible mapping for which both f and its inverse f −1 are continuously differentiable on M . In an Anosov system the tangent space Tx M , isomorphic to Rm at each point x ∈ M , splits into a direct sum of stable and unstable subspaces of the tangent mapping T fx at x ∈ M with the separation δ(Exs , Exu ) = inf{ku − vk : u ∈ Exs , v ∈ Exu , kuk = kvk = 1} between these linear subspaces assumed uniformly bounded away from zero. Definition 2.1 (Anosov System). A diffeomorphism f : M → M where M is an m-dimensional closed differentiable manifold in Rd generates an Anosov system on M if A1:

for each x ∈ M there exists a splitting Tx M = Exs ⊕ Exu such that: dim Exs = k 6= 0,

dim Exu = m − k 6= 0,

for some integer k independent of x ∈ M , and T fx Exs = Efs (x) ,

T fx Exu = Efu(x) ;

A2: there exist constants λ > 1 and ρ0 > 0 independent of x ∈ M such that: kT fxn uk ≤ ρ0 λ−n kuk,

n kT fxn vk ≥ ρ−1 0 λ kvk,

n ≥ 0,

u ∈ Exs and v ∈ Exu ; A3: there exists a constant ρ1 > 0 independent of x ∈ M such that: δ (Exs , Exu ) ≥ ρ1

10


for all x ∈ M . An example of a linear mapping f which generates an Anosov system and is called an Anosov automorphism will be considered in Chap. 3. From A3 the splitting and consequent direct sum depend continuously on x, that is, in a neighbourhood of any point x0 ∈ M there exists a set of vectors e1 (x), e2 (x), . . ., em (x) ∈ Tx M which depend continuously on x ∈ M , which can be chosen so that the vectors e1 (x), e2 (x), . . ., ek (x) form a basis of the subspace Exs ⊆ Tx M and the vectors ek+1 (x), ek+2 (x), . . ., em (x) form a basis of the subspace Exu ⊆ Tx M (cf. [8, 24]). Theorem 2.2. For an Anosov system generated by a diffeomorphism on a closed differentiable manifold M of Rd the splitting Tx M = Exs ⊕ Exu depends continuously on x ∈ M . Note that only compact Riemannian manifolds M were considered in [8, 24], but the proof of Theorem 2.2 there does not make use of this extra structure and holds for any closed differentiable manifold.

2.1.3 Hyperbolic Diffeomorphisms In an Anosov system the whole manifold M is a non-cyclic hyperbolic set. Smale extended the idea to an f -invariant subset K of the manifold. The term hyperbolic mapping will often be used for any mapping f that has a hyperbolic invariant set K. The usual definition of hyperbolicity now used refers to a diffeomorphism f : M → M where M is a compact manifold and a compact f -invariant subset K of M . As for an Anosov system, the splitting is of the tangent space Tx M at each point x ∈ K, but the expansion and contraction on the stable and unstable subspaces of the tangent mapping T fx are expressed in terms of a Riemannian norm k · kx on Tx M , which may vary with on the point x ∈ K, allowing the constant ρ0 in Condition A2 of an Anosov system to be set equal to 1. Definition 2.3 (Hyperbolicity on a Manifold). A closed subset K of a compact manifold M which is invariant for a diffeomorphism f : M → M is said to be hyperbolic if there exists a splitting Tx M = Exs ⊕ Exu for each x ∈ K, a constant λ > 1 and a Riemannian norm k · kx on Tx M such that H1∗ :

For all x ∈ K: T fx (Exs ) = Efs (x) ,

H2∗ :

T fx (Exu ) = Efu(x) ,

For all x ∈ K, u ∈ Exs and v ∈ Exu : kT fx ukf (x) ≤ λ−1 kukx ,

kT fx vkf (x) ≥ λkvkx .

2.1 Hyperbolic Mappings

11

Conditions H1∗ and H2∗ in Definition 2.3 together imply that the splitting is continuous as in Theorem 2.2, as well as the constancy of the dimension of the splitting subspaces along a trajectory of the mapping f . Remark 2.4. The use of an equivalent adapted norm in Condition H2∗ in Definition 2.3 allows the constant ρ0 in Condition A2 of an Anosov system to be set equal to 1. This is convenient for many theoretical purposes, but can be cumbersome for specific practical examples. The Whitney Embedding Theorem provides one way of avoiding the problem and using the same norm at each point of x ∈ K in the inequalities in Condition H2∗ . Essentially the 0 manifold M is smoothly embedded in a higher dimensional space Rd and the norm of this space can be used everywhere. Alternatively, Smale has shown that there is an iterate f l of a hyperbolic diffeomorphism f , such that f l is hyperbolic on the invariant set K, with the inequalities in Condition H2∗ satisfied with respect to a norm that does not depend on the point of K. The previous remark motivates the following modification of the definition of hyperbolicity for a mapping f : X → Rd , where X is an open subset of Rd containing the f -invariant compact set K under consideration, which is a diffeomorphism on a neighbourhood of the set K. Definition 2.5 (Hyperbolicity). A compact subset K ⊂ X which is invariant for a diffeomorphism f : X → Rd is said to be hyperbolic if there exists a splitting Rd = Exs ⊕ Exu for each x ∈ K, varying continuously in x ∈ K, constants λ > 1 and ρ0 > 0, and a norm k · k on Rd such that H1:

For all x ∈ K: Dfx (Exs ) = Efs (x) ,

H2:

Dfx (Exu ) = Efu(x) ,

For all u ∈ Exs and v ∈ Exu : kDfxn uk ≤ ρ0 λ−n kuk,

n kDfxn vk ≥ ρ−1 0 λ kvk,

n ≥ 0.

Given a γs , γu ≥ 0, associate with the split Rd = Exs ⊕ Exu the family of cones called (the cone field) Kxs = Kxs (γs ) = {x ∈ Rd : kPxu xk ≤ γs kPxs xk}, Kxu = Kxu (γu ) = {x ∈ Rd : kPxs xk ≤ γu kPxu xk}. Definition 2.6 (Hyperbolic Cone Field). A differentiable mapping f : X → Rd is said to have hyperbolic cone field on a compact K ⊂ X if there exists a splitting Rd = Exs ⊕ Exu with projectors Pxs , Pxu for each x ∈ K and a norm k · k on Rd such that HCF1: For some family of cones Kxs , Kxu associated with the split Rd = Exs ⊕ Exu hold the conditions

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Dfx Kxu ⊆ Int Kfu(x) ∪ {0}, HCF2:

Dfx−1 Kfs(x) ⊆ Int Kxs ∪ {0};

There exist λ < 1 < µ such that kDfx xk ≥ µkxk, kDfx−1 xk ≥ λ−1 kxk,

x ∈ Kxu , x ∈ Kfs(x) .

The splitting Rd = Exs ⊕ Exu in Definition 2.6 is not supposed to be equivariant with respect to Dfx . Conditions HCF1 and HCF2 mean that the cones Kxu are invariant under Dfx which is expanding on Kxu while the cones Kfs(x) are invariant under Dfx−1 which is expanding on Kfs(x) . As is known [65], the existence of the Hyperbolic Cone Field for a differentiable mapping enables the hyperbolicity of this mapping.

2.1.4 Generalizations Various generalizations of hyperbolicity to mappings other than diffeomorphisms have been proposed, in particular for homeomorphisms and to mappings which may be neither invertible nor continuous. Instead of a splitting into stable and unstable linear spaces, dynamically defined nonlinear stable and unstable sets are used. Let (X, ρ) be a compact metric space and let f : X → X be a homeomorphism of X onto itself. For an arbitrary point x ∈ X and any ε > 0, the local stable and unstable sets of f of size ε at x are defined as Wεs (x) = {y ∈ X : ρ(f n x, f n y) ≤ ε, n ≥ 0} and Wεu (x) = {y ∈ X : ρ(f n x, f n y) ≤ ε, n ≤ 0}, respectively, while the stable and unstable sets of f at x are defined as W s (x) = {y ∈ X : ρ(f n x, f n y) → 0, n → ∞} and W u (x) = {y ∈ X : ρ(f n x, f n y) → 0, n → −∞}, respectively. Note that cardinality # (Wεs (x) ∩ Wεu (x)) ≥ 1 for any ε > 0. Definition 2.7 (Hyperbolic Homeomorphism). A homeomorphism f of X onto itself is hyperbolic if there exist constants ε0 > 0, K > 0 and λ > 1 such that

2.2 Fundamental Properties

13

ρ(f n x, f n y) ≤ Kλ−n ρ(f

−n

x, f

−n

−n

y) ≤ Kλ

for all

x ∈ X, y ∈ Wεs0 (x)

and n ≥ 0,

for all

Wεu0 (x)

and n ≥ 0,

x ∈ X, y ∈

and there exists δ0 > 0 such that # Wεs0 (x) ∩ Wεu0 (y) = 1, for all x, y ∈ X with ρ(x, y) ≤ δ0 . The case of noninvertible f can be handled by a suggestion of Ruelle [128]. Let f : X → X be a continuous mapping and for each x ∈ X define the sequence set ( ) 0 Y † 0 X = {xn }n=−∞ ∈ X : f (x−n ) = x−n+1 , n = 1, 2, . . . n=−∞

and a mapping f † : X † → X † by f † ({xn }) = {f (xn )}. Then f † is a homeomorphism on X † with respect to the metric D({xn }, {yn }) = supn≤0 ρ(xn , yn ) and f is said to be hyperbolic on X if f † is hyperbolic on X † .

2.2 Fundamental Properties Robustness properties of a dynamical system near a hyperbolic cycle is also a feature of dynamical systems with nontrivial hyperbolic sets, but the dynamical behaviour can be much more complicated. Some fundamental properties of hyperbolic diffeomorphisms are summarized here without proof, focussing on those properties that extend, perhaps modified, to semi-hyperbolic mappings. Proofs of these results will be presented in later chapters. To simplify the exposition, results will be stated in terms of a diffeomorphism f : X → X ⊂ Rd with a compact invariant hyperbolic subset K. Homeomorphisms will also be considered below.

2.2.1 Expansivity and Shadowing An immediate consequence of hyperbolicity and is that a diffeomorphism is expansive on a hyperbolic set. Given that a hyperbolic set is compact and invariant, this suggests complex dynamics within the set itself. Definition 2.8 (Expansivity). A homeomorphism f : K → K is expansive on an invariant set K if there exists an ε > 0 such that for all x, y ∈ K condition kf n (x) − f n (y)k ≤ ε for all n ∈ Z

14


implies that x = y. Theorem 2.9. A diffeomorphism f with a hyperbolic set K is expansive on K. Thus no two distinct trajectories {f n (x) : n ∈ Z}, {f n (x) : n ∈ Z} in a hyperbolic set can be ε-close to each other, no matter how close the points x, y. Nevertheless, within the hyperbolic set there is always a true trajectory close to an approximate or pseudo-trajectory. A sequence {yn } ⊂ X is called a δ-pseudo-trajectory of a dynamical system generated by a mapping f if kyn+1 − f (yn )k < δ

for all

n∈Z

and a true trajectory {xn } is said to ε-shadow a pseudo-trajectory {yn } if kyn − xn k < ε for all

n ∈ Z.

Theorem 2.10 (Shadowing Theorem). If f : X → X is a diffeomorphism with an f -invariant hyperbolic set K ⊂ X, then for every ε > 0 there exists a δ > 0 and an open neighbourhood U of K in X such that every δ-pseudo- trajectory of f in U is ε-shadowed by a true trajectory of f in K. A dynamical system satisfying the assertion of Theorem 2.10 will be said to have the shadowing property. The following characterization of hyperbolic homeomorphisms holds. Theorem 2.11. A homeomorphism f on a compact metric space (X, ρ) is hyperbolic if and only if f is expansive and has the shadowing property. The pseudo-trajectories in the Shadowing Theorem are often interpreted as being the true trajectories of a nearby approximating system, for example generated by computation of the given system on a digital computer. Furthermore, hyperbolic diffeomorphisms enjoy a stronger relationship with systems generated by nearby mappings.

2.2.2 Conjugate Systems Let H (K) denote the space of homeomorphisms f : K → K, where K is a compact subset of Rd . Note that (H (K), ρ0 ) is a complete metric space with the metric

ρ0 (f, g) = max kf (x) − g(x)k , f −1 (x) − g −1 (x) . x∈K

A mapping g ∈ H (K) is said to be topologically semi-conjugate to a mapping f ∈ H (K) if there exists a continuous mapping h of K onto K such that


15

f ◦ h = h ◦ g and topologically conjugate when the mapping h : K → K is a homeomorphism. Many important dynamical properties are preserved under conjugacy. One illustration is the Hartman–Grobman theorem in which the mapping and its linearization about a hyperbolic point have locally similar behaviour. Moreover, for a hyperbolic diffeomorphism f there is a neighbourhood of homeomorphisms in (H (K), ρ0 ) homeomorphisms, each of which is semi-conjugate to f under a common semi-conjugacy h close to the identity mapping on K. Theorem 2.12 (Topological Stability). Let f ∈ (H (K), ρ0 ) be a hyperbolic diffeomorphism on K. Then, given ε > 0 there exists a unique continuous mapping h : K → K with kx−h(x)k, < ε for all x ∈ K and a δ = δ(ε) > 0 such that h ◦ g = f ◦ h for all g ∈ (H (K), ρ0 ) with ρ0 (f, g) < δ. Moreover, if ε is small enough and K is a compact manifold, then h maps K onto K. This result shows the topological stability of the mapping f when ε is small enough to ensure that h is a surjective mapping. Neighbouring dynamical systems g in (H (K), ρ0 ) are then also expansive mappings with the shadowing property. Whether or not diffeomorphisms g in this δ-neighbourhood of f in the space (H (K), ρ0 ) are also hyperbolic on K or a subset of K is a much deeper question and has lead to an extensive structural theory of differential dynamics which involves the stronger concept of structural stability in the space D(K) of diffeomorphisms on K with a C 1 -metric ρ1 . In this theory generic properties of diffeomorphisms in a residual subset, that is in a countable intersection of open dense subsets of (D(K), ρ1 ), are of primary interest. Topological conjugacies between mappings defined on different compact sets Ki , which need not be subsets of a common space, are also useful in comparing the dynamics of systems when one of them is relatively simple enough to be well understood. For example, the topological conjugacy between the horseshoe diffeomorphism on the sphere, restricted to the hyperbolic horseshoe subset, and the shift operator σ on the space Σ of bi-infinite sequences x = {xn }n∈Z where xn ∈ {0, 1} with the metric X ˜) = ρ(x, x 2−|n| kxn − x ñ k, n∈Z

where (σx)n = xn+1 for each n ∈ Z. The dynamics of the system generated on Σ by the homeomorphism σ is expansive on Σ, the set Per(σ) of all periodic points of σ is dense in Σ and there is a trajectory of σ which is dense in Σ. These properties are carried over to the horseshoe diffeomorphism on its hyperbolic horseshoe set by the conjugacy homeomorphism. This is but one illustration of how symbolic dynamics has become an indispensable tool in the theory of dynamical systems.

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2.2.3 Hyperbolic Sets Often a hyperbolic subset K of a mapping f defined on a set X is not unknown in advance and must be determined. There are several natural candidates for such a subset, the most obvious of which is the set Per(f, X) of all periodic points x ∈ X of f whose cycles belong to X. More interesting are Ω(f, X), the set of non-wandering points of f , and CR(f, X) of chain recurrent points of f . A point x ∈ X is a non-wandering point of f if for every neighbourhood U of x in X there exists an integer n ≥ 1 such that U ∩ Un 6= ∅ for a sequence of nonempty sets {Uk } defined by U0 = U,

Uk+1 = X ∩ f (Uk ),

k = 0, 1, . . . .

A point x ∈ X is chain recurrent for f if for every ε > 0 there exists an ε-pseudo-trajectory {xn } ⊆ X of f with x0 = x and kxN − xk ≤ ε for some N > 0.2 The sets Per(f, X), Ω(f, X) and CR(f, X) are closed in X, and Per(f, X) ⊆ Ω(f, X) ⊆ CR(f, X), where each is f -invariant. Much of the structural theory of differential dynamics has focussed on the Axiom A diffeomorphisms, where Ω(f ) is a hyperbolic set and Per(f ) is dense in Ω(f ), since a diffeomorphism is structurally stable if and only if it is an Axiom A diffeomorphism and its stable and unstable manifolds satisfy a strong transversality condition [132]. Note that Per(f ) = Ω(f ) = CR(f ) for a generic system f ∈ H (X) and the mapping f 7→ Ω(f ) is uppersemicontinuous on (H (X), ρ0 ) at each such f , in the sense that for every ε > 0 there exists a δ = δ(ε) > 0 such that Ω(g) ⊂ Oε (Ω(f ))

for all ρ0 (g, f ) < δ,

where Oε (Ω(f )) is the open ε-neighbourhood of the subset Ω(f ), [ Oε (Ω(f )) = Int B(ε, x), x∈K

where Int B(ε, x) is the open ball of radius ε centered at x. A modified form of structural stability is Ω-stability in which a topological conjugacy is established between the restricted mappings f |Ω(f ) and g|Ω(g) . For nonsmooth mappings to be considered later in this book conjugacies between f |CR(f ) and g|CR(g) , and between the sets of their trajectories, is important. 2

Sometimes a slightly different, though equivalent, definition of a chain recurrent point is used: a point x ∈ X is called chain recurrent for f if for every ε, N > 0 there exists an ε-pseudo-trajectory {xn } ⊆ X of f with x0 = xK = x for some K ≥ N (cf. [128]).


17

2.2.4 Entropy The term entropy refers to a variety of related concepts in mathematics. Here, the principal concern is that of topological entropy, an invariant of topological conjugacy [9]. Like other concepts of entropy, topological entropy provides an index of how complicated the dynamics of a system are. There are several equivalent definitions [21, 90, 138] of topological entropy and that used here is based on the Bowen scheme which allows entropy to be defined for a uniformly continuous mapping on a subset of a metric space that need not be compact (cf. [21, 138]). Let f : X → X be a continuous mapping where X is an open bounded subset of Rd and let K be a compact subset of X. For a fixed positive integer N denote by TrN (f, K) the totality of finite trajectories x = {x−N . . . , x0 , . . . , xN } of f that are contained entirely in K and introduce on TrN (f, K) the metric ˜) = ρN (x, x

sup

kxn − x ñ k.

−N ≤n≤N

Let ε > 0 and denote by Cε (Tr±N (f, K)) the ε-capacity of the set of of elements x(1) , . . . , x(p) in TrN (f, K) such that3 ρN (x(i) , x(j) ) ≥ ε for all

i 6= j.

Definition 2.13. The topological ε-entropy hε (f, K) of f : X → X on a compact subset K of X is defined by hε (f, K) = lim sup N →∞

1 Cε (TrN (f, K)). 2N + 1

The topological entropy h(f, K) of f on K is defined to be the limit h(f, K) = lim hε (f, K) = sup hε (f, K). ε→0

ε>0

Clearly, hε (f, K) is nonincreasing in ε, so h(f, K) is well defined. One of the most difficult problems in investigating topological entropy is explicit evaluation. For an ξ-expansive mapping f with f (K) = K the topological entropy in some cases is given by the following formula h(f, K) = hθ (f, K),

θ 0 n→∞

for every x, y ∈ S with x 6= y, and for every x ∈ S and any periodic point y; CH3: There exists an uncountable subset S0 of S such that: lim inf kf n (x) − f n (y)k = 0 n→∞

for every x, y ∈ S0 . A sufficient condition for the chaos in the sense of Definition 2.17 for onedimsenional mappings gives the following Li and Yorke [84] criterion called ‘period three implies chaos’. Theorem 2.18 (Period Three Implies Chaos). Let I be an interval in R1 and f : I → I be a continuous mapping. Let there is a point, a ∈ I, for which the points b = f (a), c = f 2 (a) and d = f 3 (a) satisfy d≤a c.

Then the mapping f is chaotic. The following theorem due to Shiraiwa and Kurata [130] expands the ‘Period Three Implies Chaos Theorem’ in R1 to higher dimensional mappings. See also Kloeden [71]. Theorem 2.19. Let X be a subset of Rd and f : X → X be a C 1 -map. Let x0 ∈ X be a hyperbolic fixed point of f and assume that the following three conditions are satisfied: (i) dim Exu0 > 0; (ii) there exist an ε > 0, a point x1 ∈ Wεu (x0 ), x1 6= x0 , and a positive integer m such that f m (x1 ) ∈ Wεs (x0 ; (iii) there exists a disk4 Du ⊂ Wεu (x0 ) such that Du is a neighbourhood of x1 in Wεu (x0 ), f m |Du : Du → X is an embedding, and f m (Du ) intersects Wεu (x0 ) transversally at f m (x1 ). Then the mapping f is chaotic. 4

A disk is a homeomorphic image of a ball in a normed linear space.

20


Motivated by the work of Li and Yorke [84], Marotto [89] further generalized Theorem 2.18 to the multidimensional setting. Consider the following d-dimensional system: xn+1 = f (xn ),

xn ∈ Rd , n = 0, 1, 2, . . . ,

(2.3)

where the map f : Rd → Rd is continuously differentiable. Denote by Br (x) the closed ball in Rd of radius r centered at a point x ∈ Rd and let f be differentiable in the interior of Br (x). The point x ∈ Rd is an expanding fixed point of f in Br (x), if f (x) = x and all eigenvalues of Df (y) exceed 1 in absolute value for all y ∈ Br (x). Definition 2.20 (Snap-Back Repeller). Let x be an expanding fixed point of f in Br (x) for some r > 0. Then, x is said to be a snap-back repeller of f , if there exists a point x0 ∈ Br (x) with x0 6= x, such that f m (x0 ) = x and the determinant |Df m (x0 )| = 6 0 for an integer m > 0. The following theorem is due to Marotto [89]. Theorem 2.21. If f possesses a snap-back repeller, then the system (2.3) is chaotic. A simpler definition of chaos was given by Devaney [40]. Definition 2.22 (Devaney Chaos). Let S be a set in a metric space (X, ρ). A mapping f : S → S is said to be chaotic, if DCH1: The map f has sensitive dependence on initial conditions, in the sense that there exists δ > 0, such that for any x ∈ S and any neighbourhood U of x in S, ρ(f m (x), f m (y)) > δ for some y ∈ U and some m ≥ 0; DCH2: The map f is topologically transitive, in the sense that for any pair of nonempty open subsets U, V ∈ S, there exists an integer m > 0, such that f m (U ) ∩ V 6= ∅; DCH3: The periodic points of the map f are dense in S. This definition has some redundancy and can be further simplified. If Condition DC3 above is dropped, then it is called chaos in the sense of Wiggins [139]. For further references and discussion of chaos concepts see in [27, 69, 71].

Chapter 3

Semi-Hyperbolic Mappings

The concept of semi-hyperbolicity is introduced in this chapter and some examples are considered. The first definition generalizes Definition 2.3 to a differentiable mapping, which need not be invertible, on a compact manifold and involves variable norms at each point of the compact invariant subset of the manifold. The second and third definitions apply to a general compact subset of Rd as in Definition 2.5, first for a differentiable mapping and then for a Lipschitz mapping. In both cases a fixed norm is used and invariance of the subset is not assumed. Further generalizations of the concept of semi-hyperbolicity to mappings in Banach spaces will be given in Chaps. 8 and 9, and Appendix A.

3.1 Definitions Consider a mapping f : X → Rd where X is an open subset of Rd and let K be a nonempty compact subset of X such that K ∩ f (K) 6= ∅, or consider a mapping f : M → M where M is a compact manifold and let K be an f -invariant subset of M . The following definition is for a splitting of Rd . An analogous definition holds for a splitting Tx M = Exs ⊕Exu of the tangent space Tx M of a compact manifold M , where the projectors are Pxs : Tx M → Exs and Pxu : Tx M → Exu . Definition 3.1. A splitting or decomposition Rd = Exs ⊕Exu with corresponding projectors Pxs : Rd → Exs and Pxu : Rd → Exu defined by Pxs Rd = Exs ,

Pxs Exu = 0,

(3.1)

Pxu Rd

Pxu Exs

(3.2)

=

Exu ,

= 0,

is said to be uniform on the compact subset K with respect to a mapping f if

21

22

SH0:

3 Semi-Hyperbolic Mappings

dim Exu = dim Efu(x) for x ∈ K with f (x) ∈ K;

there exists a positive real number h such that SH1:

supx∈K {kPxs k, kPxu k} ≤ h.

Note that neither the invariance of the set K with respect to the mapping f nor the continuity in x of the splitting subspaces Exs , Exu or of the projectors Pxs , Pxu are assumed in Definition 3.1. To explore further ramifications of this definition, rates of expansion and contraction for the projections are needed, in the form of a split. See Definition 3.2 below. Definition 3.2. A 4-tuple s = (λs , λu , µs , µu ) of nonnegative real numbers is called a split if λs < 1 < λu ,

(1 − λs )(λu − 1) > µs µu .

The split s is called positive if all the values λs , λu , µs and µu are positive. Clearly, for any given λs and λu satisfying λs < 1 < λu a 4-tuple of nonnegative numbers s = (λs , λu , µs , µu ) will be a split if the product µs µu is small enough. An alternative definition of a split with a geometrical interpretation is possible: a 4-tuple (λs , λu , µs , µu ) is a split if and only if the eigenvalues ∆1 and ∆2 of every matrix δ δ ∆ = 11 12 δ21 δ22 with components satisfying |δ11 | ≤ λs , |δ12 | ≤ µs , |δ21 | ≤ µu , |δ22 | ≥ λu are real and satisfy |∆1 | < 1 < |∆2 |. It is of practical importance to know how the elements of a split can be perturbed and remain a split. Set (1 − λs )(λu − 1) − µs µu . λu − λs + µs + µu

(3.3)

(1 − λs )(λu − 1) ≤ min{1 − λs , λu − 1}, λu − λs

(3.4)

ν(s) = Note that ν(s) ≤ so ν(s) < 1.

˜ s = λs + δ1 , λ ˜u = Lemma 3.3. If s = (λs , λu , µs , µu ) is a split, then for λ λu − δ2 , µ ˜s = µs + δ3 , µ ˜u = µu + δ4 and real δ1 , δ2 , δ3 , δ4 satisfying 0 ≤ ˜s, λ ˜u, µ ˜ = (λ δ1 , δ2 , δ3 , δ4 < ν(s) the 4-tuple s ˜s , µ ˜u ) is also a split. ˜ are non-negative and by (3.3), Proof. Clearly the elements of the 4-tuple s (3.4) we have δ1 < 1 − λs and δ2 < λu − 1. Hence

3.1 Definitions

23

˜ s = λs + δ1 < λs + 1 − λs = 1, λ

˜ u = λu − δ2 > λu − (λu − 1) = 1, λ

and by (3.3) ˜ s )(λ ˜ u − 1) − µ (1 − λ ˜s µ ˜u > (1 − λs − ν(s))(λu − 1 − ν(s)) − (µs + ν(s))(µu + ν(s)) = (1 − λs )(λu − 1) − µs µu − ν(s)(λu − λs + µs + µu ) = 0. The lemma is proved.

t u

3.1.1 Differentiable Mappings on Compact Manifolds Here, the variable norm version of hyperbolicity on a manifold of Definition 2.3 is generalized to semihyperbolic differentiable mappings on a manifold. Definition 3.4. Let s = (λs , λu , µs , µu ) be a split and let M be a compact manifold in Rd . A differentiable mapping f : M → M is said to be s-semihyperbolicon a compact invariant subset K of M if there exists a uniform splitting Tx M = Exs ⊕ Exu with projectors Pxs , Pxu and a norm k · kx on Tx M for each x ∈ K such that SH2(Diff)∗ :

For all x ∈ K: kPfs(x) T fx ukf (x) ≤ λs kukx ,

u ∈ Exs ,

(3.5)

Exu , Exs , Exu ,

(3.6)

kPfs(x) T fx vkf (x) kPfu(x) T fx ukf (x)

≤ µs kvkx ,

v∈

≤ µu kukx ,

u∈

kPfu(x) T fx vkf (x)

≥ λu kvkx ,

v∈

(3.7) (3.8)

where T fx denotes the tangent mapping of f at the point x. Nonzero values of µs and µu allow for a possible leakage away from the strict equivariance of the splitting subspaces Exs and Exs of Condition H1, that holds when the f is a hyperbolic diffeomorphism on K as in Definition 2.3. Essentially, the notion of semi-hyperbolicity is very close to that of hyperbolic cone field, see Definition 2.6. Comparing Definition 3.4 with Definition 2.3, it is clear that a differentiable hyperbolic mapping on K with expansivity parameter λ > 1 is s-semihyperbolic on K with the same splitting and split s = (λ−1 , λ, 0, 0), that is with µs = µu = 0.

24


3.1.2 Differentiable Mappings on Compact Sets Consider now a differentiable mapping f : X → Rd where X is an open subset of Rd . Definition 2.5 of a hyperbolic diffeomorphism on a compact subset K of Rd will be generalized to a differentiable mapping, which need not be invertible, and to a nonempty subset K of X such that only satisfies K ∩ f (K) 6= ∅ rather than being f -invariant. Definition 3.5. Let s = (λs , λu , µs , µu ) be a split K a compact subset of an open subset X of Rd . A differentiable mapping f : X → Rd is said to be s-semi-hyperbolic on K if there exists a uniform splitting Rd = Exs ⊕ Exu with projectors Pxs , Pxu for each x ∈ K and a norm k · k on Rd such that SH2(Diff):

For all x ∈ K: kPfs(x) Dfx uk ≤ λs kuk,

u ∈ Exs ,

(3.9)

kPfs(x) Dfx vk ≤ µs kvk,

v ∈ Exu ,

(3.10)

Exs , Exu ,

(3.11)

kPfu(x) Dfx uk kPfu(x) Dfx vk

≤ µu kuk,

u∈

≥ λu kvk,

v∈

(3.12)

where Dfx denotes the derivative of the mapping f at the point x. Comparing Definitions 3.5, Definition 2.5, clearly a hyperbolic diffeomorphism on K with expansivity parameter λ > 1 is s-semi-hyperbolic on K with the same splitting and split s = (λ−1 , λ, 0, 0), that is µs = µu = 0. Since the splitting Rd = Exs ⊕ Exu in Definition 3.5 is uniform, then projectors Pxs , Pxu satisfy SH1 of Definition 3.1 for some constant h. To stress this dependency on h, the s-semi-hyperbolic map f : X → Rd may occasionally be called (s, h)-semi-hyperbolic.

3.1.3 Lipschitz Mappings on Compact Sets Turning to nonsmooth applications, the following generalization of Definition 2.5 to Lipschitz mappings f : X → Rd is given. Generalization of Definition 2.3 to Lipschitz mappings on a manifold, while possible is rather cumbersome and is omitted as it is not be used elsewhere below. As in Definition 3.5, the subset K is a nonempty compact subset of X such that K ∩ f (K) 6= ∅ and a fixed norm is used. Definition 3.6. Let s = (λs , λu , µs , µu ) be a split and K a compact subset of an open set X ⊆ Rd . A Lipschitz mapping f : X → Rd is said to be s-semi-hyperbolic on K if there exists a splitting Rd = Exs ⊕ Exu on K with projectors Pxs and Pxu for each x ∈ K, a norm k · k on Rd and a positive real number δ such that

3.1 Definitions

SH0(Lip): SH1(Lip): SH2(Lip):

25

dim Exu = dim Eyu for all x, y ∈ K with kf (x) − yk ≤ δ; supx∈K {kPxs k, kPxu k} ≤ h; The inclusion x+u+v ∈X

and the inequalities kPys (f (x + u + v) − f (x + u ˜ + v)) k ≤ λs ku − u ˜k,

(3.13)

kPys kPyu kPyu

(f (x + u + v) − f (x + u + v˜)) k ≤ µs kv − v˜k,

(3.14)

(f (x + u + v) − f (x + u ˜ + v)) k ≤ µu ku − u ˜k,

(3.15)

(f (x + u + v) − f (x + u + v˜)) k ≥ λu kv − v˜k.

(3.16)

hold for all x, y ∈ K with kf (x) − yk ≤ δ and all u, u ˜ ∈ Exs and v, v˜ ∈ Exu such that kuk, k˜ uk, kvk, k˜ v k ≤ δ. The first three inequalities in SH2(Lip) of are just local Lipschitz conditions on the projections of f while the last is an expansivity condition which implies local invertibility in the unstable direction of f at x. As in the two differentiable cases above (Definitions 3.4 and 3.5), nonzero values of µs and µu allow possible leakage away from the equivariance of the splitting subspaces Exs and Exs . Remark 3.7. Condition SH0(Lip) of Definition 3.6 is more restrictive than in the uniform splitting Definition 3.1 while SH1(Lip) is the same as SH1. Comparing the inequalities (3.13)–(3.16) with the corresponding inequalities (3.5)–(3.8) or (3.9)–(3.12), note that in the first case, Lipschitz mappings on compact manifolds, projectors Pys and Pyu are considered for y from some neighbourhood of the point f (x), while in (3.5)–(3.8) or (3.9)–(3.12) they are considered only at the point y = f (x). This additional restriction can be regarded as a weaker form of continuous dependence of splitting Rd = Exs ⊕ Exu in x. See also Sect. 3.1.4 and Remark 3.11.

3.1.4 Continuous Splittings Continuous dependence on x of the subspaces Exs and Exu of a splitting Tx M = Exs ⊕ Exu of a compact smooth manifold M , or of a splitting Rm = Exs ⊕ Exu of the space Rm is a fundamental property of hyperbolic mappings, but does not follow automatically for semi-hyperbolic mappings. Definition 3.8. A splitting Tx M = Exs ⊕ Exu of a compact smooth manifold M is said to depend continuously on x if in a neighbourhood of any point x0 ∈ M a set of vectors e1 (x), e2 (x), . . . , em (x) ∈ Tx M in local coordinates, depending continuously on x, can be chosen such that the vectors e1 (x), e2 (x), . . . , ek (x) form a basis of the subspace Exs ⊆ Tx M and the vectors ek+1 (x), ek+2 (x), . . . , em (x) form a basis of the subspace Exu ⊆ Tx M .

26


For a splitting of the space Rm an alternative definition, which is sometimes more convenient to use, is: Definition 3.9. The splitting Rm = Exs ⊕ Exu is said to depend continuously on x if Pxs and Pxu of (3.1) and (3.2) respectively, are continuous in x as linear operator-valued functions. Note that if the linear operator-valued functions x 7→ Pxs , x 7→ Pxu , neither the invariance of the set K with respect to f , nor continuity in x of the splitting subspaces Exs , Exu and of the projectors Pxs , Pxu , are assumed in Definition 3.1 of uniform splitting. For hyperbolic mappings continuity of the splitting is a direct consequence of the definition, but this does not always follow from the weaker assumptions of semi-hyperbolicity. In some cases continuity may hold and, as will be seen in later chapters, stronger results are obtained. The following definition applies for semi-hyperbolic mappings with respect to either a fixed norm or variable adapted norms. Definition 3.10. A mapping f is said to be continuously semi-hyperbolic on K if it is s-semi-hyperbolic on K as in any of Definitions 3.4–3.6 for some split s for which the uniform splitting is continuous in x on K in the sense of Definitions 3.8 or 3.9. Remark 3.11. A continuously s-semi-hyperbolic differentiable mapping is sε semi-hyperbolic as in Definition 3.6 for sε = (λs + ε, λu − ε, µs + ε, µu + ε) for any sufficiently small ε > 0 and an appropriate choice of δ = δ(ε) > 0. Continuously semi-hyperbolic mappings form an open set in the class of all semi-hyperbolic mappings. A stronger result holds for Lipschitz semi-hyperbolic mappings and is stated below to illustrate the property overall. Denote by C(X, Rd ), X ⊆ Rd , the Banach space of all bounded continuous mappings f : X → Rd with norm kf kC = sup kf (x)k. x∈X

Let Lip(X, Rd ), X ⊆ Rd be the Banach space of all bounded Lipschitz mappings f : X → Rd with norm kf kLip = kf kC +

sup x,y∈X, x6=y

kf (x) − f (y)k . kx − yk

Denote by Oε (K) the open ε-neighbourhood of the subset K, that is [ Oε (K) = Int B(ε, x), x∈K

where Int B(ε, x) is the open ball of radius ε centered at x.

3.1 Definitions

27

Lemma 3.12. Let X ⊆ Rd be an open set and let the mapping f ∈ Lip(X, Rd ) be continuously semi-hyperbolic on a compact set K ⊂ X. Then there exists an η = η(f, X) > 0, a split s, constants h, δ and a uniform splitting Rd = Exs ⊕ Exu such that Oη (K) ⊂ X and every mapping from the set Fη = {g ∈ Lip(X, Rd ) : kg − f kLip ≤ η} is semi-hyperbolic on Oη (K) with the split s, constants h, δ and the splitting Rd = Exs ⊕ Exu . Proof. Let Exs ⊕ Exu be a continuous splitting of Rd , for the mapping f , satisfying Conditions SH0(Lip)–SH2(Lip) of Definition 3.6 f with the split s∗ and positive constants h∗ and δ ∗ . For each x ∈ X let π(x) be any one of closest points in K to x. Since K is compact, such a function π(x) is well defined, possibly nonuniquely, and generally will not be continuous. For what follows it is important that π(x) = x, and kπ(x) − xk < η, d

Extend the splitting R =

Exs

s Exs = Eπ(x) ,

⊕

Exu

x ∈ K, x ∈ Oη (K). to the whole of X by defining

u Exu = Eπ(x) ,

x ∈ X.

Then the extended splitting s u Rd = Exs ⊕ Exu ≡ Eπ(x) ⊕ Eπ(x) ,

x ∈ X,

(3.17)

will satisfy Conditions SH0(Lip)–SH1(Lip) of Definition 3.6 with the constant h = h∗ . Let η0 > 0 be such that Oη0 (K) ⊂ X, take 1 η < min η0 , δ ∗ , h−1 ν(s) 2 where ν(s) is as in (3.3), and set δ = δ ∗ − 2η,

s = {λ∗s + hη, λ∗u − hη, µ∗s + hη, µ∗u + hη}

Then each mapping g in the set Fη satisfies Condition SH2(Lip) of Definition 3.6 on the compact set Oη (K) with the splitting (3.17), the split s and constants h, δ. t u Remark 3.13. In spite of continuity of the splitting Rd = Exs ⊕ Exu for f in Lemma 3.12, the splitting (3.17) is generally not continuous. Continuity in (3.17) may be possible, for example, when the compact K is a retract of a neighbourhood of itself.

28


3.2 Examples All diffeomorphisms with a hyperbolic invariant set are semi-hyperbolic on that set provided the appropriate corresponding definitions are used. Consequently, mappings which are semihyperbolic, but not hyperbolic, illustrate the significance of the theory, especially when the invariant set is uncountable.

3.2.1 Elementary Examples Differential dynamics concentrates on the behaviour of a diffeomorphism within an invariant set. When dynamical behaviour of a mapping s approximated by a computational method, what occurs close to the invariant set is also of some interest, particularly when the set quite simple. Consider, for example, the problem of replicating a phase portrait about a saddle point using finite machine arithmetic. The shadowing results for semi-hyperbolic mappings in Chap. 6 are useful here, while those for hyperbolic diffeomorphisms are not applicable. Example 3.14. Let A be a hyperbolic matrix, that is |λi | 6= 1 for the eigenvalues λ1 , . . . , λd , and consider the splitting Rd = E s ⊕ E u

(3.18)

where E s is the eigenspace of A of the eigenvalues satisfying |λi | < 1 and E u the eigenspace of the eigenvalues satisfying |λi | > 1. For simplicity of exposition, let d = 2. The linear mapping f : R2 → R2 defined by f (x) = Ax is hyperbolic on the singleton set {0} with splitting (3.18), but is only semihyperbolic on, say, the unit disk D2 since D2 is not invariant under f . The splitting on D2 is obtained by translating that at {0} to each point x ∈ D2 , and is thus both uniform and continuous. Nonsmooth perturbations of hyperbolic mappings are also another source of examples of semi-hyperbolic mappings. These are easily described as in the following example where the hyperbolic set is a saddle point. Example 3.15. Let A be a 2 × 2 hyperbolic matrix as in Example 3.14 and consider a nonlinear mapping fε : R2 → R2 obtained as a perturbation of the linear mapping f0 (x) = Ax, such as x1 x ε|x1 | fε =A 1 + x2 x2 ε|x2 | for all x = (x1 , x2 )T ∈ R2 and some sufficiently small ε > 0. This mapping is Lipschitz everywhere, but is not differentiable at the origin. It is semi-hyper-

3.2 Examples

29

bolic in any set containing the origin in its interior with Exs = {0} ⊕ R1 and Exu = R1 ⊕ {0} for all x. Mappings defined by switching between members of a set of maps, often linear, on adjoining subsets are common in electronic and control systems and can behave very erratically. The switching points or sets are not necessarily semi-hyperbolic. Example 3.16. The piecewise linear mapping f : R1 → R1 with ( 1 x, x < 0, f (x) = 2 4x, x ≥ 0 is not semi-hyperbolic on any set K containing the switching point x = 0 because there is no appropriate splitting of R1 at x = 0 satisfying the inequalities of Condition SH2(Lip). Each component mapping is semi-hyperbolic on its subset of definition, with the splitting Exs ⊕ Exu = {0} ⊕ R1 for x ≤ 0 and with Exs ⊕ Exu = R1 ⊕ {0} for x ≥ 0. The piecewise linear mapping f : R1 → R1 with ( − 21 x, x < 0, f (x) = 4x, x ≥ 0 is also not semi-hyperbolic on any set K containing the switching point x = 0, but any iterate f j , j ≥ 2, is semi-hyperbolic. An admissible splitting is Exs ⊕ Exu = {0} ⊕ R1 for x ∈ K and split s is (λs , λu , µs , µu ) = (0, 22j−3 , 0, 0). Here Pxs is the zero mapping and Pxu the identity mapping. Observe that the tent mapping f (x) = 1 − |1 − 2x|, which is Lipschitz, is not semi-hyperbolic on any subset of R1 containing the point x = 21 .

3.2.2 Anosov Endomorphisms Examples of semi-hyperbolic mappings with a uniform splitting that is either not continuous or lacks equivariance of a splitting for a hyperbolic mapping are more difficult to describe explicitly. In the following examples, algebraic Anosov automorphisms and noninvertible Anosov endomorphisms are introduced and an example with the desired properties is constructed as a differentiable perturbation of an Anosov endomorphism on a torus. Structural stability is not contradicted here since neither the endomorphism which is being perturbed nor the perturbations themselves are diffeomorphisms, although differentiable. Write the elements of Rd as vectors x with coordinates x1 , x2 , . . ., xd and let Td be the standard d-dimensional torus, that is the quotient Rd /Zd . Td is a closed differentiable manifold in Rd with respect to the locally Euclidean metric

30


ρ(x, y) =

q

|x1 − y1 |2mod 1 + |x2 − y2 |2mod 1 + · · · + |xd − yd |2mod 1

on Td where |t − s| mod 1 = min {|t − s + 2k| : k = 0, ±1} ,

0 ≤ t, s < 1.

The tangent space Tx Td can then be identified with Rd by an appropriate choice of natural coordinates generated by those of Rd , so Tx Td = Rd for each x ∈ Td . Denote the natural projection from Rd onto Td by Π(x) = (x1 mod 1, x2 mod 1, . . . , xd mod 1),

x = (x1 , x2 , . . . , xd )

and consider the mapping f : Td → Td defined by f (x) = Π(Ax)

(3.19)

for some fixed d × d matrix A with integer components aij . Further, suppose that the matrix A is hyperbolic and Rd = E s ⊕ E u is the splitting of Example 3.14. Since the eigenvalues of the restriction A|E s lie inside the unit circle while those of the restriction A|E u lie outside the unit circle, there exist constants C > 0 and λ > 1 such that kAn uk ≤ Cλ−n kuk,

kAn vk ≥ C −1 λn kvk

(3.20)

for all u ∈ E s and v ∈ E u , where k · k is the Euclidean norm on Rd . Example 3.17 (Anosov Automorphisms and Endomorphisms). Let A be an invertible hyperbolic matrix with integer components. The mapping f defined by (3.19) is called an algebraic hyperbolic automorphism of the torus Td if |det A| = 1 and an the algebraic hyperbolic endomorphism of Td otherwise. An algebraic hyperbolic automorphism of a torus Td is clearly a globally invertible mapping and is a diffeomorphism on Td , but an algebraic hyperbolic endomorphism is generally not a diffeomorphism since it is only locally invertible. Since Tx Td has been identified with Rd , the tangent mapping T fx can be identified with A and the linear subspaces Exs = E s , Exu = E u for x ∈ Td are invariant under T fx ≡ A. Thus Tx Td = Rd = E s ⊕ E u = Exs ⊕ Exu

(3.21)

is an equivariant splitting for the tangent mapping T fx . Let Pxs and Pxu be projectors corresponding to the splitting (3.21) (see Definition 3.1) and for each x ∈ Td define a norm kvkx = max λn kAn Pxs vk + min λ−n kAn Pxu vk, n≥0

n≥0

3.2 Examples

31

on Tx Td = Rd , where λ is as in (3.20) (in fact, this norm k · kx does not depend on x). Then kT fx ukx ≤ λ−1 kukx ,

kT fx vkx ≥ λkvkx

(3.22)

for all u ∈ Exs , v ∈ Exu and x ∈ K, that is Conditions H1∗ and H2∗ of Definition 2.3 are valid for the endomorphism f on the torus Td ; if the mapping f is an automorphism, then it is a hyperbolic diffeomorphism on Td . The relations (3.21) and (3.22) for the Anosov hyperbolic endomorphism f defined in Example 3.17 can be rewritten in the form kPfs(x) T fx ukf (x) ≤ λ−1 kukx ,

u ∈ Exs ,

kPfs(x) T fx vkf (x) = 0,

v ∈ Exu ,

kPfu(x) T fx vkf (x) = 0,

u ∈ Exs ,

kPfu(x) T fx vkf (x) ≥ λkvkx ,

v ∈ Exu .

By Definition 3.4 this means that every Anosov hyperbolic endomorphism f is semi-hyperbolic on the torus Td . This property of semi-hyperbolicity is robust under small C 1 -perturbations, as is shown in the more general context of Theorem 5.2 in Chap. 5. Lemma 3.18. Given an Anosov hyperbolic endomorphism f : Td → Td , there exists an ε > 0 such that every differentiable mapping fε : Td → Td which is ε-close to the f in C 1 -metric is semi-hyperbolic. This extends a classical result of Anosov on the robustness of the hyperbolicity property when f is an automorphism, that is, an invertible endomorphism, where every nearby diffeomorphism fε : Td → Td is hyperbolic and in particular has a hyperbolic splitting. Theorem 5.1 will show that invertibility is essential in this case. It is sometimes more convenient to consider semi-hyperbolic mappings on a compact subset of Rd rather than on a manifold. This is also possible for Anosov mappings and their perturbations, but the transition from one approach to the other is often not trivial in practice. The following example illustrates the procedure for one particular mapping, and the general construction is later used in the proof of Theorem 5.1. Example 3.19. Consider the 3 × 3 matrix  23 A = 1 2 00

 0 0 2

√ √ which has eigenvalues λ1 = 2 − √ 3, λ2 = 2 + 3 and √ λ3 = 2 with corresponding eigenvectors v1 = (1, −1/ 3, 0), v2 = (1, 1/ 3, 0) and v3 = (0, 0, 1),

32


respectively. Denote by f : T3 → T3 the continuously differentiable mapping defined by f (x) = Π(Ax). Interpret a point z ∈ R6pas an ordered triplet (z1 , z2 , z3 ) of complex numbers with the norm kzk = |z1 |2 + |z2 |2 + |z3 |2 and let J : T3 → R6 be the immersion which maps the point ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ T3 to the point z = (e2πiϕ1 , e2πiϕ2 , e2πiϕ2 ) ∈ K, where K = z ∈ R6 : |z1 | = |z2 | = |z3 | = 1 . Consider the mapping F (z) = f (P (z)),

z1 , z2 , z3 6= 0,

where P is the natural projection on K defined by z1 z2 z3 , , , z1 , z2 , z3 6= 0. P (z) = |z1 | |z2 | |z3 | The restriction of the mapping F to K is then topologically conjugate by the immersion J to the mappings f and fε , respectively. It is not easy to find an explicit expression for F , but its restriction to K has the form F (z) = (z1a11 z2a12 z3a13 , z1a21 z2a22 z3a23 , z1a31 z2a32 z3a33 ) ,

z ∈ K,

where the aij are the components of the matrix A generating the Anosov endomorphism f . For each point z ∈ K the sets of vectors i i t t e1 (z) = iz1 , − √ z2 , 0 , e2 (z) = iz1 , √ z2 , 0 , et3 (z) = (0, 0, iz3 ) , 3 3 en1 (z) = (z1 , 0, 0) ,

en2 (z) = (0, z2 , 0) ,

en3 (z) = (0, 0, z3 )

are a basis for R6 , for which the differential DFz of F satisfies √ DFz et1 (z) = (2 − 3)et1 (F (z)), √ DFz et2 (z) = (2 + 3)et2 (F (z)), DFz et3 (z) = 2et3 (F (z)), DFz en1 (z) = DFz en2 (z) = DFz en3 (z) = 0. Consider the splitting Tz R6 = Ezs ⊕ Ezu for z ∈ K with subspaces

(3.23)

3.2 Examples

33

Ezs = span et1 (z), en1 (z), en2 (z), en3 (z) , Ezu = span et2 (z), et3 (z) with corresponding projectors Pzs and Pzu satisfying Pzs R6 = Ezs ,

Pzs Ezu = 0,

Pzu R6 = Ezu ,

Pzu Ezs = 0.

The relations (3.23) imply that the splitting Tz R6 = Ezs ⊕ Ezu is invariant for the differential DFz . On the other hand, the inequalities √ u ∈ Ezs , kPFs (z) DFz uk ≤ (2 − 3)kuk, kPFs (z) DFz vk = 0,

v ∈ Ezu ,

kPFu(z) DFz vk = 0,

u ∈ Ezs ,

kPFu(z) DFz vk ≥ 2kvk,

v ∈ Ezu ,

hold, so the differentiable mapping F is both hyperbolic and semi-hyperbolic on the compact subset set K of R6 in the sense of Definitions 2.5, 3.5.

Chapter 4

Semi-Hyperbolic Sequences of Matrices

The utility of first approximation methods in investigating the properties of semi-hyperbolic mappings and their trajectories naturally leads to consideration of linear operators in spaces of sequences generated by derivatives of semi-hyperbolic mappings. Some elementary properties of such linear operators that will be needed, in this and later chapters, are considered here. Throughout this chapter k · k will denote a fixed but otherwise arbitrary norm on Rd and `∞ (I, Rd ) will denote the space of all bounded sequences of vectors xn ∈ Rd with indices n taking values in a subset, or interval I in Z which can be finite, uni- or bi-directionally infinite depending on context. The norm on `∞ (I, Rd ) is defined as k{xn }k∞ = sup kxn k. n∈I

4.1 The Split Matrix Recall from Definition 3.2 that a split is a four-tuple s = (λs , λu , µs , µu ) of nonnegative real numbers for which λs < 1 < λu ,

(1 − λs )(λu − 1) > µs µu .

(4.1)

Recall also that the split s is positive if all of the numbers λs , λu , µs and µu are positive. Clearly, the splits in Definitions 3.4, 3.5 and 3.6 can also have positive counterparts. For a given positive split s define the 2 × 2 split matrix M (s) by λs µs M (s) := , (4.2) µu /λu 1/λu with spectral radius σ(s)

35

36

4 Semi-Hyperbolic Sequences of Matrices

  s 2 1 1 1 4µs µu  σ(s) = + λs + − λs + . 2 λu λu λu

(4.3)

A simple but cumbersome calculation shows that σ(s) < 1 − ν(s) < 1 where ν(s) =

(4.4)

(1 − λs )(λu − 1) − µs µu λu − λs + µs + µu

(see (3.3)). Moreover, since the entries of the matrix M (s) are positive for a positive split s, it follows by the Perron–Frobenius Theorem that σ(s) is the maximal eigenvalue of M (s) and that the corresponding eigenvector has positive components. This eigenvector will be written here as (1, γ(s))T , where   s 2 1 4µs µu  1  1 . (4.5) − λs + − λs + γ(s) := 2µs λu λu λu so

When the split s is fixed, we shall write M = M (s), γ = γ(s) and σ = σ(s), 1 λs µs 1 1 M = =σ , (4.6) γ µu /λu 1/λu γ γ

and introduce the norm k · k∗ on R2 defined by k(y1 , y2 )k∗ := max {γ|y1 |, |y2 |} . The corresponding norm kM k∗ of the linear operator with the matrix (4.2) clearly coincides with the spectral radius σ of M , so kM k∗ = σ < 1 by (4.4) and kM xk∗ ≤ σkxk∗ for all x ∈ R2 . Lemma 4.1. The following hold min{γ, 1} k(y1 , y2 )k∞ ≤ k(y1 , y2 )k∗ ≤ max{γ, 1} k(y1 , y2 )k∞ . Proof. By direct calculation.

t u

The next lemma will play an important role in what follows. Lemma 4.2. For m = 1, 2, . . . let the following inequalities hold: asm ≤ λs asm−1 + µs aum−1 + hs , 1 u µu s a + a + hu , aum ≤ λu m−1 λu m−1 with asm , aum , hs , hu ≥ 0 and as0 = au0 = 0. Then

(4.7)

4.1 The Split Matrix

37

lim sup(asm + aum ) ≤ m→∞

max{hs , λu hu } . ν(s)

(4.8)

Proof. Define vectors am = (asm , aum )T for m = 0, 1, . . . and h = (hs , hu )T . Then the inequalities (4.7) can be rewritten in the vector form am ≤ M am−1 + h,

m = 1, 2, . . .

(4.9)

where a0 = 0 and the inequality between vectors is interpreted componentwise. Since the entries of the matrix M = M (s) are non-negative, we preserve inequality direction when applying the matrix M to the both sides of inequality (4.9). Hence am ≤ M m a0 + (I + M + · · · + M m−1 )h,

m = 1, 2, . . . ,

from which, in view of equality a0 = 0, we obtain lim sup am ≤ (I − M )−1 h

(4.10)

m→∞

where the lim sup and inequality apply to the respective components. By direct calculation 1 λu − 1 µs λu , (I − M )−1 = µu (1 − λs )λu (1 − λs )(λu − 1) − µs µu so (4.10) can be rewritten componentwise as (λu − 1)hs + µs λu hu , (1 − λs )(λu − 1) − µs µu µu hs + (1 − λs )λu hu ≤ , (1 − λs )(λu − 1) − µs µu

lim sup asm ≤ m→∞

lim sup aum m→∞

from which inequality (4.8) is an immediate consequence.

t u

4.1.1 A Perturbation Theorem Various proofs for semi-hyperbolic mappings will involve a matrix or a sequence of matrices that have similar semi-hyperbolic properties. This motivates the next definition where s = (λs , λu , µs , µu ) is an arbitrary but otherwise fixed split. Definition 4.3 (Semi-Hyperbolic Matrix). A d × d matrix A is called an (s, h)-semi-hyperbolic matrix if the corresponding linear mapping is s-semihyperbolic on Rd with respect to a splitting Rd = E s ⊕ E u and projectors P s and P u satisfying kP s k ≤ h and kP u k ≤ h.

38


The following Perturbation Theorem, which asserts that the class of semihyperbolic matrices is an open set in the space of all matrices, indicates the robustness of semi-hyperbolic matrices and provides explicit estimates of possible perturbations which preserve the semi-hyperbolicity of a matrix. Theorem 4.4 (Perturbation Theorem). Let a d × d matrix A be (s, h)hyperbolic. Then every d × d matrix A˜ satisfying kA˜ − Ak ≤ δ
0, k ∈ Z and v ∈ Pks Rd , mapping a sequence x = {xn } ∈ `∞ ([k, ∞), Rd ) to the sequence v = {vn } ∈ `∞ ([k, ∞), Rd ) defined by Pks vk = v, Pns vn = αPns An−1 xn−1 ,

n ≥ k + 1,

(4.14)

u u xn+1 An Pnu )−1 Pn+1 Pnu vn = α−1 (Pn+1 u u An Pns xn , − (Pn+1 An Pnu )−1 Pn+1

n ≥ k.

From the split inequalities (4.1) there exists ω > 0 such that for all α ∈ [1 − ω, 1 + ω] the strict inequalities αλs < 1 < αλu ,

(1 − αλs )(αλu − 1) > α2 µs µu

(4.15)

hold. Hence ρ=

sup α∈[1−ω,1+ω]

max

αµu + αλu − 1 1 − αλs + αµs , αλu − 1 1 − αλs

is well defined and for all α ∈ [1 − ω, 1 + ω] the spectral radius σ(Mα ) of the matrix αλs αµs Mα = µu /λu 1/(αλu ) is, from ((4.3)),   s 2 1 1 1 4αµs µu  σ(Mα ) = + αλs + − αλs + < 1. 2 αλu αλu λu Since the entries of the matrix Mα are positive (recall that we have restricted ourselves to positive splits s), it follows by the Perron–Frobenius

4.2 Semi-Hyperbolicity Implies Hyperbolicity

41

Theorem that the spectral radius σ(Mα ) is an eigenvalue and that the corresponding eigenvector has positive coordinates. Without loss of generality this vector has the form (1, γα )T . Now, note that for each α ∈ [1 − ω, 1 + ω] the norm s u kxkα ∞ = sup {γα kPn xn k, kPn xn k} n≥k

on the space `∞ ([k, ∞), Rd ) is equivalent to the norm k · k∞ . Lemma 4.6. For any α ∈ [1 − ω, 1 + ω], any integer k and any v ∈ Pks Rd , the operator R[α,k,v] is contracting with constant σ(Mα ) in the norm k · kα ∞. Moreover, the set αµu s u kvk, n ≥ k (4.16) V[α,k,v] = x : kPn xn k ≤ kvk, kPn xn k ≤ αλu − 1 is invariant under R[α,k,v] . Corollary 4.7. For any v ∈ P s Rd the operator R[α,k,v] has a unique fixed point x∗ = x[α,k,v] = {x∗n } for which Pks x∗k = v and kPns x∗n k ≤ kvk,

kPnu x∗n k ≤

αµu kvk, αλu − 1

n ≥ k.

Corollary 4.7 follows immediately from Lemma 4.6 by the Banach Contraction Mapping Theorem, so it suffices to prove Lemma 4.6. Proof. Fix a value of α ∈ [1 − ω, 1 + ω]. The definition of the vector (1, γα )T gives the equality αλs αµs 1 1 = σ(Mα ) , µu /λu 1/(αλu ) γα γα from which it follows that αλs + αµs γα = σ(Mα ),

1 µu + γα = σ(Mα )γα . λu αλu

(4.17)

˜ ∈ `∞ ([k, ∞), Rd ) write Given arbitrary x, x v = R[α,k,v] (x),

˜ = R[α,k,v] (˜ v x).

Then from (4.14) it follows that Pks (vk − v˜k ) = 0, Pns (vn − vñ ) = αPns An−1 (xn−1 − x ñ−1 ), Pnu (vn

− vñ ) = α

−

u u (Pn+1 An Pnu )−1 Pn+1 (xn+1 − x ñ+1 ) u u −1 u s (Pn+1 An Pn ) Pn+1 An Pn (xn − x ñ ),

n ≥ k + 1,

−1

n ≥ k.

42


From (4.13) the following relations hold. kPks (vk − v˜k )k = 0, kPns (vn − vñ )k ≤ αλs kPns (xn−1 − x ñ−1 )k + αµs kPnu (xn−1 − x ñ−1 )k, µ u kPnu (vn − vñ )k ≤ kP s (xn − x ñ )k λu n 1 + kP u (xn+1 − x ñ+1 )k, αλu n+1

n ≥ k + 1,

n ≥ k.

From these and (4.17) it is clear that ˜ kα kR[α,k,v] (x) − R[α,k,v] (˜ x)kα ∞ ≤ σ(Mα )kx − x ∞, and the operator R[α,k,v] is contracting in the norm k · kα ∞ with contraction constant σ(Mα ). It remains only to show that the set (4.16) is invariant for R[α,k,v] . Given an arbitrary x = {xn } ∈ V[α,k,v] , denote v = R[α,k,v] (x). From (4.13) and (4.14) it is apparent that kPks vk k = kvk, kPns vn k ≤ αλs kPns xn−1 k + αµs kPnu xn−1 k, 1 µu s kPn xn k + kP u xn+1 k, kPnu vn k ≤ λu αλu n+1

n ≥ k + 1, n ≥ k.

Hence, in view of the definition (4.16) of the set V[α,k,v] , we have kPks vk k = kvk, α2 µs µu kvk, αλu − 1 µu αµu kPnu vn k ≤ kvk, kvk + λu αλu (αλu − 1) kPns vn k ≤ αλs kvk +

n ≥ k + 1, n ≥ k,

and thus, from (4.15), can conclude that kPks vk k = kvk, kPns vn k ≤ kvk + kPnu vn k ≤

α2 µs µu − (1 − αλs )(αλu − 1) kvk ≤ kvk, αλu − 1


n ≥ k + 1, n ≥ k.

This completes the proof of invariance of the set (4.16) under R[α,k,v] . Lemma 4.6 is proved. t u The next lemma follows immediately from Lemma 4.6 and from the definition of the operator R[α,k,v] .


43

Lemma 4.8. The only bounded solution of the difference equation xn+1 = αAn xn ,

n ≥ k,

(4.18)

which satisfies the condition Pks xk = v is the fixed point x[α,k,v] of the operator R[α,k,v] . Lemmas 4.6 and 4.8 can be combined to show that the fixed point sequence x[1,k,v] of R[1,k,v] is not only bounded one but is also exponentially convergent to 0 ∈ Rd . Corollary 4.9. The fixed point sequence x∗ = x[α,k,v] = {x∗n } of the operator R[1,k,v] satisfies kx∗n k ≤

ρ kvk, (1 + ω)|n−k|

n ≥ k.

(4.19)

Proof. By the definition of the constant ω > 0 the operator R[1+ω,k,v] has a fixed point x∗1+ω = {x∗1+ω,n }. By Lemma 4.8 {x∗1+ω,n } is the bounded solution of equation (4.18) for α = 1 + ω. Hence the sequence x∗ = {x∗n }, where x∗n = (1 + ω)k−n x∗1+ω,n , n ≥ k, is the solution of equation (4.18) for α = 1. In view of Corollary 4.7 we have for any n ≥ k kx∗1+ω,n k ≤ kPns x∗1+ω,n k + kPnu x∗1+ω,n k ≤

αµu + αλu − 1 kvk ≤ ρkvk αλu − 1

and so by the definition of x∗ the inequalities (4.19) are valid. Thus, x∗ is the bounded solution of equation (4.18) for α = 1 and by Lemma 4.8 it is the fixed point of the operator R[1,k,v] for which estimate (4.19) is valid. t u Define the operator L := L[α,k,w] : `∞ ((−∞, k], Rd ) → `∞ ((−∞, k], Rd ), again with parameters α > 0, k ∈ Z and w ∈ Pku Rd , mapping a sequence y = {yn } ∈ `∞ ((−∞, k], Rd ) to the sequence w = {wn } ∈ `∞ ((−∞, k], Rd ) defined by Pns wn = αPns An−1 yn−1 , Pku wk Pnu wn

n ≤ k,

= w, u u = α−1 (Pn+1 An Pnu )−1 Pn+1 yn+1 u u − (Pn+1 An Pnu )−1 Pn+1 An Pns yn ,

n ≤ k − 1.

44


The proofs of the following two lemmas and corollaries are essentially repetitions of those for Lemmas 4.6 and 4.8 and for Corollaries 4.7 and 4.9 and are omitted. Lemma 4.10. For any α ∈ [1 − ω, 1 + ω], any integer k and any w ∈ Pku Rd the operator L[α,k,w] is contracting in the norm k · kα ∞ with constant σ(Mα ). Moreover, the set αµs kwk, kPnu xn k ≤ kwk, n ≤ k W[α,k,w] = x : kPns xn k ≤ 1 − αλs is invariant under L[α,k,w] . Corollary 4.11. For any w ∈ P u Rd the operator L[α,k,w] has a unique fixed point y ∗ = y [α,k,w] = {yn∗ } for which Pku yk∗ = w and kPns yn∗ k ≤

αµs kwk, 1 − αλs

kPnu yn∗ k ≤ kwk,

n ≤ k.

Lemma 4.12. The only bounded solution of the difference equation yn+1 = αAn yn ,

n ≤ k − 1,

which satisfies the condition Pku yk = w is the fixed point y [α,k,w] of the operator L[α,k,w] . Corollary 4.13. The fixed point y ∗ = y [α,k,w] = {yn∗ } of the operator L[1,k,w] satisfies kyn∗ k ≤

ρ kwk, (1 + ω)|n−k|

n ≤ k.

4.2.2 An Equivariant Splitting Let x[α,k,v] = {x[α,k,v],n } denote the fixed point of the operator R[α,k,v] and define the operator, X[α,k] : Pks Rd → Rd by X[α,k] v = x[α,k,v],k .

Lemma 4.14. The operator X[α,k] is linear, Pks X[α,k] Pks = Pks and kPku X[α,k] vk ≤


v ∈ Pks Rd .

(4.20)

Proof. Let x∗ = {xn } = x[α,k,v] be the fixed point of the operator R[α,k,v] ˜ ∗ = {˜ and x xn } = x[α,k,˜v] be the fixed point of the operator R[α,k,˜v] for v,


45

v˜ ∈ Pks Rd . Then by Lemma 4.8 x∗ = {xn } is the solution of equation (4.18) ˜ ∗ = {˜ satisfying the condition Pks xk = v, while x xn } is the solution of equation s (4.18) satisfying the condition Pk x ˜k = v˜. Hence for any real numbers ξ and ζ, the sequence ˜ ∗ = {ξxn + ζ x z ∗ = ξx∗ + ζ x ñ } ∈ `∞ ([k, ∞), Rd ) is the solution of equation (4.18) satisfying the condition Pks zk = ξv + ζ v˜. Using Lemma 4.8 again, we find that z ∗ is the fixed point of the operator R[α,k,ξv+ζ v˜] , that is z ∗ = x[α,k,ξv+ζ v˜] . Thus ξx[α,k,v] + ζx[α,k,˜v] = x[α,k,ξv+ζ v˜] , so the mapping x[α,k,·] is linear and hence the operator X[α,k] (·) = x[α,k,·] is linear. Now note that Pks X[α,k] v = Pks x[α,k,v],k = v for v ∈ Pks Rd , which means that Pks X[α,k] Pks = Pks , as required. Estimate (4.20) then follows immediately from the definition of the operator X[α,k] and from Corollary 4.7. t u Similarly, denote the fixed point of the operator L[α,k,w] by y [α,k,w] = {y[α,k,w],n } and define an operator Y[α,k] : Pku Rd → Rd by Y[α,k] w = y[α,k,w],k . The proof of the next lemma is essentially the same as that of Lemma 4.14. Lemma 4.15. The operator Y[α,k] is linear, Pku Y[α,k] Pku = Pku and kPks Y[α,k] wk ≤

αµs kwk, 1 − αλs

w ∈ Pku Rd .

˜ s as the set of those v ∈ Rd for which the equation Now define E k vn+1 = An vn ,

n ≥ k,

(4.21)

˜ u as the set of those has a bounded solution satisfying vk = v, and define E k d w ∈ R for which the equation wn+1 = An wn ,

n < k,

(4.22)

ˆ s and has a bounded solution satisfying wk = w. Further, define subspaces E k s d ˆ of R by E k

ˆks = X[1,k] Pks Rd , E

ˆku = Y[1,k] Pku Rd , E

k ∈ Z.

(4.23)

46


Finally, recall that the separation δ(E1 , E2 ) between two subspaces E1 , E2 ⊆ Rd is defined as δ(E1 , E2 ) = inf{kx + yk : x ∈ E1 , y ∈ E2 , kxk = kyk = 1}. Lemma 4.16. Suppose that matrices An , n ∈ Z, are invertible. Then for any integer k the following relations are valid: ˆs = E ˜s, E k k s u ˆ ˆ Ek ∩ Ek = {0}, ˆs = E ˆs , Ak E k k+1 ˆks , E ˆku ) ≥ δ(E

û = E ˜u, E k k s u d ˆ ˆ Ek ⊕ Ek = R , u û = E ˆk+1 Ak E . k

(1 − λs )(λu − 1) − µs µu > 0. h(λu − 1 + µu )(1 − λs + µs )

(4.24) (4.25) (4.26) (4.27)

ˆs = E ˜ s from (4.24). Indeed, if v ∈ E ˆ s then, Proof. We first prove that E k k k ˆ s , there exists v ∗ ∈ P s Rd such that v = X[1,k] v ∗ and, by by definition of E k k definition of the operator X[1,k] , v ∗ = Pks vk where {vn }, n ≥ k, is the fixed point of the operator R[1,k,v∗ ] satisfying vk = v. Hence by Lemma 4.8, {vn } ˜ s , from which it follows is a bounded solution of equation (4.21) and so v ∈ E k s s s ˆ ⊆E ˜ . Similarly, if v ∈ E ˜ then there exist a bounded solution {vn } that E k k k of equation (4.21) satisfying vk = v. So by Lemma 4.8 {vn } is the fixed point of the operator R[1,k,Pks v] . Therefore, by the definition of the operator X[1,k] , ˆs, the vector v can be represented in the form v = X[1,k] Pks v and so v ∈ E k ˜s ⊆ E ˆ s . Combining the two results gives the from which it follows that E k k ˜s = E ˆ s . That E ˜u = E ˆ u follows similarly. desired identity E k k k k ˆ u 6= {0} for some k. ˆs ∩ E To prove the first part of (4.25) suppose that E k k s d u d Then there exist nonzero v ∈ Pk R , w ∈ Pk R and {vn } ∈ `∞ ([k, ∞), Rd ),

{wn } ∈ `∞ ((−∞, k], Rd )

such that {vn } is the fixed point of the operator R[1,k,v] , {wn } is the fixed point of the operator L[1,k,w] and ˆks ∩ E ˆku , vk = w k ∈ E

vk = wk 6= 0.

(4.28)

By Lemmas 4.8 and 4.14 the sequence {vn } is a bounded solution of equation (4.21), and the sequence {wn } is a bounded solution of equation (4.22). From (4.28), the sequence {xn } defined for all integer k by ( vn , n ≥ k, xn = wn , n < k is thus a nonzero bounded solution of


47

xn+1 = An xn ,

n ∈ Z.

Set χ = supn∈Z kxn k. Then by Corollary 4.9 kxk k ≤

ρχ ρ kxn k ≤ (1 + ω)|k−n| (1 + ω)|k−n|

for any n < k. Letting n converge to −∞, from the above relations we obtain xk = vk = wk = 0, which contradicts (4.28). The first equation of (4.25) is proved. To prove the second of (4.25) it suffices to note that by Lemmas 4.14 and 4.15 ˆks ≥ dim Pks Rd , dim E ˆku ≥ dim Pku Rd , dim E and so ˆks + dim E ˆku ≥ dim Pks Rd + dim Pku Rd = dim Rd = d. dim E From this and from the first of (4.25), the second of (4.25) follows. ˜s, E ˜ u and from invertibility of Now from the definition of the subspaces E k k the matrices An , n ∈ Z, it follows immediately that s ˜ks = E ˜k+1 Ak E ,

u ˜ku = E ˜k+1 Ak E .

Then by (4.24) ˆ s = Ak E ˜s = E ˜s = E ˆs , Ak E k k k+1 k+1 u u u ˆ = Ak E ˜ =E ˜ û Ak E k k k+1 = Ek+1 . from which (4.26) follows. Finally, turn to the inequality (4.27). Set θs =

µs , 1 − λs

θu =

µu . λu − 1

ˆ s , kxk = 1, and denote v = P s x. Then, by the Fix an arbitrary vector x ∈ E k k ˆ s , x = X[1,k] v and from Lemma 4.14 definition of the subspace E k kPku xk ≤ θu kvk = θu kPks xk.

(4.29)

ˆ u , kyk = 1, from the definition of the Similarly, for an arbitrary vector y ∈ E k u ˆ subspace Ek and from Lemma 4.15 the inequality kPks yk ≤ θs kPku yk can be easily obtained. By (4.12) and (4.30)

(4.30)

48


1 s kP (x + y)k h k 1 1 ≥ (kPks xk − kPks yk) ≥ (kPks xk − θs kPku yk) , h h

kx + yk ≥

and also from (4.12) and (4.29) 1 u kP (x + y)k h k 1 1 ≥ (kPku yk − kPku xk) ≥ (kPku yk − θu kPks xk) . h h

kx + yk ≥

From (4.29) and (4.30) it follows that in the above inequalities the terms kPks xk and kPku yk satisfy kPks xk ≥

1 1 kxk = , 1 + θu 1 + θu

kPku yk ≥

1 1 kyk = . 1 + θs 1 + θs

Hence kx + yk ≥

1 inf max{s − θs t, t − θu s}. h s≥(1+θu )−1 , t≥(1+θs )−1

Note that the function max{s − θs t, t − θu s} can attain its infimum on the set s ≥ (1 + θu )−1 , t ≥ (1 + θs )−1 only in the case when s − θs t = t − θu s, that is when (1 + θu )s = (1 + θs )t. Let τ = (1 + θu )s = (1 + θs )t, so that s=

τ , 1 + θu

t=

τ . 1 + θs

Then 1 kx + yk ≥ inf h τ ≥1

τ τ − θs 1 + θu 1 + θs

≥

1 − θu θs . h(1 + θu )(1 + θs )

ˆs, E ˆ u ), θs and θu , (4.27) follows. From this and the definitions of δ(E k k

t u

Remark 4.17. If the invertibility of the matrices An , n ∈ Z, in Lemma 4.16 is not assumed, then (4.26) is, in general, replaced by ˆs ⊆ E ˆs , Ak E k k+1

û = E û . Ak E k k+1

ˆ s and E ˆ u do not necessarily have symmetric propThat is, both the spaces E k k erties if invertibility is not present. The results of this section are summarized by the following theorem.


49

Theorem 4.18 (Equivariant Splitting Theorem). Let {An }, n ∈ Z, be an (s, h)-semi-hyperbolic sequence of invertible d × d matrices. Then the ˆns , E ˆnu defined by (4.23) form a splitting Rd = E ˆns ⊕ E ˆnu with subspaces E s d s u s d ˆ ˆ ˆ ˆ ˆnu corresponding projections Pn : R → En and Pn = I − Pn : R → E satisfying the uniform matrix bounds ˆ kPˆns k ≤ h,

ˆ kPˆnu k ≤ h.

(4.31)

This splitting is equivariant with respect to the matrix sequence {An }, s ˆns = E ˆn+1 An E ,

u ˆnu = E ˆn+1 An E .

Moreover, there exist constants ρ > 0 and 0 < λ < 1, depending only on the ˆ s the split s and the parameter h, such that for any integer k and vk ∈ E k sequence {vn } defined by vn+1 = An vn ,

n ≥ k,

satisfies ˆns , vn ∈ E

kvn k ≤ ρλ−|n−k| kvk k,

Similarly, for any integer k and wk ∈

û E k

wn+1 = An wn ,

n ≥ k.

(4.32)

the sequence {wn } defined by n < k,

satisfies ˆw, wn ∈ E n

kwn k ≤ ρλ−|n−k| kwk k,

n ≤ k.

(4.33)

ˆnu and inclusions in ˆns ⊕ E Proof. Existence of the equivariant splitting Rd = E (4.32), (4.33) follow immediately from Lemma 4.16. The inequality for kvn k in (4.32) follows from Corollary 4.9, while that for kwn k in (4.33) follows from Corollary 4.13. It remains only to prove the inequalities (4.31) for the ˆnu . ˆns and Pˆnu = I − Pˆns : Rd → E projections Pˆns : Rd → E Let ˆ = 2(λu − 1 + µu )(1 − λs + µs ) h h (1 − λs )(λu − 1) − µs µu and consider (4.31) for this constant. Clearly kxk = kPˆns x + Pˆnu xk ≥ kPˆns xk − kPˆnu xk .

(4.34)

If

s kPˆn xk − kPˆnu xk ≥ 1 max{kPˆns xk, kPˆnu xk} ˆ h then from this and from (4.34), obtain (4.31). So, suppose that s kPˆn xk − kPˆnu xk < 1 max{kPˆns xk, kPˆnu xk}. ˆ h

(4.35)

50


Without loss of generality it may be assumed that Pˆns x 6= 0, which implies

Pˆnu x 6= 0

and kPˆns xk ≥ kPˆnu xk,

kPˆns xk = max{kPˆns xk, kPˆnu xk}.

(4.36)

Now consider kxk in the following way kxk = kPˆns x + Pˆnu xk ! Pˆ s x Pˆnu x Pˆnu x ˆ s n s u ˆ ˆ + kPn xk − kPn xk − kPn xk = kPˆns xk kPˆnu xk kPˆnu xk Pˆ s x Pˆ u x ≥ n + n kPˆns xk − kPˆns xk − kPˆnu xk . kPˆns xk kPˆnu xk Here, by (4.27) and (4.36), the first term in the right-hand part can be ˆ −1 max{kPˆ s xk, kPˆ u xk}, while for the bounded from below by the value 2h n n second term the inequality (4.35) holds. Thus kxk ≥

1 max{kPˆns xk, kPˆnu xk}, ˆ h

which completes the proof of (4.31).

t u

Remark 4.19. By statements (4.32) and (4.33) of Theorem 4.18, the subspaces ˆ s and E ˆ u in the splitting Rd = E ˆ s ⊕E ˆ u can be interpreted as asymptotically E n n n n stable and asymptotically unstable, respectively. In such a situation it seems not unreasonable to conjecture from Theorem 4.18 that if if {An } is an ˆ (s, h)-semi-hyperbolic sequence of invertible matrices, then it is (ˆ s, h)-semiˆs, λ û, µ ˆ = (λ hyperbolic with a split s ˆs , µ û ) satisfying ˆ s < 1, λ

ˆ u > 1, λ

µ ˆs = µ û = 0,

ˆ s and E ˆ u in the equivariant splitting Rd = E ˆs ⊕ E û so that the subspaces E n n n n are stable and unstable, respectively, with respect to a unique fixed norm k · k in Rd . Unfortunately, a proof of such a generalization of Theorem 4.18 is not available. Indeed, Example 4.21 suggests invalidity if the class of semihyperbolic mappings is too restrictive. A possible way to generalize Theorem 4.18 in a similar way, by widening the class of semi-hyperbolic mappings by introducing a slightly weaker condition, is discussed in the next section.


51

4.2.3 Hyperbolicity of Sequences of Matrices The results of Sects. 4.2.1 and 4.2.2 can be generalized by defining a version of the semi-hyperbolic matrix sequence that uses a variable norm replacing that of Definition 3.4 and (3.5)–(3.8) Definition 4.20 (Semi-Hyperbolic Sequence of Matrices). A bounded sequence of d × d matrices {An } will be called (s, h)-semi-hyperbolic if there exists a set of norms {k · kn } on Rd satisfying the uniform boundedness condition q −1 kxk ≤ kxkn ≤ qkxk, x ∈ Rd (4.37) for some constant q ≥ 1 such that for each matrix An , n ∈ I, there exists a splitting Rd = Ens ⊕ Enu with s dim Ens = dim En+1 ,

u dim Enu = dim En+1

(4.38)

and projections Pns : Rd → Ens , Pnu = I − Pns : Rd → Enu , with the uniform matrix norm bounds, in the usual Rd operator norm, kPns k ≤ h,

kPnu k ≤ h,

(4.39)

such that, in the norms k · k, s kPn+1 An Pns xkn+1 ≤ λs kPns xkn , s kPn+1 An Pnu xkn+1 ≤ µs kPnu xkn , u kPn+1 An Pns xkn+1 ≤ µu kPns xkn ,

(4.40)

u kPn+1 An Pnu xkn+1 ≥ λu kPnu xkn .

The following simple example clearly illustrates that this transition from the fixed norm definition to one involving variable norms broadens the defined class of semi-hyperbolic mappings. Example 4.21. Consider the following sequence of linear operators A2n x = 2x,

A2n+1 x =

1 x, 4

x ∈ R1 , n ∈ Z,

acting in the one-dimensional space R1 . Clearly, the matrix sequence {An } is not semi-hyperbolic in the sense of Definition 4.5 for any possible choice of fixed norm in R1 , but if we define the norms k · kn on R1 by kxk2n = |x|,

kxk2n+1 =

1 |x|, 3

n ∈ Z,

where |x| denotes the absolute value of the number x, then kA2n xk2n+1 =

1 2 2 |2x| = |x| = kxk2n 3 3 3

52


and

1 1 3 kA2n−1 xk2n = | x| = |x| = kxk2n−1 . 4 4 4 Hence the matrix sequence {An } is semi-hyperbolic in the sense of Definition 4.20. The special case of a semi-hyperbolic sequence of matrices corresponding to the situation in which the parameters µs and µu vanish is called a hyperbolic sequence of matrices. In this case the spaces Ens and Enu are equivariant for the sequence {An }, so the formal definition is simpler than Definition 4.20. Definition 4.22 (Hyperbolic Sequence of Matrices). A bounded sequence {An } of d × d matrices will be called hyperbolic if for each n ∈ I there exists a norm k·kn in Rd satisfying the uniform boundedness condition (4.37) and a splitting Rd = Ens ⊕ Enu with corresponding projections Pns : Rd → Ens , Pnu = I − Pns : Rd → Enu satisfying (4.38) and (4.39) such that s An Ens = En+1 ,

u An Enu = En+1

(4.41)

and the inequalities kAn Pns xkn+1 ≤ λs kPns xkn ,

kAn Pnu xkn+1 ≥ λu kPnu xkn

(4.42)

hold for some constants 0 ≤ λs < 1 < λm . Using Definitions 4.20 and 4.22, the statement of Theorem 4.18 now becomes more compact and elegant. Theorem 4.23. Every semi-hyperbolic sequence of invertible matrices {An }, n ∈ Z, is hyperbolic and conversely. Proof. Clearly, every hyperbolic sequence of invertible matrices {An }, n ∈ Z, is semi-hyperbolic, so we need only prove that the semi-hyperbolicity of a matrix sequence implies its hyperbolicity. An adjustment of the proof of Theorem 4.18 to the ‘variable norm’case can be done in the same manner as was explained in the proof of Theorem 4.34. In this way we obtain that for a given (s, h)-semi-hyperbolic sequence of invertible matrices {An }, n ∈ Z, in the sense of Definition 4.20 there exists ˆns ⊕ E ˆnu with corresponding projections Pˆns : Rd → E ˆns and a splitting Rd = E u s d u ˆ ˆ ˆ Pn = I − Pn : R → En with uniform matrix bounds (4.31). This splitting is equivariant with respect to the matrix sequence {An }, that is it satisfies (4.41). Moreover, there exist constants ρ > 0 and λ ∈ (0, 1) such that for any ˆ s the sequence {vn } defined by integer k ∈ Z and vk ∈ E k vn+1 = An vn , also satisfies

n ≥ k,

(4.43)

4.3 A More Abstract Approach

ˆns , vn ∈ E

53

kvn kn ≤ ρλ−|n−k| kvk kk ,

n ≥ k.

(4.44)

ˆ u the sequence {wn } defined by Similarly, for any integer k ∈ Z and wk ∈ E k wn+1 = An wn ,

n < k,

(4.45)

also satisfies ˆnw , wn ∈ E

kwn kn ≤ ρλ−|n−k| kwk kk ,

n ≤ k.

(4.46)

Thus, to complete the proof of theorem it suffices to establish the existence of norms k · k∗n in Rd satisfying uniform boundedness condition 1 kxk ≤ kxk∗n ≤ q ∗ kxk, q∗

x ∈ Rd ,

(4.47)

with some constant q ∗ ≥ 1 such that kAn Pˆns xk∗n+1 ≤ λkPˆns xk∗n ,

kAn Pˆnu xk∗n+1 ≥ λ−1 kPˆnu xk∗n

(4.48)

To construct the required norms, fix an integer k ∈ Z and x ∈ Rd and choose vk = Pˆks x, wk = Pˆku x. Then define sequences {vn }, n ≤ k, and {wn }, n ≥ k, satisfying (4.43) and (4.45), respectively. Finally, define ∗ k−n n−k kxkk = max sup λ kvn kn , sup λ kwn kn . n≥k

n≤k

From (4.44) and (4.46), inequalities (4.48) and (4.47) immediately follow with an appropriately chosen q ∗ . The theorem is proved. t u

4.3 A More Abstract Approach In this section an abstract approach in terms of linear operators in Banach spaces is used to investigate properties of sequences of semi-hyperbolic mappings.

4.3.1 Semi-Hyperbolic Linear Operators An abstract approach is able to clarify to a deeper extent the basic relationships between conditions of semi-hyperbolicity and hyperbolicity for sequences of matrices. To implement such an approach, begin by investigating how the property of semi-hyperbolicity of a linear operator on a Banach space is related to that of hyperbolicity.

54


Let A : E → E be a bounded linear operator on a Banach space E and let s = (λs , λu , µs , µu ) be a split. Definition 4.24 (Semi-Hyperbolic Linear Operator). A linear bounded operator A on a Banach space E with a norm k · k is called semi-hyperbolic, with respect to a split s = (λs , λu , µs , µu ), a constant h and the norm k · k, if there is a splitting E = E s ⊕ E u with corresponding bounded projectors P s : E → E s and P u = I − P s : E → E u satisfying kP s k ≤ h,

kP u k ≤ h,

(4.49)

and kP s AP s xk ≤ λs kP s xk, kP s AP u xk ≤ µs kP u xk,

(4.50)

kP u AP s xk ≤ µu kP s xk, kP u AP u xk ≥ λu kP u xk.

Remark 4.25. Above, it is assumed that the linear operator A is real, that is, acting on a real Banach space E. Often, however in studying spectral properties of linear operators it is convenient to treat them as if they were acting on complex Banach spaces. The natural way to pass from real to complex operators is the complexification of a linear operator. For the sake of completness of presentation the necessary details are briefly reviewed. For a real Banach space E with a norm k · k define the complexification of E, the complex Banach space Ec , as the set of complex vectors z = x + iy with x, y ∈ E endowed with the norm kzkc = kx + iykc :=

max kαx + βyk,

|α+iβ|=1

α, β ∈ R1 .

For a linear operator A : E → E we then define its complexification Ac : Ec → Ec by Ac (x + iy) := Ax + iAy, x, y ∈ E. With such a definition of complexification, spectral properties of a real operator A are exactly the same as those of its complexification Ac , both operators have the same spectrum, the same set of eigenvalues, and so on. For what follows, it is important that the passage from a real linear operator A to its complexification Ac preserves all the essential features of semi-hyperbolicity (see Definition 4.24). More precisely, if A is a semi-hyperbolic linear operator satisfying (4.49) and (4.50) then its complexification Ac : Ec → Ec must satisfy


55

kPcs Ac Pcs zkc ≤ λs kPcs zkc , kPcs Ac Pcu zkc ≤ µs kPcu zkc , kPcu Ac Pcs zkc ≤ µu kPcs zkc , kPcu Ac Pcu zkc ≥ λu kPcu zkc , for all z ∈ Ec with corresponding bounded projectors Pcs : Ec → Ecs and Pcu = I − Pcs : Ec → Ecu satisfying kPcs kc ≤ h,

kPcu kc ≤ h.

It seems quite reasonable now to define the concept of a hyperbolic linear operator by introducing the additional requirement of invariance of the splitting subspaces E s and E u with respect to the operator A. Such an invariance condition can be expressed by one of the following three equivalent conditions: AE s ⊆ E s , AE s ⊆ E s , or P s AP u = 0,

P u AP s = 0,

P s AP u x ≡ 0,

P u AP s x ≡ 0.

or Definition 4.26 (Hyperbolic Linear Operator: Metric Definition). A linear bounded operator A on a Banach space E with a norm k · k is called hyperbolic in the norm k · k if there is an invariant splitting E = E s ⊕ E u with corresponding bounded projectors P s : E → E s and P u = I − P s : E → E u satisfying kP s AP s xk ≤ λs kP s xk,

kP u AP u xk ≥ λu kP u xk,

with appropriate constants λs < 1 and λu > 1. The concept of hyperbolicity for a linear operator A on a Banach space can also be defined in a clearer and more constructive way via a description of the spectral properties of the operator A. Definition 4.27 (Hyperbolic Linear Operator: Spectral Definition). A linear bounded operator A on a Banach space is called spectrally hyperbolic if its spectrum does not intersect the unit disc |λ| = 1 in the complex plane. The link between the concepts of semi-hyperbolicity and hyperbolicity, in both metric and spectral versions, is established by the following theorem. For linear operators the concepts of semi-hyperbolicity and hyperbolicity, metric or spectral, are equivalent. The following theorem will be proved in Appendix A and is that of [93]. Theorem 4.28. If the operator A is semi-hyperbolic with respect to a split s = (λs , λu , µs , µu ) and a norm k · k, then A is spectrally hyperbolic.

56


Theorem 4.29. For a linear bounded operator A on a Banach space E with a norm k · k the following three statements are equivalent: The operator A is semi-hyperbolic with respect to some split s, constant h and a norm k · k∗ equivalent to the norm k · k. (ii) The operator A is metrically hyperbolic in some norm k · k∗ equivalent to the norm k · k. (iii) The operator A is spectrally hyperbolic.

(i)

Proof. As was remarked earlier, throughout the proof of theorem we can treat E as a complex Banach space. To prove the theorem the following implications are shown. (i) ⇒ (iii) ⇒ (ii) ⇒ (i). Note that (i) ⇒ (iii) is given by the previously stated result, the proof of which is in Appendix A. Now turning to (iii) ⇒ (ii), that spectral hyperbolicity of the operator A implies hyperbolicity in the sense of Definition 4.26. By supposition, the spectrum σ of the operator A does not intersect the unit disc |λ| = 1. Then, since the spectrum of any operator is a closed set, the set σ can be represented as the union σ = σs ∪ σu of closed subsets σs and σu where σs lies entirely inside the unit disc and σu lies entirely outside the unit disc. Denote by E s ⊆ E the invariant spectral subspace of the operator A corresponding to the part σs of its spectrum, and denote by E u ⊆ E the invariant spectral subspace of the operator A corresponding to the part σu of its spectrum. Then, from spectral theory, E s ∩ E u = ∅,

Es ⊕ Eu = E

hold. Denote by P s : E → E s and P u = I − P s : E → E u the projections corresponding to the splitting E = E s ⊕ E u , which are bounded in this case. Now, the spectral radius σ(A) of a linear operator A on a Banach space satisfies σ(A) = lim sup kAn k1/n , n→∞

from which it follows that for any ε > 0 there exists a number cε such that kAn k ≤ cε (σ(A) + ε)n ,

n ≥ 0.

(4.51)

By definition, the closed set σs lies inside the disc |λ| < 1. Thus numbers λs ∈ (0, 1) and ε > 0 can be found such that σs , is contained in the disc |λ| ≤ λs − ε. Then, applying the formula (4.51) to the restriction A|E s , of the linear operator A to its invariant subspace E s , obtain k(A|E s )n k ≤ cs,ε λns ,

n ≥ 0.

(4.52)


57

Similarly, numbers λu ∈ (0, 1) and ε > 0 can be found such that σu is contained in the exterior of the disc |λ| ≤ λu + ε. The restriction A|E s , of the linear operator A to its invariant subspace E s , is an invertible operator, and applying the formula (4.51) to (A|E s )−1 , obtain k(A|E s )−n k ≤ cu,ε λ−n u ,

n ≥ 0.

(4.53)

Define the norm k · k∗ in E by n s n −n u kxk∗ = max λ−n P xk. s k(A|E s ) P xk + max λu k(A|E u ) n≥0

n≥0

Clearly kxk∗ ≥ kP s xk + kP u xk ≥ kP s x + P u xk = kxk, and, at the same time, by (4.52), (4.53) kxk∗ ≤ cs,ε kP s xk + cu,ε kP u xk ≤ max {cs,ε kP s k, cu,ε kP u k} kxk. Hence the norms k · k∗ and k · k are equivalent. To complete the proof of the implication (iii) ⇒ (ii), it remains only to note from (4.52), (4.53) that the inequalities kP s AP s xk∗ ≤ λs kP s xk∗ , kP u AP u xk∗ ≥ λu kP u xk∗ , then follow, which, together with invariance of the subspaces E s and E u with respect to A, mean that A is a linear hyperbolic operator in the sense of Definition 4.26. The implication (iii) ⇒ (ii) is then proved. Finally, the implication (ii) ⇒ (i) is clear by definition of the split s and constant h, namely s = (λs , λu , 0, 0), h = max {kP s k, kP u k}. The proof of the theorem is complete. t u

4.3.2 Equivalent Operators in Sequence Spaces Let {An } be a sequence of d × d matrices defined for n belonging to some interval I ⊆ Z of the form [0, N ) or [0, ∞) or (−∞, 0] or Z. Now, define some linear operators induced by the sequence {An }, acting on an appropriate sequence space. Consider first the linear operator A : X → Y defined by (A x)n+1 = An xn ,

n ∈ I.

(4.54)

Here the definition of the spaces X and Y depend on I. If I = [0, N ], then it is convenient to set X = `∞ ([0, N + 1], Rd ),

Y = `∞ ([1, N + 1], Rd );

(4.55)

58


if I = (−∞, 0], then

X = Y = `∞ ((−∞, 1], Rd );

(4.56)

if I = [0, ∞), then set X = `∞ ([0, ∞), Rd ),

Y = `∞ ([1, ∞), Rd );

(4.57)

and, finally, if I = Z set X = Y = `∞ (Z, Rd ).

(4.58)

In each case Y ⊆ X, but only in the cases I = (−∞, 0] and I = Z do we have Y = X. Note that the domain of A is strictly contained in X in (4.57) and (4.58), although its range is all of Y . Consider also the linear operator D : X → Y defined by (Dx)n+1 = xn+1 − An xn ,

n ∈ I,

(4.59)

that is, D = I − A .

4.3.3 An Inversion Theorem Consider the invertibility problem for the operator D. It turns out that this problem depends heavily on which set of indices, I = [0, N ] or I = [0, ∞) or I = (−∞, 0] or I = Z, the sequence of matrices An is defined. For example, to find the inverse operator of D in the case I = [0, N ] we need to find for each z = {zn } ∈ `∞ ([1, N +1], Rd ) an x = {xn } ∈ `∞ ([0, N +1], Rd ) which satisfies xn+1 = An xn + zn+1 , n ∈ I. (4.60) Clearly, the set of equations (4.60) is not sufficient to define x uniquely when I = [0, N ], as x0 can be chosen arbitrarily here. A similar kind of problem, not quite so obviously stated, also occurs when we try to find a bounded sequence x satisfying (4.60) in the case I = [0, ∞) or I = (−∞, 0]. Let Rd = Ens ⊕ Enu be the splitting associated with the matrix An , for which the projectors are Pns : Rd → Ens and Pnu = I − Pns : Rd → Enu . Rewrite (4.60) in the decomposed form s s s s Pn+1 xn+1 = Pn+1 An Pns xn + Pn+1 An Pnu xn + Pn+1 zn+1 , u u u u Pn+1 xn+1 = Pn+1 An Pns xn + Pn+1 An Pnu xn + Pn+1 zn+1 . u u Note that the linear operator Un = Pn+1 An Pnu : Enu → En+1 is invertible by (4.11) and by the last condition (4.13) so we can rewrite these equations as


59

s s Pn+1 xn+1 = Pn+1 (An Pns xn + An Pnu xn + zn+1 ) ,

Pnu xn

=

u Un−1 Pn+1

(xn+1 −

An Pns xn

− zn+1 ) .

(4.61) (4.62)

Now introduce the linear operator Hz : X → X, which, for a fixed sequence z = {zn } ∈ Y , transforms every sequence x = {xn } ∈ X into a sequence w = {wn } ∈ X satisfying s s Pn+1 wn+1 = Pn+1 (An Pns xn + An Pnu xn + zn+1 ) ,

Pnu wn

=

u Un−1 Pn+1

(xn+1 −

An Pns xn

− zn+1 ) ,

(4.63) (4.64)

where both equations hold for n ∈ I. Observe that (4.63) does not define P0s w0 in the case when 0 is the left end of the interval I, while (4.64) fails to define PNu +1 wN +1 when N is the right end of the interval I. Hence, the operator Hz is well-defined only if equations (4.63) and (4.64) are supplemented by boundary conditions, which may be chosen to be one of the following: P0s w0 = 0,

PNu +1 wN +1 = 0

if I = [0, N ],

(4.65)

or P1u w1 = 0

if I = (−∞, 0],

(4.66)

or P0s w0 = 0,

if

I = [0, ∞).

(4.67)

The system of equations (4.63)–(4.64) is sufficient to well-define Hz without boundary conditions only in the case when I = Z. Now introduce the norm k · k◦∞ on the space X by kxk◦∞ = sup max {γkPns xn k, kPnu xn k} ,

(4.68)

n

where γ = γ(s) as in (4.5) and the indices n are taken from [0, N + 1] if I = [0, N ], from (−∞, 1] if I = (−∞, 0], and from I if I = [0, ∞) or I = Z. As mentioned in Sect. 4.1, the split s under consideration may be treated without loss of generality as positive, in which case the norm k · k◦∞ in X is equivalent to the norm k · k∞ of (4.12) for any admissible n. To see this, note that for any xn ∈ Rd we have 1 kxn k ≤ max {kPns xn k, kPnu xn k} ≤ hkxn k, 2 so by Lemma 4.1 1 min{γ, 1}kxn k ≤ max {γkPns xn k, kPnu xn k} ≤ h max{γ, 1}kxn k, 2 from which it follows that

60


1 min{γ, 1}kxk∞ ≤ kxk◦∞ ≤ h max{γ, 1}kxk∞ . 2

(4.69)

Lemma 4.30. For each z ∈ Y the operator Hz : X → X has a unique fixed point x := Bz ∈ X for which kxk∞ ≤

h kzk∞ . ν(s)

(4.70)

Proof. We show first that the operator Hz : X → X is contracting in the norm k · k◦∞ with contraction constant σ = σ(s) as in (4.3). ˜ ∈ X, write y = x − x ˜ and v = Hz (x) − Hz (˜ Given x, x x). Then by (4.63), (4.64) s s Pn+1 vn+1 = Pn+1 (An Pns yn + An Pnu yn ) , u Pnu vn = Un−1 Pn+1 (yn+1 − An Pns yn ) ,

to which may be added, if needed, one or the both following boundary conditions P0s v0 = 0, PNu +1 vN +1 = 0, by (4.68) if I = (−∞, 0], or (4.69) if I = [0, ∞). Hence, by definition of the operator Un and by (4.13) obtain s kPn+1 vn+1 k ≤ λs kPns yn k + µs kPnu yn k, 1 µu s kP yn k + kP u yn+1 k, kPnu vn k ≤ λu n λu n+1

(4.71) (4.72)

again to be supplemented, if necessary, by kP0s v0 k = 0,

kPNu +1 vN +1 k = 0.

(4.73)

Write V1 = sup {γkPns vn k} ,

V2 = sup {kPnu vn k}

{γkPns yn k} ,

Y2 = sup {kPnu yn k}

n

Y1 = sup n

n

n

Then kvk◦∞ = max {V1 , V2 } ,

kyk◦∞ = max {Y1 , Y2 }

and so, from (4.71), (4.72), and from (4.73), if necessary, obtain V1 ≤ λs Y1 + γµs Y2 ≤ (λs + γµs ) kyk◦∞ , µu 1 µu 1 V2 ≤ Y1 + V2 ≤ + kyk◦∞ . γλu λu γλu λu But from (4.6)

(4.74) (4.75)


61

λs + γµs = σ,

µu 1 + = σ, γλu λu

so, by (4.74) and (4.75) obtain kvk◦∞ = max {V1 , V2 } ≤ σkyk◦∞ , that is

˜ k◦∞ . kHz (x) − Hz (˜ x)k◦∞ ≤ σkx − x

By the Banach Contraction Mapping Theorem the operator Hz for any z ∈ Y has a unique fixed point x := Bz ∈ X for which kBzk◦∞ ≤

1 kHz (0)k◦∞ . 1−σ

Unfortunately, the obvious bound on kxk∞ provided by Lemma 4.1 is only a rough bound, so another approach is required to give the tighter inequality (4.72). Since by (4.69) the norms k · k◦∞ and k · k∞ are equivalent then, from the Contraction Mapping Theorem, the sequence of successive iterates x(m) = Hz (x(m−1) ),

m = 1, 2, . . . ,

with x(0) = 0, converges in the norm k · k∞ to the fixed point x = Bz of Hz . In particular, kxk∞ ≤ lim sup kx(m) k∞ . (4.76) m→∞

Set am = sup {kPns x(m) n k}, n

bm = sup {kPnu x(m) n k}

(4.77)

n

and s hs = sup {kPn+1 zn+1 k}, n

u hu = sup {kUn−1 Pn+1 zn+1 k}.

(4.78)

n

Then, by taking the supremum of the norms of the left and right hand sides of equations (4.63), (4.64) and, if necessary, of one or both boundary conditions of (4.65), obtain the system of inequalities am ≤ λs am−1 + µs bm−1 + hs , µu 1 am−1 + bm−1 + hu , bm ≤ λu λu from which, by Lemma 4.2, it follows that lim sup kx(m) k∞ ≤ lim sup(am + bm ) ≤ m→∞

m→∞

max{hs , λu hu } . ν(s)

(4.79)

62


But clearly, by (4.12) and the definition of the operator Un , hs ≤ hkzk∞ ,

hu ≤

h kzk∞ . λu

(4.80)

The required bound (4.70) follows from this and from (4.76) and (4.79). The lemma is proved. t u Now consider the invertibility of the bounded linear operator D that was defined by (4.59) for a semi-hyperbolic sequence of matrices {An }. Theorem 4.31. If {An } is a (s, h)-semi-hyperbolic sequence of d × d matrices, then the linear operator D : X → Y defined by (4.59), with spaces X, Y as in any of (4.55), (4.56), (4.57) or (4.58), is bounded and has a bounded right inverse D −1 : Y → X satisfying

−1

D ≤ ∞

h . ν(s)

(4.81)

Proof. By Lemma 4.30, for each z ∈ Y the operator Hz has a unique fixed point x = Bz for which kBzk∞ ≤

h kzk∞ ν(s)

(4.82)

holds. The operator B is obviously linear. By comparing equations (4.63) and (4.64) with (4.61) and (4.62), conclude that x is the solution of (4.60). Thus, the operator B is the right inverse to D, that is B = D −1 , and the inequality (4.81) follows from (4.82). t u Remark 4.32. When the sequence of matrices An is defined for the set of indices I = Z the bounded right inverse to D operator D −1 satisfying (4.81) is defined uniquely, while in the cases I = [0, N ], I = [0, ∞) and I = (−∞, 0] it is not. Remark 4.33. In Definition 4.5, and hence in Theorem 4.31, the matrix sequence {An } is supposed to be bounded. This requirement has been imposed only to simplify proofs and can be easily removed, with obvious changes in the formulation of assertions and proofs. With Definition 4.20 of semi-hyperbolic sequences of matrices an analogue of Theorem 4.31 is valid with some insignificant changes resulting from using the variable norm inequalities (4.40). Theorem 4.34. If {An } is a (s, h)-semi-hyperbolic sequence of d×d matrices as in Definition 4.20), then the linear operator D : X → Y defined by (4.59), with spaces X, Y defined by any of (4.55), (4.56), (4.57) or (4.58), is bounded and has a bounded right inverse D −1 : Y → X satisfying


63 2

−1

D ≤ q h , ∞ ν(s)

where q is as in the inequalities (4.37) of Definition 4.20. Proof. In the proof of Theorem 4.31, we adjust definition (4.68) of the norm k · k◦∞ and definitions (4.77), (4.78) of an , bn , hs , hu by replacing the norms kPns xn k, kPnu xn k by the norms kPns xn kn , kPnu xn kn in the following way kxk◦∞ = sup max {γkPns xn kn , kPnu xn kn } . n

With such an adjustment the proof of Theorem 4.34 is essentially the same as that of Theorem 4.31. It needs only to multiply by the factor q the corresponding sides of inequalities (4.79) and (4.80) involving estimation of the k · k∞ -norms of elements using an , bn , h, and conversely. t u

4.3.4 Hyperbolicity of the Linear Operator A To complete Sect. 4.3 consider the relationship between the hyperbolicity of a matrix sequence and hyperbolicity of the linear operator A defined by (4.54) in Sect. 4.3.2. Then, using Theorem 4.23, reveals the relationship between the semi-hyperbolicity of a matrix sequence and hyperbolicity of the linear operator A and, by Theorem 4.29, the relationship between the semi-hyperbolicity of a matrix sequence and semi-hyperbolicity of the linear operator A. Theorem 4.35. The sequence of invertible matrices {An }, n ∈ Z, is hyperbolic if and only if the linear operator A : `∞ (Z, Rd ) → `∞ (Z, Rd ) defined by (4.54) is hyperbolic. Proof. To simplify notation, denote L = `∞ (Z, Rd ). Suppose that the sequence of invertible matrices {An }, n ∈ Z, is hyperbolic. For each integer n, let the projections Pns , Pnu and norm k · kn be as in Definition 4.22 and define linear operators P s and P u on L by (P s x)n = Pns xn ,

(P u x)n = Pnu xn ,

n ∈ Z.

Clearly, the operators P s and P u are bounded projections and, moreover, P s + P u = I. Hence the subspaces E s = P sL ⊆ L ,

E u = P uL ⊆ L

(4.83)

have trivial intersection and their direct sum coincides with L , E s ∩ E u = {0},

Es ⊕Eu = L.

(4.84)

64


In addition, it follows from the equivariance relations (4.41) and the definition of projections P s and P u that the subspaces E s and E u are invariant under the linear operator A , A E s ⊆ E s,

A E u ⊆ E u.

Introduce the norm kxk∗∞ = sup {kPns xn kn , kPnu xn kn } n∈Z

on the space L . Using (4.37) and (4.39) obtain the chain of inequalities 1 1 1 kxn k ≤ kxn kn ≤ (kPns xn kn + kPnu xn kn ) 2q 2 2 ≤ max {kPns xn kn , kPnu xn kn } ≤ hkxn kn ≤ hqkxn k for any integer n, from which it follows that 1 kxk∞ ≤ kxk∗∞ ≤ hqkxk∞ , 2q and the norms k · k∞ and k · k∗∞ in the space L are equivalent. From the inequalities (4.42), which hold for the matrix sequence {An } because of hyperbolicity, it follows immediately that kA P s xk∗∞ ≤ λs kP s xk∗∞ ,

kA P u xk∗∞ ≥ λu kP u xk∗∞ .

Together with (4.83), these inequalities imply that the spectrum σs , of the restriction A |E s of the linear operator A to its invariant subspace E s , lies entirely in the disc |λ| ≤ λs < 1. Similarly, the spectrum σu , of the restriction A |E u of the linear operator A to its invariant subspace E u , lies entirely outside the disc |λ| ≤ λu . Now, from (4.83) and (4.84) it follows that the spectrum of the operator A coincides with the union of the sets σs and σu , and thus does not intersect the disc |λ| = 1. Hence, the linear operator A is hyperbolic. Now consider the converse, namely that hyperbolicity of the operator A implies hyperbolicity of the matrix sequence {An }. By supposition, the spectrum σ of the operator A does not intersect the unit disc |λ| = 1. Since the spectrum of any operator is a closed set, the set σ can then be represented as union σ = σs ∪ σu of closed subsets, σs which lies entirely inside the unit disk, and σu which lies entirely outside the unit disc. Denote by E s ⊆ L the invariant spectral subspace of the operator A corresponding to the subset σs of its spectrum, and denote by E u ⊆ L the invariant spectral subspace of the operator A corresponding to σu . Then the relations (4.84) hold. Denote the projections corresponding to the splitting


65

(4.84) by P s : L → E s and P u = I − P s : L → E u , which are bounded in this case. The spectral radius σ(A) of a linear operator A on a Banach space satisfies σ(A) = lim sup kAn k1/n , n→∞

from which follows the existence for each ε > 0 of a number cε such that kAn k ≤ cε (σ(A) + ε)n ,

n ≥ 0.

(4.85)

By definition, the closed set σs lies in the disc |λ| < 1, so numbers λs ∈ (0, 1) and ε > 0 can be found such that σs will, in fact, belong to the disc |λ| ≤ λs − ε. Applying (4.85) to the restriction A |E s , of the linear operator A to its invariant subspace E s , and taking k · k∞ as the operator norm k · k in (4.85), it follows that k(A |E s )n k∞ ≤ cε λns ,

n ≥ 0,

for an appropriate constant cε , from which kA n P s xk∞ ≤ cε λns kxk∞ ,

x ∈ L , n ≥ 0.

(4.86)

Similarly, numbers λu > 1 and ε > 0 can be found such that the closed set σu belongs to the exterior of the disc |λ| ≤ λu + ε. Then the restriction A |E u , of the linear operator A to its invariant subspace E u , is invertible and, applying the formula (4.85) to (A |E u )−1 , obtain k(A |E u )−n k∞ ≤ cε λ−n u ,

n ≥ 0,

for an appropriate constant cε , from which it follows that n kA n P u xk∞ ≥ c−1 ε λu kxk∞ ,

x ∈ L , n ≥ 0.

(4.87)

Define the norm on L by n s −n n u kxk∗ = sup λ−n s kA P xk∞ + inf λu kA P xk∞ . n≥0

n≥0

(4.88)

From (4.86) and (4.87) the norms k · k∞ and k · k∗ are equivalent, so there exists a number Q such that Q−1 kxk∞ ≤ kxk∗ ≤ Qkxk∞

(4.89)

for all x ∈ L . From (4.88) it thus follows that kA P s xk∗ ≤ λs kP s xk∗ ,

kA P u xk∗ ≥ λu kP u xk∗ .

(4.90)

66


Now fix an integer k ∈ Z and define Eks ⊆ Rd to be the set of those v ∈ Rd for which the equation vn+1 = An vn ,

n ≥ k,

(4.91)

has a bounded solution satisfying vk = v. Similarly, define Eku ⊆ Rd to be the set of those w ∈ Rd for which the equation wn+1 = An wn ,

n < k,

(4.92)

has a bounded solution satisfying wk = w. It is not hard to see that Eks and Eku are subspaces of Rd . It remains to show that these form a splitting of Rd , that is with Eks ∩ Eku = {0}, Eks ⊕ Eku = Rd . (4.93) Let x ∈ Eks ∩ Eku . By definition of the subspace Eks there exists a bounded sequence {vn }, n ≥ k, satisfying (4.91) and vk = x. Similarly, by definition of the subspace Eku there exists a bounded sequence {wn }, n ≤ k, satisfying (4.92) and wk = x. Then the sequence x = {xn } defined by ( vn , n ≥ k, xn = wn , n ≤ k will satisfy the equation xn+1 = An xn , and hence

n ∈ Z,

x = A x.

In view of hyperbolicity of the operator A , this implies that x = 0. The first part of (4.93) is thus proved. To prove the remainder of (4.93) choose an arbitrary but fixed x ∈ Rd and for any fixed k > 0 define the sequence x = {xn } ∈ L by xk = x, and let

xn = 0

xs = P s x,

for n 6= k,

xu = P u x.

(4.94) (4.95)

Then x = xs + xu and x = xsk + xuk . It suffices to show that xsk ∈ Eks ,

xuk ∈ Eku .

Write v (k) = xs and define elements v (n) ∈ L recursively by v (n+1) = A v (n) ,

n ≥ k.

(4.96)


67

From (4.90) obtain kv (k) k∗ ≥ kv (k+1) k∗ ≥ . . . ≥ kv (n) k∗ ≥ . . . ,

n ≥ k,

and the sequence with components v (n) ∈ L , n ≥ k, is bounded in the norm k · k∗ and thus, by (4.89), also in the norm k · k∞ . Hence the sequence with elements vn ∈ Rd defined by vn = vn(n) ,

n ≥ k,

is also bounded. Clearly, the sequence {vn } satisfies (4.91) and vk = xsk . Hence, in view of the definition of the subspace Eks , the first inclusion of (4.96) is proved. The second inclusion of (4.96) is proved similarly. The second part of (4.93) is now proved. Define now operators Pks and Pku by Pks x = xsk ,

Pku x = xuk .

By definition of the points xsk and xuk , these operators are linear and satisfy Pks Rd ⊆ Eks ,

Pku Rd ⊆ Eku ,

Pks + Pku = I.

From these and (4.93) it follows that the operators Pks and Pku are projections. To estimate the norms kPks k and kPku k we use the chain of inequalities kPks xk = kxsk k ≤ kxs k∞ = kP s xk∞ ≤ kP s k∞ kxk∞ = kP s k∞ kxk from which it follows that kPks k ≤ kP s k∞ ≤ max{kP s k∞ , kP u k∞ }. Similarly, kPku k ≤ kP u k∞ ≤ max{kP s k∞ , kP u k∞ }, so a uniform norm bound exists for the projections Pks and Pku with k ∈ Z (cf. (4.39)). The equivariant identities (4.41) for the matrix sequence {An } then follows immediately from the definition of subspaces Ens and Enu and from invertibility of the matrices An . It remains only to construct norms k · kn which ensure that the inequalities (4.42) are valid. To do this, more detailed information about properties of the projections P s and P u is required. Given a point x ∈ Rd construct points x, xs , xu ∈ L by formulae (4.94) and (4.95). It suffices to show that x = xs

if x ∈ Eks ,

(4.97)

x = xu

if x ∈ Eku .

(4.98)

68


For definiteness, let x ∈ Eks . Then, in the same way as the inclusions (4.96) were obtained, xsn ∈ Ens , xun ∈ Enu , n ∈ Z. (4.99) But since x = xs + xu , from the definition (4.94) of the point x, it follows immediately that xsn = −xun , n 6= k, which, in view of (4.99) and (4.93), can only be satisfied if xsn = xun = 0,

n 6= k.

The required relations (4.97) and (4.98) follow from this. Now define the norm k · kk on Rd by kxkk = kxk∗ ,

x ∈ Rd ,

where x ∈ L is the element defined by (4.94). From (4.89) and from the obvious equality kxk = kxk∞ obtain Q−1 kxk = Q−1 kxk∞ ≤ kxkk = kxk∗ ≤ Qkxk∞ = Qkxk, and the uniform boundedness conditions (cf. (4.37)) are satisfied by the norms k · kk , k ∈ Z. s Now, given x ∈ Eks , denote y = Ak x. Then, as previously shown, y ∈ Ek+1 . Define the element x ∈ L by (4.94) and the element y ∈ L in the same manner by yk+1 = y, yn = 0 for n 6= k + 1. s Then clearly y = A x, and from the inclusions x ∈ Eks , y ∈ Ek+1 and (4.97) it follows that x ∈ E s, y = A x ∈ E s,

from which, by (4.90), kAk xkk+1 = kA xk∗ ≤ λs kxk∗ = λs kxkk ,

x ∈ Eks .

The inequality kAk xkk+1 = kA xk∗ ≥ λu kxk∗ = λu kxkk ,

x ∈ Eku

can be proved in similar fashion. The proof of the theorem is now complete. t u

Chapter 5

Semi-Hyperbolicity and Hyperbolicity

In Chapter 4 it was shown that a semi-hyperbolic sequence of matrices is hyperbolic, that is semi-hyperbolicity implies hyperbolicity in the linear case. This is generally not true for nonlinear mappings, which is shown below by an example in Sect. 5.1, which demonstrates that a semi-hyperbolic mapping may not possess an invariant splitting and so cannot be a hyperbolic. Nevertheless, it is shown in Sect. 5.2 that a semi-hyperbolic mapping which is smooth and invertible in a neighbourhood of a compact invariant set is hyperbolic on that set. The proofs depend substantially on background material and results presented in Chap. 4.

5.1 Perturbation of Anosov Endomorphisms In this section an example is constructed which demonstrates that generally a semi-hyperbolic mapping may not possess an invariant splitting, and thus cannot be hyperbolic. Briefly recall some properties of Anosov mappings from Sect. 3.2.2. Write the elements of Rd as vectors x with coordinates x1 , x2 , . . ., xd and let Td be the standard d-dimensional torus. This torus Td is a compact differentiable manifold in Rd with respect to the locally Euclidean metric q ρ(x, y) = |x1 − y1 |2mod 1 + |x2 − y2 |2mod 1 + · · · + |xd − yd |2mod 1 on Td where |t − s| mod 1 = min {|t − s + 2k| : k = 0, ±1} ,

0 ≤ t, s < 1.

The tangent space Tx Td can then be identified with Rd by an appropriate choice of natural coordinates generated by those of Rd , so Tx Td = Rd for each x ∈ Td . Denote the natural projection from Rd onto Td by

69

70

5 Semi-Hyperbolicity and Hyperbolicity

Π(x) = (x1 mod 1, x2 mod 1, . . . , xd mod 1),

x = (x1 , x2 , . . . , xd ).

Let A be a given d × d matrix with integer components aij and define the mapping f : Td → Td defined by f (x) = Π(Ax). Theorem 5.1. Let d ≥ 3. There exists a hyperbolic d × d matrix A with integer components such that for every ε > 0 there can be found a differentiable semi-hyperbolic mapping fε : Td → Td which is ε-close in the C 1 -norm to the mapping f (x) = Π(Ax), but for which there is no continuous hyperbolic splitting. Proof. First, let d = 3. Consider the 3 × 3 matrix   230 A = 1 2 0 002 √ √ which has eigenvalues λ1 = 2 −√ 3, λ2 = 2 + 3 and √ λ3 = 2 with corresponding eigenvectors v1 = (1, −1/ 3, 0), v2 = (1, 1/ 3, 0) and v3 = (0, 0, 1), respectively. Let B be a fixed 3 × 3 matrix   b11 b12 0 B = b21 b22 0 0 0 0 such that v2 is not an eigenvector, but otherwise arbitrary. Consider the sequence of points x(0) = (0, 0, 0),

x(n) = (0, 0, 2n )

for

n = −1, −2, . . .

and for ε ≥ 0 denote by fε : T3 → T3 a continuously differentiable mapping satisfying (i) fε is ε-close to f in the C 1 -norm; (ii) fε (x(−1) ) = f (x(−1) ) = 0 and (T fε )x(−1) = A + εB, where T is the tangent mapping; (iii) fε (x) ≡ f (x) for all x such that kx − x(−1) k ≥ 18 . Such a mapping fε exists for sufficiently small ε > 0. It satisfies x(n) = fε (x(n−1) ),

n = 0, −1, −2, . . .

(5.1)

and is semi-hyperbolic by Lemma 3.18. Since the mapping f = f0 is an Anosov endomorphism, see Example 3.17, it has a hyperbolic splitting at every point x independent of x, and can thus be written as

5.1 Perturbation of Anosov Endomorphisms

71

Tx T3 = E s ⊕ E u ,

x ∈ T3 ,

(5.2)

where E s is the eigenspace of the matrix A corresponding to the eigenvalue λ1 and E u is the eigenspace of the matrix A corresponding to the eigenvalues λ2 and λ3 . Proof by contradiction is used to show that the mapping fε , for sufficiently small ε > 0, does not have a hyperbolic splitting. Fix ε > 0 sufficiently small and suppose that s u Tx T3 = Ex,ε ⊕ Ex,ε ,

x ∈ T3 ,

(5.3)

is a hyperbolic splitting for fε . By Property (iii) fε (x) ≡ f (x) in a neighbourhood of the point x = 0, so x = 0 is a fixed point of fε and by the Hartman–Grobman Theorem [88] the splitting (5.3) at the point x = 0 coincides with the splitting (5.2), that is s E0,ε ≡ Es,

u E0,ε ≡ Eu.

In fact, to these equivalences can be added Exu(n) ,ε ≡ E u ,

n = −1, −2, . . . .

Indeed, by Properties (ii) and (iii) (T fε )x(n) = A for n = −2, −3, . . . ,

(5.4)

u are equivariant for T fε , then and since by supposition the subspaces Ex,ε from (5.4) and (5.1) it follows that

Exu(n) ,ε = (T fεn−k )x(k) Exu(k) ,ε = (T fε )x(n−1) (T fε )x(n−2) . . . (T fε )x(k) Exu(k) ,ε = An−k Exu(k) ,ε

(5.5)

for all k ≤ n. But, by the continuity of the splitting (5.3), the subspace Exu(k) ,ε is close to E u for k → −∞ and thus does not contain vectors from E s . Hence, taking the limit as k → −∞ in the right hand side of (5.5), we obtain Exu(n) ,ε = lim An−k Exu(k) ,ε = E u . k→−∞

It remains to observe that (T fε )x(−1) Exu(−1) ,ε = (T fε )x(−1) E u = (A + εB)E u u 6= E u = E0,ε = Exu(0) ,ε = Efuε (x(−1) ),ε

since (T fε )x(−1) = A + εB, and this contradicts the assumed equivariance of the splitting (5.3) with respect to T fε . This contradiction completes the proof in the case when d = 3.

72


To prove the theorem in the case when d > d × d block diagonal matrices A0 B ˆ ˆ A= , B= 0 0 0

3 it suffices to consider the 0 0

ˆ and repeat the argument above for the mapping f˜(x) = Π(Ax).

t u

5.2 Converse Case Any hyperbolic mapping is clearly semi-hyperbolic. Theorem 5.2 below shows that the converse is also valid for invertible smooth mappings, so establishing equivalence between semi-hyperbolicity and hyperbolicity for invertible mappings. The proof is valid for each of the various definitions of semi-hyperbolicity considered earlier, namely in the variable norm sense of Definitions 3.4, 4.5, or in the single norm sense of Definitions 3.5 and 4.20. Theorem 5.2. Let f : X → Rd be a continuously differentiable mapping which is semi-hyperbolic either in the variable norm Definition 3.4 or the single norm Definition 3.5, on a bounded open set X ⊆ Rd . Further, suppose that f is invertible in a neighbourhood of some compact set K ⊂ X and that K is an invariant set for both f and f −1 , that is f (K) = f −1 (K) = K. Then f is hyperbolic on K. Proof. First we consider an equivariant splitting of Rd for the mapping f on K. For a point x ∈ K define the sequence of linear operators An,x = (Df n+1 )x = (Df )f n (x) .

(5.6)

From the semi-hyperbolicity of f , for any x ∈ K the sequence of linear operators {An,x } will be also semi-hyperbolic in the same sense. Then by Theorem 4.18, for the point x and for any integer n there exists a splitting u s ˆn,x ˆn,x satisfying and E of the space RN into the direct sum of subspaces E s s ˆn,x ˆn+1,x An,x E =E ,

that is,

s s ˆn,x ˆn+1,x (Df n+1 )x E =E ,

u u ˆn,x ˆn+1,x An,x E =E ,

u u ˆn,x ˆn+1,x (Df n+1 )x E =E .

(5.7)

ˆ s consists of the set of v ∈ Rd By Lemma 4.16 and by (5.6) the subspace E n,x for which the equation vk+1 = (Df k+1 )x vk ,

k ≥ n,

u ˆn,x has a bounded solution satisfying vn = v, and the subspace E is the set of d those w ∈ R for which the equation

5.2 Converse Case

73

wk+1 = (Df k+1 )x wk ,

k < n,

has a bounded solution satisfying wn = w. From these properties of the s u ˆn,x ˆn,x subspaces E , E and from invertibility of the linear operators Dfx at any point x ∈ K it follows that s ˆn,x ˆs n , E =E 0,f (x)

u ˆn,x û n , E =E 0,f (x)

n ∈ Z.

ˆ s , Exu = E ˆ u obtain a splitting Rd = Exs ⊕ Exu , So, denoting Exs = E 0,x 0,x x ∈ K, which by (5.7) satisfies the equivariance property Dfx Exs = Efs(x) ,

Dfx Exu = Efu(x) ,

x ∈ K.

Thus f satisfies Condition A1 of Definition 2.1. Now from Theorem 4.18, in particular the inequalities (4.32) and (4.33), it follows that the mapping f satisfies Condition A2 of Definition 2.1, and from the inequality (4.27) of Lemma 4.16 Condition A3 of Definition 2.1 also holds for f . The mapping f thus satisfies all the conditions of an Anosov system in Definition 2.1 except that f is defined not on a closed Riemann manifold, but on a neighbourhood of a compact invariant set. This distinction formally excludes the use of Theorem 2.2 for proving continuous dependence of the splitting Rd = Exs ⊕ Exu on x ∈ K. But a closer inspection reveals that f being defined on a closed Riemann manifold is not essential to the proof of Theorem 2.2. The statement of that theorem remains valid under the supposition that f is defined on a compact invariant set. Hence, by this reasoning, the splitting Rd = Exs ⊕ Exu depends continuously on x ∈ K, and thus, by Definition 2.5, the mapping f is hyperbolic on K. t u

5.2.1 Alternative Proof The investigation of the relationship between semi-hyperbolicity and hyperbolicity in Chap. 4 and in Sect. 5.2 was intentionally done in a more traditional analytical manner, via an explicit description of splitting subspaces, in order that their properties could be determined more fully. Modern techniques based on differential geometry and smooth dynamics are also applicable here and provide an alternative proof of Theorem 5.2, based on the Mather Projection Lemma, by adapting the proof of Hirsch and Pugh [60]. See that reference and similar works for an explanation of the terms used below. Let E be a vector bundle over a compact set K ⊂ Rd and let f : K → K be a C 1 diffeomorphism on a neighbourhood of K. Denote by C(E) the Banach space of bounded continuous sections of E over K with the sup-norm topology. Let f˜ : C(E) → C(E) be the continuous linear mapping

74


f˜(ϕ) = Df ◦ ϕ ◦ f −1 .

(5.8)

Lemma 5.3 (Mather Projection Lemma). If f˜ − I is a hyperbolic isomorphism, then f is hyperbolic on K. Now let C(TK Rd ) = C be the Banach space of bounded continuous sections of TK Rd and let f˜ : C → C be the map (5.8) induced by f . Write C = C (E1 |K ) × C (E2 |K ). Then, making use of Theorem 4.29, it can be shown that the conditions of Theorem 5.2 are satisfied by f˜, so f˜ is a hyperbolic linear operator. Hence, by the Mather Projection Lemma 5.3, f is hyperbolic on the set K.

Chapter 6

Expansivity and Shadowing

Section 2.2 demonstrated that hyperbolic systems possess some rather strong and striking properties such as expansivity, see Definition 2.8 and Theorem 2.9, and shadowing, see Theorem 2.10. This chapter shows that these and other properties remain valid for semi-hyperbolic systems. Moreover, explicit values or sharp estimates of relevant parameters and intervals of validity will be obtained.

6.1 Expansivity In this section, semi-hyperbolic dynamical systems are shown to be exponentially expansive, at least locally, and explicit rates of expansion are determined. Differentiability is not required, so the proof is given only for Lipschitz mappings.

6.1.1 Definitions Let (X, ρ) be a metric space. Definition 6.1. A trajectory of a dynamical system generated by a mapping f : X → X on a state space X is a sequence x = {xn } ⊂ X satisfying xn+1 = f (xn ) for all n belonging to some interval I ⊆ Z. The trajectory is said to be finite when I is of finite length and infinite otherwise. Recall from Definition 2.8 that the dynamical system generated by a homeomorphism f : X → X is said to be ξ-expansive on X if the inequalities

75

76

6 Expansivity and Shadowing

ρ(f n (x0 ), f n (y0 )) ≤ ξ

for n = 0, ±1, ±2, . . .

imply that x0 = y0 , that is any two bi-infinite trajectories {xn } = {f n (x0 )} and {yn } = {f n (y0 )} which always remain within a threshold ξ of each other are identical. The more interesting cases occur when the metric space (X, ρ) is compact since then with expansivity complicated dynamical behaviour arises. An important characteristic of ξ-expansivity is the rate of divergence of different trajectories. Let Tr±k (f, X) denote the set of all trajectories x = {x−k , . . . , x0 , . . . , xk } ˜∈ of the mapping f that are contained entirely in a set X, and for any x, x Tr±k (f, K) define ˜ ) = max kxn − x ρk (x, x ñ k. −k≤n≤k

Lemma 6.2. Let K be a compact subset of Rd and let f : K → K be a continuous ξ-expansive mapping. Then for every 0 < ε, θ < ξ there exists a positive integer κ(ε, θ) such that ˜) > θ ρk (x, x ˜ ∈ Tr±k (f, K) with kx0 − x holds for all x, x ˜0 k ≥ ε and k ≥ κ(ε, θ). Proof. Suppose the contrary. Then for any positive integer k there exist tra˜ (k) ∈ Tr±k (f, K) satisfying jectories x(k) , x (k)

(k)

˜ (k) ) ≤ θ < ξ. kx0 − x ˜0 k ≥ ε and ρk (x(k) , x By compactness of the set K, the sequences {x(k) } and {˜ x(k) } can be assumed ∗ ∗ ˜ ∈ Tr±∞ (f, K). Then to converge componentwise to trajectories x and x kx∗0 − x ˜∗0 k ≥ ε, but at the same time kx∗n − x ˜∗n k ≤ θ < ξ,

n = 0, ±1, ±2, . . . ,

which contradicts the ξ-expansivity of f .

t u

In expansive systems distinct trajectories often separate exponentially, at least locally, as is seen from the next Example. Example 6.3. Let (Σ, ρ) be the compact metric space of bi-infinite binary +∞ sequences x = {xi }i=−∞ , xi ∈ {0, 1}, endowed with the metric ρ (x, y) =

+∞ X i=−∞

2−|i| |xi − yi | ,

6.1 Expansivity

77

and consider the shift mapping σ : Σ → Σ on Σ defined by (σx)i = xi+1 ,

i ∈ Z.

The inequality ρ (σ n x, σ n y) ≤ ξ < 1

(6.1)

can be rewritten as |xn − yn | ≤ ξ < 1, which for binary sequences x and y can only be valid if xn = yn . (6.2) By (6.2), (6.1) holds for any n and so the sequences x and y coincide. Hence the shift mapping σ is ξ-expansive for any ξ < 1. Indeed, the shift mapping σ possesses a stronger property than ξ-expansivity, namely, the inequalities max ρ (σ n x, σ n y) , ρ σ −n x, σ −n y ≥ 2n−1 ρ(x, y) (6.3) are valid for n = −n− , . . . , 0, . . . , n+ whenever ρ (x, y) ≤ ξ < 1, where n− and n+ are the largest possible integers such that ρ (σ n x, σ n y) ≤ ξ < 1

for

n = −n− , . . . , 0, . . . , n+ .

(6.4)

To see this, suppose that inequalities (6.4) hold with maximal n− and n+ . Then by (6.1) and (6.2) xn = yn ,

n = −n− , . . . , 0, . . . , n+ ,

so ρ (x, y) can be represented as X X ρ (x, y) = 2i |xi − yi | + 2−i |xi − yi | , in+

where at least one of summands on the right hand side must be greater than or equal to 21 ρ (x, y). For definiteness, suppose that X i>n+

2−i |xi − yi | ≥

1 ρ (x, y) . 2

Similarly for any n = −n− , . . . , 0, . . . , n+ , obtain

(6.5)

78


ρ (σ n x, σ n y) =

X

2−|i| |xi+n − yi+n | =

i∈Z

=

2−|i−n| |xi − yi |

i∈Z

X

−|i−n|

2

|xi − yi | +

in+

X

i−n

2

|xi − yi | +

in+

≥ 2n

X

2−i |xi − yi | .

i>n+

Then by (6.5)

1 n 2 ρ (x, y) , 2 and the inequalities (6.3) for n = −n− , . . . , 0, . . . , n+ are proved. ρ (σ n x, σ n y) ≥

Example 6.3 is of fundamental interest because the shift mapping σ is conjugate to the diffeomorphism of the Horseshoe which is one of the best known examples of a chaotic hyperbolic system. The example motivates the following definition of exponential expansivity. Let Oε (S) denote the open ε-neighbourhood of a nonempty subset S ⊂ Rd . Definition 6.4. Let X ⊆ Rd be an open bounded set and let K be a compact subset of X. A continuous mapping f : X → Rd is said to be exponentially expansive on K with exponent r > 1 if there exist constants ξ and c > 0 such that Oξ (K) ⊆ X and for any (finite) trajectories x = x−n− , . . . , x0 , . . . , xn+ , y = y−n− , . . . , y0 , . . . , yn+ satisfying x ⊆ K and kyi − xi k ≤ ξ for n = −n− , . . . , 0, . . . , n+ at least one of the following groups of inequalities holds: kxn − yn k ≥ crn kx0 − y0 k,

n = 1, 2, . . . , n+ ,

(6.6)

or kxn − yn k ≥ cr−n kx0 − y0 k,

n = −1, −2, . . . , −n− .

(6.7)

The exponential expansivity of diffeomorphisms and homeomorphisms on a compact hyperbolic set is well known. It also holds under the weaker assumptions of semi-hyperbolicity, which is proved in the next section for Lipschitz mappings satisfying an even weaker version of the inequalities in Condition SH2(Lip). Explicit values of the exponential expansivity parameters can be determined in terms of the split coefficients and the other semi-hyperbolicity parameters.

6.1 Expansivity

79

6.1.2 Lipschitz Mappings Conditions SH0(Lip), SH1(Lip) and SH2(Lip) of semi-hyperbolicity Definition 3.6 for a Lipschitz mapping are stronger than necessary for establishing expansivity. In particular, the inequalities of Condition SH2(Lip) can be weakened. As before, let s = (λs , λu , µs , µu ) be a split which, without loss of generality, may be assumed positive. Consider a mapping f : X → Rd , where X is an open subset of Rd , and let K be a nonempty compact subset of X such that K ∩ f (K) 6= ∅. Theorem 6.5. Let f be a continuous mapping defined on X ⊆ Rd and suppose that for each x ∈ K there exists a uniform splitting Rd = Exs ⊕ Exs satisfying Condition SH1(Lip) and the following modification of Condition SH2(Lip) of Definition 3.6: SH2(Mod): For all x satisfying x, f (x) ∈ K and for all z ∈ Rd satisfying kPxs zk, kPxu zk ≤ δ, the inclusion x + z ∈ X and the inequalities kPfs(x) (f (x + z) − f (x)) k ≤ λs kPxs zk + µs kPxu zk, kPfu(x) (f (x + z) − f (x)) k ≥ λu kPxu zk − µu kPxs zk

(6.8)

hold. Then the mapping f is exponentially expansive in K with exponent 

r = σ(s)−1

1 + λs + = 2 λu

s

1 − λs λu

2

−1 4µs µu  , + λu

as in (4.3) and constants c=

1 −1 h min{γ(s), γ(s)−1 } 2

and

ξ = h−1 δ,

(6.9)

where h as in Condition SH1(Lip) of Definition 3.6. Proof. Two lemmas will form the basis of the proof. Since the split s is fixed write M = M (s), σ = σ(s) and γ = γ(s), see equations (4.2), (4.3) and (4.5) in Sect. 4.1. In addition, the norm k · k∗ on R2 defined by k(y1 , y2 )k∗ = max{γ|y1 |, |y2 |}, is used. Since the split s is positive, γ > 0. Lemma 6.6. The inequalities ηkzk ≤ k(kPxs zk, kPxu zk)k∗ ≤ 2ηh max{γ, γ −1 }kzk

80


with η = 12 min{γ, 1} hold for all z ∈ Rd and all x from the semi-hyperbolicity set K of the mapping f . Proof. By Lemma 4.1 min{γ, 1} max{kPxs zk, kPxu zk} ≤ k(kPxs zk, kPxu zk)k∗ ≤ max{γ, 1} max{kPxs zk, kPxu zk}, so it suffices to check that the inequalities 1 kzk ≤ max{kPxs zk, kPxu zk} ≤ hkzk 2 hold for all z ∈ Rd and all x ∈ K. The inequality on the right follows from the semi-hyperbolicity Condition SH1(Lip), while that on the left is just kzk = kPxs z + Pxu zk ≤ kPxs zk + kPxu zk ≤ 2 max{kPxs zk, kPxu zk}. The result now follows with η =

1 2

min{γ, 1}.

t u

For the remaining lemma note that semi-hyperbolicity does not require the sets K and X to be invariant under the mapping f . Lemma 6.7. Suppose that x, f (x), f 2 (x) ∈ K and y, f (y), f 2 (y) ∈ X with kx − yk, kf (x) − f (y)k, kf 2 (x) − f 2 (y)k < h−1 δ. Then at least one of the following pair of inequalities σk(kPxs r0 k, kPxu r0 k)k∗ ≥ k(kPfs(x) r1 k, kPfu(x) r1 k)k∗ , σk(kPfs2 (x) r2 k, kPfu2 (x) r2 k)k∗ ≥ k(kPfs(x) r1 k, kPfu(x) r1 k)k∗ , holds, where r0 = x − y,

r1 = f (x) − f (y),

r2 = f 2 (x) − f 2 (y).

Proof. By (6.10) and Condition SH1(Lip) of Definition 3.6, kPxs r0 k, kPfs(x) r1 k, kPfs2 (x) r2 k < δ, kPxu r0 k, kPfu(x) r1 k, kPfu2 (x) r2 k < δ, from which, by (6.8), kPfs(x) r1 k ≤ λs kPxs r0 k + µs kPxu r0 k, kPfu2 (x) r2 k ≥ λu kPfu(x) r1 k − µu kPfs(x) r1 k. That is,

(6.10)

6.1 Expansivity

81

kPfs(x) r1 k ≤ λs kPxs r0 k + µs kPxu r0 k 1 µu s r1 k + kP kP u2 r2 k. kPfu(x) r1 k ≤ λu f (x) λu f (x) Here, by definition of the norm k · k∗ , kPxs r0 k ≤ γ −1 k(kPxs r0 k, kPxu r0 k)k∗ , kPxu r0 k ≤ k(kPxs r0 k, kPxu r0 k)k∗ , kPfs(x) r1 k ≤ γ −1 k(kPfs(x) r1 k, kPfu2 (x) r2 k)k∗ , kPfu2 (x) r2 k ≤ k(kPfs(x) r1 k, kPfu2 (x) r2 k)k∗ , and therefore kPfs(x) r1 k ≤ γ −1 (λs + µs γ)k(kPxs r0 k, kPxu r0 k)k∗ , µu 1 kPfu(x) r1 k ≤ γ −1 + γ k(kPfs(x) r1 k, kPfu2 (x) r2 k)k∗ . λu λu From the definition of σ = σ(s) and γ = γ(s) in (4.6), λs + µs γ = σ,

1 µu + γ = σγ, λu λu

so kPfs(x) r1 k ≤ σγ −1 k(kPxs r0 k, kPxu r0 k)k∗ ,

(6.11)

kPfu(x) r1 k

(6.12)

≤

σk(kPfs(x) r1 k, kPfu2 (x) r2 k)k∗ .

Suppose now that the lemma is invalid, that is both the inequalities σk(kPxs r0 k, kPxu r0 k)k∗ < k(kPfs(x) r1 k, kPfu(x) r1 k)k∗ ,

(6.13)

σk(kPfs2 (x) r2 k, kPfu2 (x) r2 k)k∗ < k(kPfs(x) r1 k, kPfu(x) r1 k)k∗

(6.14)

and

hold. Then inequalities (6.11) and (6.13) imply that the strict inequality γkPfs(x) r1 k < k(kPfs(x) r1 k, kPfu(x) r1 k)k∗ is valid from which, since σ = σ(s) < 1, σγkPfs(x) r1 k < k(kPfs(x) r1 k, kPfu(x) r1 k)k∗ This last inequality together with (6.14) shows that σk(kPfs(x) r1 k, kPfu2 (x) r2 k)k∗ < k(kPfs(x) r1 k, kPfu(x) r1 k)k∗ ,

(6.15)

82


and by (6.12) the strict inequality kPfu(x) r1 k < k(kPfs(x) r1 k, kPfu(x) r1 k)k∗

(6.16)

is valid. Now, by definition of the norm k · k∗ , the inequalities (6.15) and (6.16) imply that k(kPfs(x) r1 k, kPfu(x) r1 k)k∗ < k(kPfs(x) r1 k, kPfu(x) r1 k)k∗ , a contradiction and so the lemma is proved.

t u

Now, to complete the proof of Theorem 6.5 consider two trajectories x = x−n− , . . . , x0 , . . . , xn+ , y = y−n− , . . . , y0 , . . . , yn+ satisfying x ⊆ K, y ⊆ X and kyn − xn k ≤ ξ for n = −n− , . . . , n+ and write rn = xn − yn ,

νn = k(kPfs(xn ) rn k, kPfu(xn ) rn k)k∗

(6.17)

for n = −n− , . . . , n+ . From Lemma 6.7 it then follows that at least one of the inequalities ν−1 ≥ σ −1 ν0 , ν1 ≥ σ −1 ν0 holds. For definiteness, suppose that ν−1 ≥ σ −1 ν0 . Then, by Lemma 6.7 again, at least one of the inequalities ν−2 ≥ σ −1 ν−1 ,

ν0 ≥ σ −1 ν−1

holds. But the second of these inequalities can not be valid at the same time as the inequality ν−1 ≥ σ −1 ν0 because of σ < 1. Hence, ν−2 ≥ σ −1 ν−1 . In the same way it can be shown that νn−1 ≥ σ −1 νn ,

for n = −n− + 1, . . . , −1, 0,

and hence that νn ≥ σ −n ν0 ,

for

n = −n− + 1, . . . , −1.

Recalling definition (6.17) of the νn and using Lemma 6.6 to estimate the norms kxn − yn k via νn , obtain the inequalities (6.7) with constants c and ξ as in (6.9). When ν1 ≥ σ −1 ν0 , the proof follows along same lines. t u Remark 6.8. As mentioned above, condition (6.8) is a weakened version of Condition SH2(Lip) of Definition 3.6. Indeed, the inequalities (3.13)–(3.16) of Condition SH2(Lip), namely

6.2 Shadowing

83

kPys (f (x + u + v) − f (x + u ˜ + v)k ≤ λs ku − u ˜k, kPys kPyu kPyu

(3.13)


(3.14)


(3.15)

(f (x + u + v) − f (x + u + v˜)) k ≥ λu kv − v˜k,

(3.16)

imply (6.8). Putting y = f (x), u = P s z, v = P u z, u ˜ = 0 in (3.13) obtain kPfs(x) (f (x + z) − f (x + P u z)) k ≤ λs kP s zk, while putting y = f (x), u = 0, v = P u z, v˜ = 0 in (3.14) obtain kPfs(x) (f (x + P u z) − f (x)) k ≤ µs kP u zk, and the first inequality of (6.8) follows immediately. Similarly, the second inequality of (6.8) can be obtained by appropriate choice of elements u, v, u ˜ and v˜ in (3.15) and (3.16).

6.2 Shadowing The relationship between the behaviour of a given dynamical system and perturbations of it are also very important for nonsmooth and nonhomeomorphic systems. For example, a computer simulation of a specific dynamical system is really only an approximation of that system because computer arithmetic is finite and there is subsequent round off error. In particular, computed trajectories are only pseudo-trajectories of the original system. This relationship is, however, difficult to express in terms of conjugacy, as is possible for diffeomorphic systems. A convenient and highly useful replacement of the conjugacy property in such a situation is provided by bi-shadowing, where trajectories and pseudo-trajectories are compared rather than the mappings themselves. A typical problem can be roughly stated in the following general way: given a dynamical system xn+1 = f (xn ) generated by a mapping f on a metric or topological space X, suppose that the function f is perturbed at any iteration n so that another system yn+1 = ϕ(n, yn ) is in fact observed, where ϕ is close to f and can possibly also vary with n. Then the following Direct Shadowing Problem can be posed: for every trajectory of every small perturbation ϕ of a given mapping f does there exists a close trajectory of f ? In considering non-autonomous perturbations ϕ of

84


the mapping f we have considerably more latitude. The Direct Shadowing Problem is usually formulated in terms of relationships between the true trajectories and pseudo-trajectories of a system f , as in Sect. 6.2.2. The Inverse Shadowing Problem is also of interest namely can every trajectory of a mapping f be approximated by some trajectory of each perturbation ϕ of f from a predefined class T ? Here the class of perturbations T is usually taken to be rather appropriate size to obtain meaningful and strong results. For example, the class T may consist of all continuous autonomous mappings ϕ or of all, possibly discontinuous mappings resulting from some approximation technique or numerical method. The asymmetrical roles of the classes of perturbations T in each case should be emphasized, with the class T being as broad as possible in Direct Shadowing and as meagre as possible in indirect or Inverse Shadowing for the sharpest results in each case. In the direct shadowing of the classical Shadowing Lemma, see Chap. 2, Theorem 2.10, T consists of all possible nonautonomous perturbations of the given system, while in inverse shadowing a natural class T consists of trajectories of all continuous mappings ϕ that are sufficiently close to f .

6.2.1 Definitions Consider a mapping f : X → X, where X is an open bounded subset of Rd , and let k · k denote a fixed but otherwise arbitrary norm on Rd . Definition 6.9. A γ-pseudo-trajectory of a dynamical system generated by a mapping f on X is a sequence y = {yn } ⊂ X with kyn+1 − f (yn )k ≤ γ,

γ > 0,

(6.18)

for n = −n− , . . . , 0, . . . , n+ where n± ≤ ∞. The term finite may be appended when n± < ∞, otherwise the pseudo-trajectory is infinite. Clearly, any trajectory of a system is 0-pseudo-trajectory of itself, but not every pseudo-trajectory is a trajectory. Pseudo-trajectories arise naturally in a number of ways, such as the presence of roundoff error in computer calculations of trajectories, although accumulated roundoff error can rapidly destroy any meaningful connection between a computed pseudo-trajectory and an original trajectory. The concept of shadowing provides a way of comparing trajectories and pseudo-trajectories. Definition 6.10. A trajectory x = {xn } is said to ε-shadow a γ-pseudotrajectory y = {yn } on some finite or infinite interval I ⊆ Z if kxn − yn k ≤ ε,

n ∈ I.

6.2 Shadowing

85

Without loss of generality, suppose that the set I is one of the following intervals: I = [0, N ], I = (−∞, 0], I = [0, ∞) or I = (−∞, ∞). The gist of a shadowing Theorem like Theorem 2.10 is that, under certain assumptions on f such as hyperbolicity, for every ε > 0 there exists a δ = δ(ε) > 0 such that each δ-pseudo-trajectory is ε-shadowed by a true trajectory. This is what is meant by specifying direct shadowing.

6.2.2 Direct Shadowing The proof of the Shadowing Theorem for hyperbolic diffeomorphisms does not require the full structure of hyperbolicity. The theorem stated and proved below establishes the same result for differentiable semi-hyperbolic mappings. The proof is not only much shorter in the differentiable semi-hyperbolic case than for hyperbolic case, but is more general. Below, the continuity of Dfx for x ∈ X is needed only to ensure uniform continuity of Dfx . Theorem 6.11. Let f : X → X be differentiable and semi-hyperbolic (in the sense of Definition 3.5) on an open bounded set X ⊂ Rd with Dfx continuous on X. Then for every sufficiently small ε > 0 there exists a δ = δ(ε) > 0 such that every δ-pseudo-trajectory y = {yn } of f is ε-shadowed by a true trajectory x = {xn }. Proof. The proof adapts and slightly simplifies those of [30] and [32]. Let y = {yn } be a δ-pseudo-trajectory of f defined over some integer interval I of indices. Define the nonlinear mapping F : X → Y by (F (x))n = xn+1 − f (xn ),

n ∈ I,

(6.19)

where x = {xn } ∈ X and spaces X and Y are as follows X = `∞ ([0, N + 1], Rd ), Y = `∞ ([1, N + 1], Rd ),

if I = [0, N ],

X = ` ((−∞, 1], R ),

Y = ` ((−∞, 1], R ),

if I = (−∞, 0],

X = ` ([0, ∞), R ),

Y = ` ([1, ∞), R ),

if I = [0, ∞),

∞

∞

d

∞

d

X = ` ((−∞, ∞), R ), ∞

∞

d

d

d

Y = ` ((−∞, ∞), R ), ∞

d

if I = (−∞, ∞).

Then F is C 1 with Fréchet derivative (DF (x)u)n = un+1 − Dfxn un . Let ω(τ ), τ ≥ 0, be the modulus of continuity of Df over X, ω(τ ) = sup kDfy − Dfx k : x, y ∈ X, ky − xk ≤ τ ,

86


and note that ω(τ ) → 0+ as τ → 0+ by the continuity of Dfx on the set X. Let ε0 be the largest positive number less than or equal to ε such that ω(ε0 ) ≤ 1/(2α(s, h)), where α(s, h) =

h λu − λs + µs + µu = h. ν(s) (1 − λs )(λu − 1) − µs µu

Here ν(s) is the constant (3.3). Note that the sequence of matrices {Dfxn } is semi-hyperbolic because of semi-hyperbolicity of the mapping f . The derivative DF (x) thus has a right inverse DF (x)−1 which satisfies

DF (x)−1 ≤ ∞

h = α(s, h). ν(s)

Choose δ = δ(ε) = ε0 /(2α(s, h)). Now, if y is the δ-pseudo-trajectory sequence and x is any `∞ (I, Rd ) sequence satisfying kx − yk∞ ≤ ε0 , then kDF (x) − DF (y)k∞ ≤ sup kDfxn − Dfyn k n

≤ ω(ε0 ) ≤

1 1 ≤ . 2α(s, h) 2 kDF (y)−1 k∞

To complete the main proof, the following fixed point lemma due to Chow, Lin and Palmer [30] is used and a proof given for completeness. For simplicity of exposition, denote the norms in X , Y by the same symbol k · k. Lemma 6.12. Let X , Y be Banach spaces and suppose that F : X → Y is C 1 . Let y ∈ X be a point such that DF (y)−1 is a bounded linear right inverse of DF (y) and let ε0 > 0 be chosen so that kDF (x) − DF (y)k ≤

1 2kDF (y)−1 k

(6.20)

for kx − yk ≤ ε0 . If 0 < ε ≤ ε0 and kF (y)k ≤

ε 2kDF (y)−1 k

then the equation F (x) = 0 has a unique solution x such that kx − yk ≤ ε. Proof. Write F (x) = F (y) + DF (y)(x − y) + η(x). For kx1 − yk, kx1 − yk ≤ ε0 by (6.20)

6.2 Shadowing

87

kη(x1 ) − η(x2 )k = kF (x1 ) − F (x2 ) − DF (y)(x1 − x2 )k

1

Z

· kx1 − x2 k (DF (x + θ(x − x )) − DF (y)) dθ ≤ 2 1 2

0

≤

kx1 − x2 k . (6.21) 2kDF (y)−1 k

Rewrite the equation F (x) = 0 as x = y − DF (y)−1 {F (y) + η(x)} = T (x). For 0 < ε ≤ ε0 , define B(ε, y) = {x ∈ X : kx − yk ≤ ε}. If T is a contraction on B(ε, y), the lemma will then follow immediately. To see this first note that if x ∈ B(ε, y) then kT (x) − yk = kDF (y)−1 (F (y) + η(x)) k

kx1 − x2 k ε + ≤ DF (y)−1 2kDF (y)−1 k 2kDF (y)−1 k ε ε ε kx1 − x2 k ≤ + = ε, = + 2 2 2 2 where (6.20) and (6.21) are used with x1 = x, x2 = y. Hence T maps B(ε, y) into itself. Moreover if x1 , x2 ∈ B(ε, y) then, using (6.21), kT (x1 ) − T (x2 )k = kDF (y)−1 (η(x1 ) − η(x2 )) k

kx1 − x2 k kx1 − x2 k ≤ DF (y)−1 = . −1 2kDF (y) k 2 Thus T is indeed a contraction on B(ε, y) and the proof of Lemma 6.12 follows. t u Now apply Lemma 6.12 to the mapping F : X → Y of (6.19), with the Banach spaces X and Y introduced above and endowed with the norm k · k∞ . Then the equation F (x) = 0 has a unique solution x ∈ X satisfying kx − yk∞ ≤ ε. That is, the δ-pseudo-trajectory y = {yn } is ε-shadowed by the true trajectory x = {xn }. This completes the proof of Theorem 6.11. t u

6.2.3 Inverse Shadowing As a distance between the mappings f, ϕ : Rd → Rd consider the semi-norm kf − ϕk∞ = sup kf (x) − ϕ(x)k. x∈Rd

88


Note that this is not a norm because it may take infinite values. Definition 6.13. A finite trajectory x = {x0 , x1 , . . . , xN } of a mapping f is called α-robust for some α > 0, if there exists an ε0 > 0 such that any continuous mapping ϕ : Rd → Rd satisfying kf − ϕk∞ ≤ ε0 has at least one trajectory y = {y0 , y1 , . . . , yN } such that kyn − xn k ≤ αkf − ϕk∞ ,

n = 0, 1, . . . , N.

(6.22)

This is a form of inverse shadowing where the reference class T is the space of all continuous mappings on X. Remark 6.14. The key condition in Definition 6.13 is the existence of α > 0 independent of the length of the given trajectory. The estimates (6.22) can be obtained easily if α is allowed to depend on the length of the trajectory. For example, any trajectory x is (1 + L + · · · + LN )-robust if the mapping f is Lipschitz with Lipschitz constant L in a neighbourhood of the trajectory x. Indeed, in this case, for a given trajectory x of the mapping f and for a trajectory y of a mapping ϕ satisfying y0 = x0 , kyn+1 − yn+1 k = kϕ(yn ) − f (xn )k ≤ kϕ(yn ) − f (yn )k + kf (yn ) − f (xn )k ≤ kf − ϕk∞ + Lkyn − xn k, from which kyn − xn k ≤ (1 + L + · · · + Ln )kf − ϕk∞ ,

n = 1, 2, . . . , N.

Thus, any trajectory x = {x0 , x1 , . . . , xN } of the mapping f is αN -robust with αN = (1 + L + · · · + LN ). As the next theorem shows, semi-hyperbolicity allows the robustness constant α to be chosen independently of the particular trajectory and its length N , uniformly throughout the domain of semi-hyperbolicity X. Theorem 6.15. Let f : X → X be differentiable and (s, h)-semi-hyperbolic on an open set X ⊆ Rd . Then every finite trajectory x ⊂ X is α-robust for any λu − λs + µs + µu h, (6.23) α > α(s, h) = (1 − λs )(λu − 1) − µs µu where λu , λs , µu , µs are the parameters of the split s = (λu , λs , µu , µs ). Proof. Denote by BN the space of (N + 1)-sequences z = {z0 , z1 , . . . , zN },

zn ∈ Rd ,

n = 0, 1, . . . , N,

6.2 Shadowing

89

satisfying the boundary conditions Pxs0 z0 = PxuN zN = 0.

(6.24)

BN is a subspace of the (N + 1)d-dimensional vector space Rd × . . . × Rd (N + 1 times) with norm kzk∞ = max kzn k. 0≤n≤N

Let ϕ : Rd → Rd be a given mapping and x = {x0 , x1 , . . . , xN } be a given trajectory of the mapping f . Introduce the linear operator Un : Exun−1 → Exun defined by Un v = Pxun Dfxn−1 v. This operator is surjective from inequality (3.12) of Definition 3.5 and is thus invertible and Un−1 is well defined. Consider now the operator H : BN → BN , which transforms each sequence z = {z0 , z1 , . . . , zN } into a sequence w = {w0 , w1 , . . . , wN } defined by the boundary conditions Pxs0 w0 = PxuN wN = 0, see (6.24), and by Pxsn wn = Pxsn (ϕ(xn−1 + zn−1 ) − xn ), Pxun−1 wn−1 = Un−1 (Pxun zn − Pxun Dfxn−1 Pxsn−1 zn−1 + + Pxun (−ϕ(xn−1 + zn−1 ) + xn + Dfxn−1 zn−1 )), for n = 1, 2, . . . , N . The necessity of introducing the operator H is explained by the next lemma, whose proof is obvious. Lemma 6.16. The operator H is continuous, and for any of its fixed point z = {z0 , z1 , . . . , zN } the sequence y = {x0 + z0 , x1 + z1 , . . . , xN + zN } is a trajectory of the mapping ϕ. First, more notation and definitions are required. Given β > 0, let δβ (ε) be the largest positive value of δ such that kxn + Dfxn−1 z − f (xn−1 + z)k ≤ βε,

n = 1, 2, . . . , N,

(6.25)

for any kzk ≤ δ. For each z ∈ B define the pair of real numbers ms (z) = max kPxsn zn k, 0≤n≤N

mu (z) = max kPxun zn k, 0≤n≤N

and let m(z) be the 2-dimensional column vector with components ms (z), mu (z). Let M = M (s) be the split matrix (4.2) and define the column vector

90

6 Expansivity and Shadowing T h = h(1, λ−1 u ) .

Lemma 6.17. Let β > 0. Then the inequality, component by component, m(H (z)) ≤ M m(z) + (1 + β)kf − ϕk∞ h is valid for every continuous mapping ϕ and every z from the set W = {z ∈ B : kzk∞ ≤ δβ (kf − ϕk∞ )} . Proof. First consider the component ms (H (z)). By definition ms (H (z)) = max kvns k, 0≤n≤N

(6.26)

where vns = Pxsn (ϕ(xn−1 + zn−1 ) − xn ) . Rewrite this as vns = I1 + I2 + I3 + I4 ,

(6.27)

where I1 = Pxsn Dfxn−1 Pxsn−1 zn−1 , I2 = Pxsn Dfxn−1 Pxun−1 zn−1 , I3 = Pxsn (ϕ(xn−1 + zn−1 ) − f (xn−1 + zn−1 )), I4 = Pxsn (f (xn−1 + zn−1 ) − (f (xn−1 ) + Dfxn−1 zn−1 )). From (3.9), kI1 k ≤ λs kPxsn−1 zn−1 k,

(6.28)

while from (3.10), kI2 k ≤ µs kPxun−1 zn−1 k. Condition SH1(Lip) of Definition 3.6 implies that kI3 k ≤ hkf − ϕk∞ . Finally, Condition SH1(Lip) of Definition 3.6 and the definition of the function δβ (ε) imply that kI4 k ≤ hβkf − ϕk∞ . (6.29) From (6.27) and (6.28)–(6.29) it then follows that kvns k ≤ λs kPxsn−1 zn−1 k + µs kPxun−1 zn−1 k + (1 + β)kf − ϕk∞ h, and by (6.26) ms (H (z)) ≤ λs ms (z) + µs ms (z) + (1 + β)kf − ϕk∞ h.

(6.30)

6.2 Shadowing

91

Now consider the component mu (H (z)). By definition, mu (H (z)) = max kvnu k, 0≤n≤N

(6.31)

where u vn−1 = Un−1 (Pxun zn − Pxun Dfxn−1 Pxsn−1 zn−1

+ Pxun (−ϕ(xn−1 + zn−1 ) + f (xn−1 ) + Dfxn−1 zn−1 )). Rewrite this as u vn−1 = Un−1 (J1 + J2 + J3 + J4 ),

(6.32)

where J1 = Pxun zn , J2 = J3 =

(6.33)

−Pxun Dfxn−1 Pxsn−1 zn−1 , Pxun (−ϕ(xn−1 + zn−1 ) +

(6.34) f (xn−1 + zn−1 ),

J4 = −f (xn−1 + zn−1 ) + (f (xn−1 ) + Dfxn−1 zn−1 ).

(6.35) (6.36)

The relations (3.12) and (6.33) imply that u kUn−1 J1 k ≤ λ−1 u kPxn zn k,

(6.37)

while (3.11), (3.12) and (6.34) imply that s kUn−1 J2 k ≤ λ−1 u µu kPxn−1 zn−1 k.

In addition, (3.12), (6.35) and Condition SH1(Lip) from Definition 3.6 give kUn−1 J3 k ≤ λ−1 u hkf − ϕk∞ . Finally, (3.12), (6.36), Condition SH1(Lip) from Definition 3.6 and the definition of the function δβ (ε) imply that kUn−1 J4 k ≤ λ−1 u hβkf − ϕk∞ .

(6.38)

From (6.32) and (6.37)–(6.38) it thus follows that u u s kvn−1 k ≤ λ−1 kP z k + µ kP z k + (1 + β)kf − ϕk h , n u n−1 ∞ u xn xn−1 so by (6.31) u s mu (H (z)) ≤ λ−1 u (m (z) + µu m (z) + (1 + β)kf − ϕk∞ h) .

The lemma follows from the inequalities (6.30) and (6.39).

(6.39) t u

92


Returning to the proof of Theorem 6.15, without loss of generality assume that the split s = (λs , λu , µs , µu ) is positive. Recall from Sect. 4.1 that σ(M ) < 1 is the spectral radius (4.3) of the split matrix M and that k · k∗ is the norm on R2 defined for a given positive split s by k(y1 , y2 )k∗ = max{γ|y1 |, |y2 |}. with γ = γ(s) > 0 given by (4.5). Choose any fixed number α > α(s, h), with α(s, h) defined by (6.23), write β=

α − 1, α(s, h)

and consider the convex set V = z : km(z)k∗ ≤

1+β khk∗ kf − ϕk∞ , 1 − σ(M )

which is well defined since σ(M ) < 1. Lemma 6.18. There exists an ε0 > 0 such that V ⊆ W for

Proof. First,

kf − ϕk∞ ≤ ε0 .

(6.40)

V ⊆ V +,

(6.41)

where V+=

z : kzk∞ ≤ 2

1+β max{γ −1 , 1}khk∗ kf − ϕk∞ . 1 − σ(M )

To see this, note that km(z)k∗ = max γ max

kPxsn zn k,

kPxun zn k

max 0≤n≤N = max max γkPxsn zn k, kPxun zn k 0≤n≤N

0≤n≤N

= max k(kPxsn zn k, kPxun zn k)k∗ , 0≤n≤N

and so by Lemma 6.6 km(z)k∗ ≥

1 1 min{γ, 1} max kzn k = min{γ, 1}kzk∞ . 0≤n≤N 2 2

The inclusion (6.41) then follows from this last inequality. Now, by definition of the differential of a mapping,

6.2 Shadowing

93

lim inf ε→0+

δβ (ε) = ∞, ε

where δβ (·) is as in 6.25, and thus a number ε0 > 0 can be chosen such that 2

1+β max{γ −1 , 1}khk∗ ε ≤ δβ (ε) 1 − σ(M )

for ε ≤ ε0 .

Hence, for f and ϕ satisfying (6.40) the inclusion V + ⊆ W holds, and thus by (6.41), V ⊆ W . The lemma is proved. t u Now turn to completion of the proof of Theorem 6.15. Choose ε0 > 0 as in Lemma 6.18. Then V ⊆ W and for any vector z ∈ V , by Lemma 6.17 the component by component inequality m(H (z)) ≤ M m(z) + (1 + β)kf − ϕk∞ h holds. Hence km(H (z))k∗ ≤ kM m(z)k∗ + (1 + β)kf − ϕk∞ khk∗ . Recall from Sect. 4.1 that the corresponding matrix norm kM k∗ of the matrix M coincides with the spectral radius σ(M ) < 1 of M and that kM yk∗ ≤ σ(M )kyk∗ for all y ∈ R2 . Hence km(H (z))k∗ ≤ σ(M )km(z)k∗ + (1 + β)kf − ϕk∞ khk∗ and by definition of the set V H (z) ∈ V

for z ∈ V ,

that is, the set V is invariant under the operator H . Then, from continuity of H , see Lemma 6.16, and the Schauder Fixed Point Theorem, there exists a point z ∗ satisfying H (z ∗ ) = z ∗ such that z∗ ∈ V ⊆ W . This and Lemma 6.17 imply that m(z ∗ ) = m(H (z ∗ )) ≤ M m(z ∗ ) + (1 + β)kf − ϕk∞ h. Hence, by positivity of the components of the split matrix M , m(z ∗ ) ≤ (1 + β)kf − ϕk∞ (I − M )−1 h =

αkf − ϕk∞ (I − M )−1 h, α(s, h)

from which max kz ∗n k ≤ ms (z ∗ ) + mu (z ∗ ) ≤ αkf − ϕk∞ ,

0≤n≤N

94


see the proof of Lemma 4.2 for details. By Lemma 6.16 and by this last inequality, for any continuous mapping ϕ satisfying kf − ϕk∞ ≤ ε0 ∗ the sequence {x0 + z0∗ , x1 + z0∗ , . . . , xN + zN } is a finite trajectory of ϕ and satisfies (6.22). That is, the trajectory x = {x0 , x1 , . . . , xN } is α-robust and Theorem 6.15 is proved. t u

6.3 Bi-Shadowing The concept of ε-shadowing as in Definition 6.10 and α-robustness as in Definition 6.13 can be applied to investigate perturbations of a given system via Theorems 6.11 and 6.15. Their practical utility is somewhat limited as the proofs are of existence rather than constructive. A unified approach for investigating both direct and inverse shadowing is provided by the concept of bi-shadowing which has the added advantage of providing explicit values of the parameters involved. Let Tr(f, K, γ) denote the totality of finite or infinite γ-pseudo-trajectories (6.18) of f belonging entirely to the subset K ⊆ X. Since a true trajectory can be regarded as a γ-pseudo-trajectory for any γ ≥ 0, in particular with γ = 0, it is natural to denote the corresponding set of true trajectories by Tr(f, K, 0) or simply by Tr(f, K). Obviously Tr(f, K) ⊂ Tr(f, K, γ), with strict inclusion as there are γ-pseudo-trajectories which are not trajectories. As a distance between mappings ϕ and f on X use the semi-norm kϕ − f k∞ = sup kϕ(x) − f (x)k. x∈X

Definition 6.19. A dynamical system generated by a mapping f : X → X is said to be bi-shadowing on a subset K of X with positive parameters α and β if for any given finite pseudo-trajectory x = {xn } ∈ Tr(f, K, γ) with 0 ≤ γ ≤ β and any mapping ϕ : X → X satisfying kϕ − f k∞ ≤ β − γ

(6.42)

there exists a trajectory y = {yn } ∈ Tr(ϕ, X) such that kxn − yn k ≤ α(γ + kϕ − f k∞ )

(6.43)

for all n for which y is defined. Remark 6.20. Bi-shadowing conceptualizes the robustness of observed dynamical behaviour of a dynamical system and perturbations such as those arising from computer simulations. It can also be interpreted as a form of dy-

6.3 Bi-Shadowing

95

namical structural stability when restricted to specific classes of mappings, such as continuous mappings. Moreover, it implies both direct shadowing and inverse shadowing properties, in the form of α-robustness previously discussed. Taking ϕ ≡ f in (6.42) and (6.43) gives (αγ)-shadowing of any γpseudo-trajectory x ∈ Tr(f, K, γ) by a true trajectory y ∈ Tr(f, K), while αrobustness of any trajectory x of f follows because a trajectory y ∈ Tr(ϕ, X) can always be found which (αkϕ − f k∞ )-shadows a given true trajectory x ∈ Tr(f, K), considered here as the γ-pseudo-trajectory with γ = 0.

6.3.1 Bi-Shadowing of Finite Trajectories The main result of this section is that semi-hyperbolicity is sufficient to ensure bi-shadowing of a dynamical system generated by a Lipschitz mapping with respect to a class of perturbed systems generated by continuous mappings. It not only generalizes existing versions of the Shadowing Lemma to a far broader class of systems, but includes inverse as well as direct shadowing and provides explicit values of the shadowing parameters. Theorem 6.21. Let f : X → X be a Lipschitz mapping which is semihyperbolic on a compact subset K ⊂ X with a split s and constants h, δ as in Definition 3.6. Then it is bi-shadowing on K with respect to continuous mappings ϕ : X → X with parameters α = α(s, h) and β = β(s, h, δ) defined by λu − λs + µs + µu (6.44) α(s, h) = h (1 − λs )(λu − 1) − µs µu and β(s, h, δ) = δh−1

(1 − λs )(λu − 1) − µs µu . max{λu − 1 + µs , 1 − λs + µu }

(6.45)

The proof proceeds through a number of lemmas. Denote by Bxu (r) the closed ball of the radius r centered at 0 in the linear space Exu . For each x, y ∈ K with kf (x) − yk ≤ δ and each z ∈ Rd satisfying kPxs zk ≤ δ define the mapping Fx,y,z : Bxu (δ) → Eyu by Fx,y,z (v) = Pyu (f (x + Pxs z + v) − f (x + Pxs z)) . Lemma 6.22. Let 0 ≤ r ≤ δ. Then Byu (λu r) ⊆ Fx,y,z (Bxu (r)). Proof. The lemma is immediate for r = 0, so suppose that r > 0. Denote by ∂Bxu (r) and Int Bxu (r) the boundary and the interior of the ball Bxu (r). Clearly, Fx,y,z (0) = Pyu (f (x + Pxs z) − f (x + Pxs z)) = 0 ∈ Int Byu (λu r). On the other hand, by inequality (3.16)

(6.46)

96


kFx,y,z (v) − Fx,y,z (˜ v )k ≥ λu kv − v˜k

for v, v˜ ∈ Exu , kvk, k˜ v k ≤ δ. (6.47)

In particular, from (6.47) kFx,y,z (v)k ≥ λu r

for

v ∈ Exu , kvk = r ≤ δ,

and so Fx,y,z (∂Bxu (r)) ∩ Int Byu (λu r) = ∅.

(6.48)

By Condition SH0(Lip) of Definition 3.6, the dimensions of the linear spaces Exu and Eyu are equal for any x, y ∈ K with kf (x) − yk ≤ δ. Therefore, by (6.47), the mapping Fx,y,z (v) is homeomorphic on the ball Bxu (r) with 0 < r ≤ δ. Hence by the Principle of Domain Invariance [5, p. 396] the image of the boundary ∂Bxu (r) of the ball Bxu (r) under the map Fx,y,z (v) is also the boundary ∂Fx,y,z (Bxu (r)) of the set Fx,y,z (Bxu (r)), Fx,y,z (∂Bxu (r)) = ∂Fx,y,z (Bxu (r)). By (6.48) this last implies that ∂Fx,y,z (Bxu (r)) ∩ Int Byu (λu r) = ∅. This, together with (6.46), gives Byu (λu r) ⊆ Fx,y,z (Bxu (r)) and the lemma is proved.

t u

As an immediate consequence of Lemma 6.22 and inequality (3.16), Lemma 6.23. Let x, y ∈ K and kf (x) − yk ≤ δ. Then the operator Qx,y,z = −1 is well defined and continuous on Byu (λu δ) and satisfies kQx,y,z (v)k ≤ Fx,y,z −1 λu kvk. As in the proof of Theorem 6.11, let BN be the space of (N + 1)-sequences z = {z0 , z1 , . . . , zN },

zn ∈ Rd , n = 0, 1, . . . , N.

The set BN can be regarded as the (N + 1)d-dimensional vector space Rd × · · · × Rd (N + 1 times), with norm kzk∞ = max kzn k. 0≤n≤N

Let x = {x0 , x1 , . . . , xN } be a given γ-pseudo-trajectory of the system f and let ϕ be a given continuous mapping such that β = γ + kf − ϕk∞ satisfies the inequality

(6.49)

6.3 Bi-Shadowing

97

β ≤ β(s, h, δ)

(6.50)

with β(s, h, δ) as in (6.45). Introduce the operator H : BN → BN which transforms z = {z0 , z1 , . . . , zN } ∈ BN into a sequence w = {w0 , w1 , . . . , wN } ∈ BN defined by Pxs0 w0 = 0,

(6.51)

Pxsn wn = Pxsn (ϕ(xn−1 + zn−1 ) − xn )

(6.52)

for n = 1, 2, . . . , N , and PxuN wN = 0,

(6.53)

Pxun−1 wn−1 = Qxn−1 ,xn ,zn−1 (Pxun (−ϕ(xn−1 + zn−1 ) + f (xn−1 + zn−1 ) + xn − f (xn−1 + Pxsn−1 zn−1 ) + zn ))

(6.54)

for n = 1, 2, . . . , N . Define parameters a and b as the coordinates of the two-dimensional vector (a, b)T = (I − M (s))−1 h

T with h = h(1, λ−1 u ) ,

(6.55)

where M (s) is the 2 × 2 split matrix given by (4.2). Explicitly, λu − 1 + µs h, (1 − λs )(λu − 1) − µs µu 1 − λs + µu b= h, (1 − λs )(λu − 1) − µs µu

a=

(6.56) (6.57)

and so α(s, h) = a + b,

β(s, h, δ) = δ min a−1 , b−1 .

(6.58)

Consider the set S (β) = {z ∈ BN : kPxsn zn k ≤ aβ, kPxun zn k ≤ bβ, 0 ≤ n ≤ N }.

(6.59)

By (6.50) and (6.58) aβ ≤ aβ(s, h, δ) ≤ δ,

bβ ≤ bβ(s, h, δ) ≤ δ

(6.60)

and since x+u+v ∈ X from Condition SH2(Lip) of Definition 3.6, trajectories from S (β) belong to X. Lemma 6.24. The operator H is well defined and continuous on S (β) and for any of its fixed points z = {z0 , z1 , . . . , zN } ∈ S (β) the sequence

98


y = {x0 + z0 , x1 + z1 , . . . , xN + zN }

(6.61)

is a trajectory of the system ϕ. Proof. First the operator H is well defined and continuous at any point z ∈ S (β). For, it is clear by (6.60) that the right hand side of (6.52) is well defined and depends continuously on z ∈ S (β). Hence, It suffices to prove that for any n = 1, 2, . . . , N the right hand side of the (6.54) is well defined and continuous for z ∈ S (β). By Lemma 6.23 it is sufficient to establish that kPxun (−ϕ(xn−1 + zn−1 ) + f (xn−1 + zn−1 ) + xn − f (xn−1 + Pxsn−1 zn−1 ) + zn )k ≤ λu δ. Rewrite the last inequality in the form kJ1 + J2 + J3 k ≤ λu δ, where J1 = Pxun (−ϕ(xn−1 + zn−1 ) + f (xn−1 + zn−1 )) + Pxun (xn − f (xn−1 )), J2 = Pxun (f (xn−1 ) − f (xn−1 + Pxsn−1 zn−1 )),

(6.62)

J3 = Pxun zn . To estimate kJ1 k note that by (6.49) kϕ(xn−1 + zn−1 ) − f (xn−1 + zn−1 )k ≤ kϕ − f k∞ = β − γ and also that kxn − f (xn−1 )k ≤ γ since x is a γ-pseudo-trajectory of the mapping f , so by Condition SH1(Lip) kJ1 k ≤ βh.

(6.63)

kJ2 k ≤ µu kPxsn−1 zn−1 k.

(6.64)

kJ3 k = kPxun zn k.

(6.65)

By inequality (3.15), Clearly On the other hand, by definition (6.59) of the set S (β), z ∈ S (β) implies that kPxsn−1 zn−1 k ≤ βa, kPxun zn k ≤ βb. (6.66) Hence by (6.63)–(6.66) kJ1 + J2 + J3 k ≤ kJ1 k + kJ2 k + kJ3 k ≤ β(h + aµu + b)

6.3 Bi-Shadowing

99

so it suffices to establish that β(h + aµu + b) ≤ λu δ.

(6.67)

From (6.56) and (6.57) we have h + aµu + b = λu b, so (6.67) can be rewritten as βλu b ≤ λu δ, but this follows from (6.58). Hence the operator H is well defined and continuous at any point z ∈ S (β). It remains to prove that the sequence (6.61) is a trajectory of the system ϕ provided that z = {z0 , z1 , . . . , zN } ∈ S (β) is a fixed point of the operator H . By assumption z is a fixed point of the operator H , so equation (6.52) can be rewritten as Pxsn zn = Pxsn (ϕ(xn−1 + zn−1 ) − xn ) which can be rearranged to give Pxsn (xn + zn ) = Pxsn ϕ(xn−1 + zn−1 ).

(6.68)

Similarly, equation (6.54) can be rewritten as Pxun−1 zn−1 = Qxn−1 ,xn ,zn−1 (Pxun (−ϕ(xn−1 + zn−1 ) + f (xn−1 + zn−1 ) + xn − f (xn−1 + Pxsn−1 zn−1 ) + zn )). Applying the nonlinear operator Fxn−1 ,xn ,zn−1 = Q−1 xn−1 ,xn ,zn−1 to both sides of this last equation, obtain Fxn−1 ,xn ,zn−1 (Pxun−1 zn−1 ) = Pxun (f (xn−1 + zn−1 ) − ϕ(xn−1 + zn−1 ) + xn − f (xn−1 + Pxsn−1 zn−1 ) + zn ). By definition of Fx,y,z (v), Fxn−1 ,xn ,zn−1 (Pxun−1 zn−1 ) = Pxun (f (xn−1 + Pxsn−1 zn−1 + Pxun−1 zn−1 ) − f (xn−1 + Pxsn−1 zn−1 )), so, by comparing the last two equations, 0 = Pxun (zn − ϕ(xn−1 + zn−1 ) + xn ) or on rearranging Pxun (xn + zn ) = Pxun ϕ(xn−1 + zn−1 ). From (6.68) and (6.69) it now follows that

(6.69)

100


xn + zn = ϕ(xn−1 + zn−1 ),

n = 1, 2, . . . , N.

which means that the sequence (6.61) is a trajectory of the system ϕ. The lemma is proved. t u Lemma 6.25. The set S (β) is invariant under the operator H . Proof. Let z ∈ S (β) and write w = H (z), see (6.52) and (6.54). Write (6.52) in the form Pxsn wn = I1 + I2 where I1 = Pxsn ((ϕ(xn−1 + zn−1 ) − f (xn−1 + zn−1 )) + (f (xn−1 ) − xn )) , I2 = Pxsn (f (xn−1 + zn−1 ) − f (xn−1 )) , and rewrite (6.54) in the form Pxun−1 wn−1 = Qxn−1 ,xn ,zn−1 (J1 + J2 + J3 ) where J1 , J2 , J3 are defined in (6.62). To estimate kI1 k note that by (6.49) kϕ − f k∞ = β − γ

and kxn − f (xn−1 )k ≤ γ

since x is a γ-pseudo-trajectory of the mapping f . Hence kϕ(xn−1 + zn−1 ) − f (xn−1 + zn−1 )k + kf (xn−1 ) − xn k ≤ (β − γ) + γ = β and by Condition SH1(Lip) kI1 k ≤ hβ.

(6.70)

By (3.13), (3.14) and from Condition SH2(Lip) kI2 k ≤ λs kPxsn−1 zn−1 k + µs kPxun−1 zn−1 k.

(6.71)

Now for each z ∈ BN , define the pair of real nonnegative numbers ms (z) = max kPxsn zn k, 0≤n≤N

mu (z) = max kPxun zn k, 0≤n≤N

and denote by m(z) the two-dimensional column vector with coordinates ms (z) and mu (z). From the estimates (6.70), (6.71) and definition (6.59) of the set S (β) it follows that ms (H z) = ms (w) ≤ βh + βλs a + βµs b.

(6.72)

Similarly, from (6.63)–(6.65), the definition (6.59) of the set S (β) and Lemma 6.23 it follows that

6.3 Bi-Shadowing

101

mu (H z) = mu (w) ≤ λ−1 u (βµu a + βb + βh).

(6.73)

Inequalities (6.72) and (6.73) are equivalent to the component by component inequality z ∈ S (β),

m(H z) = m(w) ≤ βM (s)(a, b)T + βh,

(6.74)

where, by (6.55), βM (s)(a, b)T + βh = β(M (s)(I − M (s))−1 + I)h = β(I − M (s))−1 h = β(a, b)T . Hence, (6.74) is equivalent to the coordinate-wise estimate z ∈ S (β),

m(H z) = m(w) ≤ β(a, b)T ,

which, by the definition (6.59) of S (β), means that the set S (β) is invariant under the operator H . t u Proof of Theorem 6.21. By Lemma 6.24 the operator H is continuous on the convex set S (β) and by Lemma 6.25 H (S (β)) ⊆ S (β), so by the Brouwer Fixed Point Theorem H has a fixed point z = {z0 , z1 , . . . , zN } ∈ S (β). Hence, by Lemma 6.24 again, the sequence y = {x0 + z0 , x1 + z1 , . . . , xN + zN } is a trajectory of the mapping ϕ. By definition (6.59) of the set S (β), kzn k ≤ kPxsn zn k + kPxun zn k ≤ (a + b)β,

n = 0, 1, . . . , N,

holds for the fixed point z ∈ S (β) of the operator H . Thus by the definitions of β and a, b in (6.49) and (6.58) respectively, kzn k ≤ α(s, h)(γ + kϕ − f k∞ ),

n = 0, 1, . . . , N,

that is kxn − yn k = kzn k ≤ α(s, h)(γ + kϕ − f k∞ ),

n = 0, 1, . . . , N.

The proof is complete.

t u

Remark 6.26. From the Definition 6.19 of bi-shadowing, Theorem 6.21 ensures the existence of a trajectory y = {yn } ∈ Tr(ϕ, X), for a given pseudotrajectory x = {xn } ∈ Tr(f, K, γ) with 0 ≤ γ ≤ β(s, h, δ) satisfying kxn − yn k ≤ α(s, h)(γ + kϕ − f k∞ ).

102


Often it is important that the inequalities kPxsn (xn − yn )k, kPxun (xn − yn )k ≤ δ, also hold. These follow from yn − xn = zn ∈ S (β), definition (6.59) of the set S (β) and the inequalities (6.60) established in the proof of Theorem 6.21.

6.3.2 Bi-Shadowing of Infinite Trajectories Results on bi-shadowing of infinite trajectories follow from expansivity of semi-hyperbolic Lipschitz mappings and Theorem 6.21. Theorem 6.27. Let f : X → X be a Lipschitz mapping which is semihyperbolic on a compact subset K ⊂ X with split s and constants h, δ (see Definition 3.6), and let α(s, h) and β(s, h) be as in (6.44) and (6.45). Then the following statements hold. (i)

For any infinite pseudo-trajectory x = {xn } ∈ Tr(f, K, γ) and any continuous mapping ϕ : X → X satisfying kϕ − f k∞ ≤ β(s, h, δ) − γ there exists an infinite trajectory y = {yn } ∈ Tr(ϕ, X) such that kxn − yn k ≤ α(s, h)(γ + kϕ − f k∞ ),

n ∈ Z.

(ii) For any given infinite pseudo-trajectory x = {xn } ∈ Tr(f, K, γ) with γ ≤ min {δ/(2hα(s, h)), β(s, h, δ)} there exists a unique infinite trajectory y = {yn } ∈ Tr(f, X) such that kxn − yn k ≤ α(s, h)γ,

n ∈ Z.

Proof. To prove (i) for the given γ-pseudo-trajectory x = {xn } ∈ Tr(f, K, γ), consider the sequence of finite γ-pseudo-trajectories (k)

(k)

(k)

x(k) = {x−k , . . . , x0 , . . . , xk } ∈ Tr(f, K, γ),

k = 1, 2, . . .

with x(k) n = xn ,

n = 0, ±1, ±2, . . . , ±k.

Then, for a given mapping ϕ satisfying kϕ − f k∞ ≤ β(s, h, δ) − γ, by Theorem 6.21 there exists for any k = 1, 2, . . . a finite trajectory (k)

(k)

(k)

y (k) = {y−k , . . . , y0 , . . . , yk } ⊆ Tr(ϕ, X) such that kyn(k) − xn k = kyn(k) − x(k) n k ≤ α(s, h)(γ + kϕ − f k∞ )

(6.75)

6.3 Bi-Shadowing

103

for n = 0, ±1, ±2, . . . , ±k. Moreover, by Remark 6.26 the inequalities kPxsn (yn(k) − xn )k ≤ δ,

kPxun (yn(k) − xn )k ≤ δ

(6.76)

hold for n = 0, ±1, ±2, . . . , ±k. (k) From (6.75), for any integer n the sequence {yn } is bounded and thus, without loss of generality, by taking an appropriate subsequence and rela(k) belling, suppose that it converges to a limit yn , yn → yn as k → ∞. Taking the limit in (6.76), kPxsn (yn − xn )k ≤ δ,

kPxun (yn − xn )k ≤ δ,

and so, since xn ∈ K and Condition SH2(Lip) of Definition 3.6, yn ∈ X,

n ∈ Z.

(6.77)

Taking the limit k → ∞ on both sides of (k)

yn+1 = ϕ(yn(k) ) obtain yn+1 = ϕ(yn ). Together with (6.77) this means that y = {yn }∞ n=−∞ ∈ Tr(ϕ, X). Finally, taking the limit k → ∞ in (6.75) the infinite trajectory y ∈ Tr(ϕ, X) of the system ϕ satisfies kxn − yn k ≤ α(s, h)(γ + kϕ − f k∞ ),

n ∈ Z.

This completes the proof of (i). To prove (ii), note first that the existence of an infinite trajectory y ∈ Tr(f, X) satisfying kxn − yn k ≤ α(s, h)γ,

n∈Z

follows from (i). To prove the uniqueness of such an infinite trajectory y ∈ Tr(f, X) the exponential expansivity of a semi-hyperbolic mapping established in Theorem 6.5 is used. Suppose that the statement of (ii) is false. Then there exist trajectories ˜ ∈ Tr(f, X) such that y, y kyn − xn k, k˜ yn − xn k ≤ α(s, h)γ, and hence such that

n ∈ Z,

104


kyn − yñ k ≤ 2α(s, h)γ ≤ h−1 δ,

n ∈ Z.

This contradicts Theorem 6.5 and so (ii) holds.

t u

6.3.3 Cyclic Bi-Shadowing Cyclic, or periodic, behaviour is often of particular interest in dynamical systems. Here it is shown that bi-shadowing is preserved when attention is restricted to periodic trajectories, also often called cycles. Definition 6.28. A trajectory x = {xn }N n=0 ∈ Tr(f, K) is called a cycle, or periodic trajectory, of period N if xN = x0 and a pseudo-trajectory y = {yn }N n=0 ∈ Tr(f, K, γ) is called a γ-pseudo-cycle, or periodic γ-pseudo-trajectory, of period N if kyN − y0 k ≤ γ. Let Cyc(f, K, γ) ⊂ Tr(f, K, γ) denote the totality of γ-pseudo-cycles of any period belonging entirely to the subset K of X, with Cyc(f, K, 0) ⊂ Tr(f, K) or Cyc(f, K) ⊂ Tr(f, K) denoting the totality of proper cycles of any period which are contained entirely in K. Obviously Cyc(f, K) ⊂ Cyc(f, K, γ) for every γ > 0. Definition 6.29. A dynamical system generated by a mapping f : X → X is said to be cyclically bi-shadowing with positive parameters α and β on a subset K of X if for any given pseudo-cycle x ∈ Cyc(f, K, γ) with 0 ≤ γ ≤ β and any mapping ϕ : X → X satisfying (6.42) there exists a proper cycle y ∈ Cyc(ϕ, X) of period N equal to that of x such that (6.43) holds for n = 0, 1, . . . , N . Note that the cycle y in Definition 6.29 is required only to be in X rather than in the subset K. Cyclic bi-shadowing is also a consequence of semihyperbolicity. Theorem 6.30. Let f : X → X be a Lipschitz mapping which is semihyperbolic on a compact subset K ⊆ X with a split s and constants h, δ. Then it is cyclically bi-shadowing on K with respect to continuous mappings ϕ : X → X with parameters α(s, h) and β(s, h, δ) given by (6.44) and (6.45). The proof repeats verbatim that of Theorem 6.21 with the following two minor modifications. First, the boundary conditions Pxs0 (w0 ) = PxsN (wN ),

PxuN (wN ) = Pxu0 (w0 )

should be used instead of (6.51) and (6.53) in the definition of the operator H. Second, in the proof the following specifically cyclic version of Lemma 6.24 should be used.

6.3 Bi-Shadowing

105

Lemma 6.31. The operator H is well defined and continuous on S (β), and for any of its fixed points z = {z0 , z1 , . . . , zN } ∈ S (β) the sequence y = {x0 + z0 , x1 + z1 , . . . , xN + zN } is a cycle of the system ϕ.

Chapter 7

Structural Stability

This chapter further explores some structural stability properties of semihyperbolic mappings. More precisely, it is shown that topological entropy is nondecreasing with respect to continuous perturbations of a Lipschitz semihyperbolic mapping. Conjugation and factorization properties of semi-hyperbolic mappings are also studied. Lastly, chaotic phenomena of semi-hyperbolic mappings are discussed. Throughout the chapter k · k will denote a fixed but otherwise arbitrary norm on Rd .

7.1 Topological Entropy Throughout this section, X ⊆ Rd is open and K ⊂ X is a compact subset of X. Let hε (f, K) and h(f, K) denote the ε-entropy and entropy of the mapping f as defined in Definition 2.13. Recall from Chap. 6 that semi-hyperbolic mappings are expansive. The following lemma relating ε-entropy and entropy of expansive mappings, is well known [90], and is useful for providing a method of calculation for entropy. Lemma 7.1. Let K ⊂ Rd be a compact set and f : K → K be a continuous ξ-expansive mapping with f (K) = K. Then h(f, K) = hθ (f, K) for every 0 ≤ θ < ξ. Proof. Fix θ with 0 < θ < ξ. As noted in footnote 3 in Chap. 2, hε (f, K) is non-increasing in ε, so h(f, K) = sup hε (f, K) ε>0

and thus by Definition 2.13 it suffices to prove that 107

108

7 Structural Stability

hε (f, K) ≤ hθ (f, K)

(7.1)

for ε > 0 sufficiently small. From Lemma 6.2 there exists κ(ε, θ) for which the inequality ˜) ≥ ε ρN (x, x

˜ ∈ Tr±(N +κ(ε,θ)) (f, K) for x, x

˜ ) ≥ θ for any positive integer N . Now, from equalimplies that ρN +κ(ε,θ) (x, x ity f (K) = K each trajectory x ∈ Tr±N (f, K) can be extended to some trajectory in Tr±(N +κ(ε,θ)) (f, K). Hence Cε (Tr±N (f, K)) ≤ Cθ (Tr±(N +κ(ε,θ)) (f, K)), where Cε (Tr±N (f, K)) is defined in Sect. 2.2.4 as the binary logarithm of the maximal number of elements x(1) , . . . , x(p) in Tr±N (f, K) such that ρN (x(i) , x(j) ) ≥ ε for all i 6= j. By Definition 2.13 the inequality (7.1) immediately follows. t u A rich theory of topological entropy has been developed for hyperbolic mappings (see [90] and the references therein) and many of the results remain valid for semi-hyperbolic mappings. The following theorem is illustrative of such generalizations. Theorem 7.2. Let X ⊆ Rd be an open set and f : X → Rd be a Lipschitz mapping which is continuously s-semi-hyperbolic with constants h, δ on a compact invariant set f (K) = K ⊂ X. Then h(g, Oδ/2h (K)) ≥ h(f, K) for each continuous mapping g : X → X satisfying kg − f kC
0. Hence it remains only to prove the middle inequality in the following h(f, K) = hθ (f, K) ≤ hσ (g, Oδ/2h (K)) ≤ h(g, Oδ/2h (K)), with σ = θ − 2α(s, h)kg − f kC > 0. This, in turn, by Definition 2.13 will follow from

7.2 Structural Stability Properties

109

Cθ (Tr±N (f, K)) ≤ Cσ (Tr±N (g, Oδ/2h (K)),

N > 0.

(7.2)

To prove (7.2) let {x(1) , . . ., x(p) } be the maximal subset of elements from the set Tr±N (f, K) satisfying ρN (x(i) , x(j) ) ≥ θ,

i 6= j.

By Theorem 6.30, for each such x(i) ∈ Tr±N (f, K) there exists a trajectory y (i) ∈ Tr±N (g, X) for which, by the conditions of the theorem, ρN (y (i) , x(i) ) ≤ α(s, h)kg − f kC ≤ δ/2h. Hence y (i) ∈ Tr±N (g, Oδ/2h (K)) for i = 1, . . . , p, and ρN (y (i) , y (j) ) ≥ ρN (x(i) , x(j) ) − ρN (x(i) , y (i) ) − ρN (x(j) , y (j) ) ≥ θ − 2α(s, h)kg − f kC = σ for any j 6= i, and (7.2) follows.

t u

7.2 Structural Stability Properties Let X ⊂ Rd be also bounded. Denote by Σ(X) the metric space of all biinfinite sequences x = {xn }∞ n=−∞ with xn ∈ X for n = 0, ±1, ±2, . . . with norm ∞ X ˜) = ρ(x, x 2−|n| kxn − x ñ k. (7.3) n=−∞

Let σ denote the shift operator on Σ(X) defined as (σx)n = xn+1 ,

x ∈ Σ(X),

n ∈ Z.

For a set Y ⊆ X and a mapping f ∈ C(X, Rd ) let Tr±∞ (f, Y ) ⊂ Σ(X) be the totality of bi-infinite trajectories y = {yn } ⊆ Y . For completeness, note the following obvious Lemma 7.3. If the set Y ⊆ X ⊆ Rd is compact and f ∈ C(X, Rd ), then the set Tr±∞ (f, Y ) ⊂ Σ(X) is also compact in the metric space (Σ(X), ρ). Definition 7.4. Let f1 , f2 ∈ C(X, Rd ) and let Y1 and Y2 be closed subsets of X such that f1 (Y1 ) = Y1 and f2 (Y2 ) = Y2 . The restriction f1 |Y1 is said to be a weak factorization of the restriction f2 |Y2 if there exists a continuous surjection Φ : Tr±∞ (f2 , Y2 ) → Tr±∞ (f2 , Y2 ) which is shift invariant, Φ ◦ σ = σ ◦ Φ.

110


A further closely connected concept is that of weak conjugacy. Definition 7.5. Let f1 , f2 ∈ C(X, Rd ) and let Y1 and Y2 be closed subsets of X such that f1 (Y1 ) = Y1 and f2 (Y2 ) = Y2 . The restricted mappings f1 |Y1 and f2 |Y2 are said to be weakly conjugate if there exists a continuous injection Ψ : Tr±∞ (f1 , Y1 ) → Tr±∞ (f2 , Y2 ) which is shift invariant. In the above definitions Y1 and Y2 are closed subsets of the bounded set X ⊂ Rd . Hence they are compact, and so also are the metric spaces (Tr±∞ (f1 , Y1 ), ρ) and (Tr±∞ (f2 , Y2 ), ρ). Therefore the injective mapping Ψ in Definition 7.5 is a homeomorphism. Then, the restricted mappings f1 |Y1 and f2 |Y2 are weakly conjugate if the restrictions of the shift operator σ to the sets Tr±∞ (f1 , Y1 ) and Tr±∞ (f2 , Y2 ) are topologically conjugate, that is if σ1 = σ|Tr±∞ (f1 ,Y1 ) and σ2 = σ|Tr±∞ (f2 ,Y2 ) are topologically conjugate. The notion of weak factorization extends an analogue of semi-conjugacy (see Sect. 2.2.2 for definitions) to semi-hyperbolic mappings. Weak conjugacy is a generalization of topological conjugacy of mappings and reduces to it in the case of invertible mappings. The suitability of such generalizations in the analysis of noninvertible mappings is well known, see, for example [128, Sect. 15.6]. In particular, topological entropy is an invariant with respect to weak conjugacy and is nonincreasing with respect to weak factorization. The sets Y1 and Y2 above are often sets of chain recurrent points. Recall the definitions from Sect. 2.2.3 in the following form. Definition 7.6. A point x ∈ Y ⊆ X is called ε-chain recurrent in Y for the mapping f defined on X if there exists an ε-pseudo-cycle {xn } ⊆ Y of f with x0 = x. A point x ∈ Y ⊆ X is called chain recurrent for f defined if it is ε-chain recurrent in Y for any sufficiently small ε > 0. Denote the set of ε-chain recurrent points of f in Y by CR(f, ε, Y ). Then CR(f, Y ), the set of chain recurrent points of f , is \ CR(f, Y ) = CR(f, ε, Y ). ε>0

If Y ⊆ X where X is bounded, then CR(f, Y ) is compact and f -invariant, f (CR(f, Y )) = CR(f, Y ). Lemma 7.7. Let X and Y be open bounded subsets of Rd , f ∈ C(X, Rd ) and CR(f, Y ) ⊂ Y ⊂ Y ⊂ X. Then there exists a nondecreasing function q(ε, f ) of ε ≥ 0 with lim q(ε, f ) = q(0, f ) = 0,

ε→0

q(ε, f ) > 0

for

ε > 0,


111

such that CR(f, ε, Y ) ⊆ Oq(ε,f ) (CR(f, Y )). In particular, there exists an ε = ε(f, Y ) > 0 for which CR(f, ε, Y ) ⊂ Y . Proof. Suppose the contrary. Then, for some ε0 > 0, there exists a sequence (k) x(k) = {xn } of 1/m-pseudo-cycles of f with x(k) ⊆ Y ,

(k)

x0 6∈ Oε0 (CR(f, Y )).

(7.4) (k)

Since Y is bounded, the set Y is compact and the sequence {x0 } has a limit point y ∈ Y . By the first inclusion (7.4) and the definition of chain recurrence, y ∈ CR(f, Y ). But by the second relation (7.4), y 6∈ Oε0 (CR(f, Y )). A contradiction and the lemma follows. t u

7.2.1 Weak Factorization Let X and Y be open bounded subsets of Rd with Y ⊆ X and let Lipschitz mapping f ∈ Lip(X, Rd ) be continuously semi-hyperbolic on CR(f, Y ) with CR(f, Y ) ⊂ Y . By Lemma 3.12 there exists an η = η(f, Y ) > 0 such that f is semi-hyperbolic in Oη (CR(f, Y )) for some split s and constants h, δ. Denote by α, β the corresponding constants (6.44), (6.45), λu − λs + µs + µu , (1 − λs )(λu − 1) − µs µu (1 − λs )(λu − 1) − µs µu . β = β(s, h, δ) = δh−1 max{λu − 1 + µs , 1 − λs + µu } α = α(s, h) = h

Theorem 7.8. Let Y be an open set, Y ⊆ X, and let f ∈ Lip(X, Rd ) be continuously s-semi-hyperbolic in CR(f, Y ) ⊂ Y with constants h, δ. If g ∈ C(X, Rd ) with kg − f kC < ε, ε < δ/(2hα),

q(ε, f ) < η,

Oq(ε,f )+αε (CR(f, Y )) ⊂ Y.

(7.5)

Then the restriction f |CR(f,Y )) is a weak factorization of g|CR(g,Y )) . Proof. Consider a mapping g ∈ C(X, Rd ), kf − gkC < ε, where ε satisfies (7.5). Denote the ball in Σ(X) of radius ε centered at x by B(x, ε) = {y ∈ Σ(X) : kxi − yi k ≤ ε, i ∈ Z} and define the mapping Φ for x ∈ Tr±∞ (g, CR(g, Y )) by Φ(x) = B(αε, x) ∩ Tr±∞ (f, CR(f, Y )).

(7.6)

112


To prove the theorem it suffices to show that Φ is well-defined, that is (i)

the set B(αε, x) ∩ Tr±∞ (f, CR(f, Y ))

(7.7)

is non-empty; (ii) the set (7.7) contains exactly one point; (iii) the mapping Φ is a surjection of the set Tr±∞ (g, CR(g, Y )) onto the set Tr±∞ (f, CR(f, Y )); (iv) the mapping Φ is continuous; (v) the mapping Φ is shift-invariant. (i) It suffices to show that for a fixed x = {xn } ∈ Tr±∞ (g, CR(g, Y )),

(7.8)

for each positive integer m there exists a 1/m-pseudo-cycle x(m) = {x(m) n } ⊂ CR(g, 1/m, Y ) of the mapping g, satisfying kx(m) − xn k < 1/m, n

n = −m, . . . , 0, 1, . . . , m.

(7.9)

Note that by the continuity of g for a given m there can be found δ¯ = ¯ δ¯m > 0 such that each δ-pseudo-trajectory y = y−m , . . . , y0 , y1 , . . . , ym of g satisfying y0 = x−m , satisfies also the estimates sup

kyn − xn k < 1/m.

−m≤n≤m

¯ Then, since x−m ∈ CR(g, Y ) follows from (7.8), by chain recurrency such a δ(m) (m) ¯ pseudo cycle x ⊆ Y of the mapping g can be found that kx−m −x−m k < δ. (m) ¯ Hence, for this δ-pseudo cycle x the inequalities (7.9) also hold. So, the existence of x(m) is proved. Now each x(m) ∈ Tr±∞ (g, CR(g, 1/m, Y )), m = 1, 2, . . ., is simultaneously a (1/m + kf − gkC )-pseudo-cycle of f , x(m) ⊆ CR(f, 1/m + kf − gkC , Y ). Consequently, by Lemma 7.7, x(m) ⊆ Oq(1/m+kf −gkC ,f ) (CR(f, Y )) and so, by q(ε, f ) < η(f, Y ) (see (7.5)) and kf − gkC < ε,


113

x(m) ⊆ Oq(ε,f ) (CR(f, Y )) ⊆ Oη(f,Y ) (CR(f, Y ))

(7.10)

for all sufficiently large m. Hence, in view of the definition of η(f, Y ), the pseudo-cycles x(m) belong to the region of semi-hyperbolicity of f for all sufficiently large m. From (6.44), (6.45) it is seen that δ/α ≤ β, and then due to the first inequality of (7.5) ε ≤ β/2h ≤ β holds. Therefore, by the cyclic bi-shadowing Theorem 6.30, for each x(m) satisfying (7.10) for sufficiently large m there exists a cycle y (m) of f such that kx(m) − yn(m) k < αε n Hence

for all

m.

(7.11)

y (m) ⊆ Oη(f,Y )+αε (CR(f, Y )),

and y (m) ⊆ Y by (7.5). Moreover, since the trajectory y (m) is periodic, y (m) ∈ Tr±∞ (f, CR(f, Y )). Therefore, by Lemma 7.3 the sequence y (m) has a limit point y ∈ Tr±∞ (f, CR(f, Y )). By (7.9) and (7.11) kxn − yn k ≤ αε for all

n

which means that y ∈ B(αε, x). Hence the set (7.7) is non-empty for each x ∈ Tr±∞ (g, CR(g, Y )). (ii) Now the set (7.7) consists of a single point. For, note that because of the first inequality of (7.5), for any two trajectories ˜ ∈ B(αε, x) ∩ Tr±∞ (f, CR(f, Y )), y, y the estimate kyi − y˜i k < 2αε ≤ δ/h hold. By Theorem 6.5, the set (7.7) then contains no more than one element, and so the mapping Φ is well-defined. (iii) Further, the mapping Φ is a surjection of Tr±∞ (g, CR(g, Y )) onto Tr±∞ (f, CR(f, Y )). For, by definition of the mapping Φ it is sufficient, for each y ∈ Tr±∞ (f, CR(f, Y )), to construct an element x ∈ Tr±∞ (g, CR(g, Y )) with kxi − yi k < αε. As above, for each positive integer m there exist a 1/m-pseudo-cycle y (m) ⊆ CR(g, 1/m, Y ) of the mapping f satisfying kyn(m) − yn k < 1/m,

n = −m, . . . , 0, 1, . . . , m.

By the definition of chain recurrence, y is a limit, in the metric (7.3), of a sequence {y (m) }, m = 1, 2, . . ., of 1/m-pseudo-cycles of the mapping f . As

114


was mentioned above, from the first inequality of (7.5) ε ≤ β. So, by the cyclic bi-shadowing Theorem 6.30 for sufficiently large m there exist cycles x(m) of g satisfying kx(m) − yn(m) k < αε n

for all

n.

It remains to define x as a limit point of the sequence {x(m) } in the metric space (Σ(X), ρ) and such a limit point exists by Lemma 7.3. (iv) To prove the continuity of the mapping Φ, suppose the contrary. Then there exist x, x(m) ∈ Tr±∞ (g, CR(g, Y )),

m = 1, 2, . . . ,

such that ρ(x, x(m) ) → 0

(7.12)

(m)

but ρ(Φ(x), Φ(x )) ≥ ε¯ for some ε¯ > 0. In this case, without loss of generality, assume that k(Φ(x))0 − (Φ(x(m) ))0 k ≥ ε¯,

m = 1, 2, . . . .

(7.13)

˜ ) be the semi-norm in Σ(X) Now, for any positive integer N , let ρN (z, z defined by ˜ ) = sup kzn − zñ k. ρN (z, z −N ≤n≤N

Choose θ satisfying 2αε < θ < δ/h. Such a θ exists by (7.5). Since by Theorem 6.5 the mapping g is ξ-expansive with ξ = δ/h on the set CR(f, Y ), then by Lemma 6.2 there exists a positive integer κ(¯ ε, θ), independent of m, such that ρκ(¯ε,θ) (Φ(x), Φ(x(m) )) ≥ θ (7.14) holds whenever (7.13) is valid. But from the definition of the mapping Φ it follows that Φ(x) ∈ B(αε, x), and so ρκ(¯ε,θ) (Φ(x), x) ≤ αε. Hence ρκ(¯ε,θ) (Φ(x), Φ(x(m) )) ≤ ρκ(¯ε,θ) (Φ(x), x) + ρκ(¯ε,θ) (x, x(m) ) + ρκ(¯ε,θ) (x(m) , Φ(x(m) )) ≤ αε + ρκ(¯ε,θ) (x, x(m) ). Here 2αε < θ by the definition of θ and ρκ(¯ε,θ) (x, x(m) ) → 0 in view of (7.12). Therefore, ρκ(¯ε,θ) (Φ(x), Φ(x(m) )) < θ for sufficiently large m, which contradicts (7.14) and so the mapping Φ is continuous. (v) The shift invariance identity Φ ◦ σ ≡ σ ◦ Φ follows directly from the definition of the mapping Φ. Theorem 7.8 is proved. t u


115

7.2.2 Weak Conjugacy Theorem 7.8 above is a weak form of semi-conjugacy for semi-hyperbolic mappings, while Theorem 7.9 below is a version of structural stability for such mappings. The explicit estimates of the radii of semi-conjugacy and structural stability in these theorems are particularly useful in applications. Note that semi-conjugacy in Theorem 7.8 is a C 0 -robust property, while conjugacy in Theorem 7.9 below is Lipschitz-robust. Again suppose X and Y to be open bounded subsets of Rd with Y ⊆ X and let f ∈ Lip(X, Rd ) be a continuously semi-hyperbolic Lipschitz mapping on CR(f, Y ) with CR(f, Y ) ⊂ Y . Let ε(f, Y ) > 0 as defined in Lemma 7.7 such that CR(f, ε, Y ) ⊂ Y for ε ≤ ε(f, Y ). Then by Lemma 3.12 there exists an η = η(f, Y ) < ε(f, Y ) such that every mapping g ∈ Lip(X, Rd ) with kf − gkLip ≤ η is semi-hyperbolic in Oη (CR(f, Y )). Theorem 7.9. Let Y be an open set with Y ⊆ X and let f ∈ Lip(X, Rd ) be a Lipschitz mapping continuously s-semi-hyperbolic in CR(f, Y ) ⊂ Y with constants h, δ. If g ∈ Lip(X, Rd ) with kg − f kLip < η(f, Y ) then the restrictions f |CR(f,Y ) and g|CR(g,Y ) are weakly conjugate. Proof. Consider g ∈ Lip(X, Rd ) satisfying kg − f kLip < η(f, Y ). Introduce the homotopy mapping gλ (x) = λg(x) + (1 − λ)f (x), 0 ≤ λ ≤ 1, x ∈ X. Clearly, kgλ − f kLip ≤ kg − f kLip < η(f, Y ). (7.15) To prove the theorem it suffices, by transitivity of the weak conjugacy relation, to find a finite set of parameters λ0 = 0, λ1 , . . . , λk = 1 such that the mappings gλi and gλi+1 are weak conjugate for each i = 0, 1, . . . , k − 1. Since the family {gλ : 0 ≤ λ ≤ 1} is a compact subset of Lip(X, Rd ), the existence of parameters {λi } follows from the result below. Lemma 7.10. There exists ζ > 0 such that gλ |CR(gλ ,Y ) is weakly conjugate ¯ ∈ [0, 1] with |λ − λ| ¯ < ζ. to gλ¯ |CR(gλ¯ ,Y ) for all λ, λ Proof. By (7.15) and the definition of η(f, Y ), for any λ ∈ [0, 1] the mapping gλ is continuously semi-hyperbolic in CR(f, η(f, Y ), Y ). Since η(f, Y ) < ε(f, Y ), then by Lemma 7.7 CR(f, η(f, Y ), Y ) ⊂ Y . Again, from (7.15) it follows that kgλ − f kC ≤ kg − f kC < η(f, Y ), and so any chain recurrent point of gλ belonging to Y is an η(f, Y )-chain recurrent point of f . Hence, by definition of the value η(f, Y ), CR(gλ , Y ) ⊆ CR(f, η(f, Y ), Y ) ⊂ Y,

0 ≤ λ ≤ 1.

(7.16)

¯ there exist η > 0 and ζ1 > 0 Then by Lemmata 3.12 and 7.7 for a given λ ¯ such that all mappings gλ (x) with |λ − λ| < ζ1 are uniformly semi-hyperbolic in Oη (CR(gλ¯ , Y )) with the same split s and constants h, δ.

116


By (7.16) CR(gλ¯ , Y ) ⊂ Y and Theorem 7.8 there exists a positive ε satisfying ε < δ/(2hα) (7.17) such that for each gλ with kgλ − gλ¯ kC < ε the following property is true: (i)

gλ¯ |CR(gλ¯ ,Y ) is a weak factorization of gλ |CR(gλ ,Y ) with the corresponding mapping Φλ (x) = B(αε, x) ∩ Tr±∞ (gλ¯ , CR(gλ¯ , Y )), for x ∈ Tr±∞ (g, CR(g, Y )). On the other hand, take ε > 0 satisfying q(ε, gλ¯ ) < η,

(7.18)

where the function q(ε, f ) is defined as in Lemma 7.7. Then for gλ , gλ¯ with kgλ − gλ¯ kC < ε by Lemma 7.7, CR(gλ , Y ) ⊆ CR(gλ , δ, Y ) ⊆ CR(gλ¯ , ε, Y ) ⊆ Oη (CR(gλ¯ , Y )), with any δ < ε − kgλ − gλ¯ kC . Thus, (ii) CR(gλ , Y ) ⊂ Oη (CR(gλ¯ , Y )). ¯ < ζ imply that Choose ζ > 0 and λ ∈ [0, 1] such that ζ < ζ1 and |λ − λ| kgλ − gλ¯ k < ε. From property (i) above, gλ¯ |CR(gλ¯ ,Y ) is a weak factorization of gλ |CR(gλ ,Y ) . It remains to prove that Φλ is an injection and that the inverse mapping Φ−1 λ is continuous. To prove that Φλ is injective choose arbitrary but fixed ˜ ∈ Tr±∞ (gλ , CR(gλ , Y )) x, x ˜ . By property (ii) and Theorem 6.5 each gλ is δ/h-expansive such that x 6= x and so sup kxi − x ˜i k ≥ δ/h. (7.19) i∈Z

¯ < ζ, the mapping On the other hand, by property (i) and the inequality |λ−λ| Φλ satisfies sup kΦλ (x)i − xi k, sup kΦλ (˜ x)i − x ˜i k < αε i

i

from which, by (7.17), sup kΦλ (x)i − xi k, sup kΦλ (˜ x)i − x ˜i k < δ/2h. i

i

The latter inequality together with (7.19) implies that

(7.20)

7.3 Chaos

117

sup kΦλ (x)i − Φλ (˜ x)i k i

≥ sup kxi − x ˜i k − sup kΦλ (x)i − xi k − sup kΦλ (˜ x)i − x ˜i k > 0, i

i

i

and the mapping Φλ is an injection. Finally, to show continuity of the mapping Φ−1 λ (x) in the metric (7.3) is similar to the proof of continuity of the mapping Φ(x) in Theorem 7.8. Suppose the contrary. Then there exists a sequence y (m) ∈ Tr±∞ (gλ¯ , CR(gλ¯ , Y )), converging in the metric (7.3) to some y ∈ Tr±∞ (gλ¯ , CR(gλ¯ , Y )) (m) such that x(m) = Φ−1 ) does not converge to x = Φ−1 λ (y λ (y). Then without (m) loss of generality suppose that kx0 − x0 k ≥ η for some positive η. Choose θ, 0 < 2αε < θ < δ/h. Such a constant θ exists by (7.17). Then by Theorem 6.5 and Lemma 6.2 there exists a positive integer κ(η, θ) satisfying

ρκ(η,θ) (x(m) , x) =

max

−κ(η,θ)≤i≤κ(η,θ)

(m)

kxi

− xi k ≥ θ > 2αε.

From (7.19), (7.20) and this last inequality it follows that ρκ(η,θ) (y (m) , y) ≥ ρκ(η,θ) (x(m) , x) − ρκ(η,θ) (y (m) , x(m) ) − ρκ(η,θ) (y, x) ≥ θ − 2αε > 0. This contradicts lim ρ(y (m) , y) = 0,

m→∞

and thus Φ−1 λ (x) is continuous. The proofs of Lemma 7.10 and Theorem 7.9 are complete. t u

7.3 Chaos 7.3.1 Definition of Chaotic Behaviour Let X be an open bounded subset of Rd . Consider a mapping f : X → X and the corresponding discrete-time dynamical system generated by f . A trajectory of this system is a sequence x = {xn }∞ n=−n− satisfying

118


xn+1 = f (xn ),

n = −n− , . . . , 0, 1, 2, . . . ,

where 0 ≤ n− ≤ ∞, and n− = n− (x) may depend on the particular trajectory x. Let Tr(f ) denote the totality of trajectories of the dynamical system generated by f and let Tr∞ (f ) be the subset of x ∈ Tr(f ) with n− (x) = ∞. Important attributes of chaotic behaviour include sensitive dependence on initial conditions, an abundance of unstable periodic trajectories and an irregular mixing effect characterised by the existence of a finite number of separated subsets U1 , . . . , Um of Rd which can be visited by trajectories of some fixed iterate f k of f in any prescribed order. Symbolic dynamics gives a more formal description of this last property. Let Σ(m) denote the totality of all bi-infinite sequences b = {bn }∞ −∞ with bn ∈ {1, . . . , m} and let

for n = 0, ±1, ±2, . . . ,

W = {w1 , . . . , wm }

be an unordered subset of distinct points in X. Sequences in Σ(m) will be used to prescribe the order in which some disjoint balls of the form Ui = {z ∈ X : kz − wi k < ε},

i = 1, . . . , m,

are visited by trajectories. Let ε > 0, k ∈ N and let Y be a compact subset of X for which max kx − yk ≥ 2ε.

x,y∈Y

Definition 7.11. A continuous mapping f is (ε, k)-chaotic in a neighbourhood of Y if for each finite subset W = {w1 , . . . , wm } of Y with mini6=j kwi − wj k ≥ 2ε there exists a mapping Zf : Σ(m) → Tr∞ (f ) such that S1: For each b ∈ Σ(m) the trajectory z = Zf (b) of f satisfies zkj ∈ Ubj for all integers j; S2: The mapping b 7→ Zf (b) is shift invariant in the sense that a 1-shift σ of a sequence b ∈ Σ(m) induces a k-shift σ k of the corresponding trajectory Zf (b); S3: If a sequence b ∈ Σ(m) is periodic with minimal period p, then the corresponding trajectory z = Zf (b) is periodic with minimal period kp; S4: For each η > 0 there exists an uncountable subset Σ0 (η) of Σ(m) such that lim supn→∞ kZf (a)n − Zf (b)n k ≥

1 ε 2

for all

a, b ∈ Σ0 (η), a 6= b,

and lim inf n→∞ kZf (a)n − Zf (b)n k < η

for all

a, b ∈ Σ0 (η).

7.3 Chaos

119

Note that if f is (ε, k)-chaotic on Y then the topological entropy E top of f in the ε-neighbourhood of Y is positive and satisfies the inequality E top ≥ k(ε)−1 ln(Cε/4 (Y )), where Cε (Y ) denotes ε-capacity of the compact set Y [47, 65]. The above defining properties of chaotic behaviour are similar to those in the Smale transverse homoclinic trajectory theorem [129, Theorem 16.2], with an important difference being that the existence of an invariant Cantor set is not required. In addition neither the uniqueness of the trajectory Zf (b) for b ∈ Σ(m) nor the continuity of Zf is assumed, so Zf need not be a semiconjugacy. Instead, the definition includes Condition S3, which is usually a corollary of uniqueness, and Condition S4 which is a form of sensitivity and irregular mixing as in the Li–Yorke definition of chaos [84], with the subset of trajectories Zf (Σ0 (η)) corresponding to the Li–Yorke scrambled subset S0 and the whole set Zf (Σ(m)) to their whole set S. From a physical point of view the definition means that the trajectories of the system f appear to behave chaotically if the measurements are carried out with precision no better than ε at equal time intervals between subsequent measurements no shorter than k(ε). Definition 7.12. The minimal ε0 ≥ 0, with the property that for each ε > ε0 the system f is (ε, k)-chaotic for an appropriate k, is called the chaos threshold of the system f . The chaos threshold characterizes the accuracy of measurements for which the behaviour of the system in the vicinity of the subset Y appears completely chaotic if the time lapse between subsequent measurements is sufficiently large. For instance, a chaotic diffeomorphism f which is topologically conjugate on Y to the shift operator σ on Σ(2) has zero chaos threshold.

7.3.2 Perturbations of Bi-Shadowing Systems A bi-infinite trajectory x = {xn }∞ −∞ of a continuous bounded mapping f : X → X ⊂ Rd is called a homoclinic trajectory if its elements are not all identical and there exists a point x∗ ∈ X such that limn→∞ x−n = limn→∞ xn = x∗ . Theorem 7.13. Let x be a homoclinic trajectory of a continuous bounded mapping f : X → X, f is both bi-shadowing and cyclically bi-shadowing on the unordered set {x} with parameters α and β. Define γ(ε) =

1 min{β, ε/α} > 0. 2

120


Then every mapping g ∈ C satisfying kg − f kC < γ(ε) is (ε, k)-chaotic on a neighbourhood of {x} for any positive integer k ≥ k(ε), where k(ε) = max{m : ∃ an integer j0 : kxj − x∗ k ≥ γ(ε), j = j0 , . . . , j0 + m}. (7.21) Proof. In the proof below there will be fixed ε ≤ maxx,y∈x kx − yk, a positive integer k > k(ε) and a finite set W = {w1 , . . . , wm }, m > 1, satisfying min kwi − wj k ≥ 2ε. i6=j

To prove the theorem it suffices to construct a mapping Zf which satisfies Conditions S1–S4. Let Oρ (S) denote the open ρ-neighbourhood of a subset S of Rd . For each w ∈ W with w 6= x∗ there exist a positive integer m− (w), a non-negative integer m+ (w) and a finite sequence u = u(w) = u(w)−m− (w) , u(w)−m− (w)+1 , . . . , u(w)m+ (w)−1 , uniquely defined by u(w)0 = w,

u(w)j = f (u(w)j−1 ),

j = −m− (w) + 1, . . . , m+ (w) − 1

such that u(w)−m− (w) , f (u(w)m+ (w)−1 ) ∈ Oγ(ε) ({x∗ }),

u(w)j 6∈ Oγ(ε) ({x∗ })

for −m− (w) < j < m+ (w). Consider a given integer k > k(ε) and a given sequence b ∈ Σ(m). Define a sequence v = {vj } in X by vj+kn = u(wn )j

for

− m− (wn ) ≤ j < m+ (wn )

when wn 6= x∗ and by v j = x∗ for all other j. This sequence is a γ(ε)-pseudo-trajectory of f . Hence, by bishadowing of f , for any mapping g ∈ C with kg − f kC < γ(ε) the set Zg (b) of all trajectories z satisfying kzkn − wbn k < ε for all n is nonempty. Furthermore, by cyclically bi-shadowing of f , this set Zg (b) contains a trajectory of minimal period pk if b is periodic with minimal period p. Using standard constructions and Zorn’s lemma [66, pp. 31–36], a singlevalued selector Zg of the multi-valued mapping b → Zg (b) can be chosen which satisfies Conditions S1–S3 in Definition 7.11. To see this, denote by Z the totality of single-valued selectors Zg which are defined on subsets of D(Z) ⊂ Σ(m) and satisfy Conditions S1–S3 and consider this set as being partially ordered by inclusion of the corresponding graphs

7.3 Chaos

121

Gr(Zg ) = {(b, Zg (b)) : b ∈ D(Z)}. b of Z has an By construction every totally ordered subset, or chain, S of Z upper bound, the graph of which is defined as the union Zg ∈Z b Gr(Zg ). Hence by Zorn’s lemma there exists a maximal element Z∗ in the set Z. Suppose that the strict inclusion D(Z∗ ) ⊂ Σ(m) holds. Then there exist an element b∗ ∈ Σ(m) \ D(Z∗ ). If for some positive integer i the sequence b∗ is the i-th shift of a sequence b0 ∈ D(Z∗ ) then the mapping ( Z∗ (b), b ∈ D(Z∗ ), Z0 (b) = −ik σ Z∗ (b0 ), b = b∗ satisfies Conditions S1–S3 and strictly dominates Z∗ , which contradicts the definition of Z∗ . On the other hand, if the sequence b∗ cannot be represented as a shift of a sequence b ∈ D(Z∗ ) then define Z0 (b) as an arbitrary element from the nonempty set Zg (b) of all trajectories z satisfying kzkn − wbn k < ε for all n; again the mapping Z0 satisfies Conditions S1–S3 and strictly dominates Z∗ , again a contradiction. It remains to prove that the selected mapping Zg also satisfies Condition S4. This follows immediately from the next more general result. Lemma 7.14. Let (Ω, d) be an uncountable compact metric space and let S be the set of sequences s = {sj }j≥1 with sj ∈ Ω for all j ≥ 1. Then for each η > 0 there exists an uncountable subset S (η) of S such that lim inf j→∞ d (sj , tj ) < η for any s, t ∈ S (η). Proof. By compactness of (Ω, d) there exists a finite partition P of Ω such that diam(A) < η for each A ∈ P. Consider the equivalence relation Eη on S defined by: Eη (s, t) if and only if sj and tj belong to the same subset from the partition for all sufficiently large j. Denote the set of equivalence classes by T and note that each equivalence class E ∈ T contains no more than a denumerable set of elements, because each element from E differs from a chosen element {tj }j≥1 only for a finite number of indices j. On the other hand, the set S itself is uncountable by construction and so also is T . Choose a single element from each equivalence class in T , denote the set of these sequences by S∗ and say that two sequences s, t ∈ S∗ are connected if there exist arbitrarily large j for which sj and tj belong to the same subset from the partition. Since there are connected elements in every set which contains more than #(P) elements, there are connected elements in every denumerable set. The assertion of the lemma will hold if an uncountable subset S (η) ⊆ S of pairwise connected sequences can be constructed. That this can be done follows by an application of a transfinite analogue of the Ramsey Complete Graph Theorem ( [52, p. 608, Theorem 5.23]; see also [53, pp. 427–428]): If G is a graph of power m, where m is a transfinite cardinal, and if every denumerable subset of G contains two connected elements, then G contains a complete graph of power m.

122


This completes the proof of Lemma 7.14 and hence the proof of Theorem 7.13. t u The positive integer k(ε) defined by (7.21) represents the maximum number of iterations of an element of the unordered set {x} that can remain outside of the γ(ε) neighbourhood of the homoclinic point x∗ . Corollary 7.15. The chaos threshold of each continuous mapping g satisfying kg − f kC < γ(ε) does not exceed ε. The next theorem provides a simple means of locating homoclinic trajectories. Recall that a mapping f : X → X is said to be ξ-expansive in X if for any infinite trajectories x, y ∈ Tr(f ) either x = y or supn∈Z kxn − yn k ≥ ξ. Theorem 7.16. Let w = {w0 , . . . , wp−1 } be a γ-pseudo-cycle of a continuous mapping f : X → X which is bi-shadowing and cyclically bi-shadowing with parameters α, β on {w} and ξ-expansive in X . Suppose that γ ≤ β, kf (w0 ) − w0 k ≤ β and 2αγ < max kwi − wj k, i,j

α(γ + kf (w0 ) − w0 k) < ξ.

(7.22)

Then f has a homoclinic trajectory x in an open αγ neighbourhood of {w}. Proof. The point w0 is clearly an (kf (w0 ) − w0 k)-pseudo-equilibrium of f . By the assumption that kf (w0 ) − w0 k ≤ β and the cyclically bi-shadowing property there exists a proper equilibrium x∗ of f satisfying kx∗ − w0 k ≤ α kf (w0 ) − w0 k.

(7.23)

Now consider the bi-infinite sequence {yj } defined by ( w0 , j < 0 or j ≥ p, yj = wj , otherwise, which is obviously a γ-pseudo-trajectory of f . Since γ ≤ β, there exists a proper trajectory x = {xj } in the (αγ)-neighbourhood of the pseudotrajectory {yj }. The elements of this trajectory are not all identical because of the first inequality in (7.22). To complete the proof it remains to show that the trajectory x is homoclinic. That is, lim x−n = lim xn = x∗ .

(7.24)

lim xn = x∗ .

(7.25)

n→∞

n→∞

Suppose that n→∞

does not hold. Then there exists a sequence of indices jm → ∞ and an ε1 > 0 such that

7.3 Chaos

123

kxjm − x∗ k > ε1 ,

m = 1, 2, . . . .

(7.26)

Consider a coordinate-wise limit point x∗ = {x∗n }n>−∞ of the sequence of shifted trajectories n o (m) (m) x(m) = x−jm , x−jm +1 , . . . (m)

defined by xn−jm = xn for n = 0, 1, 2, . . .. Then (7.26) implies kx∗0 − x∗ k > ε1 .

(7.27)

Now every sequence x(m) is a trajectory of f , so x∗ is also a trajectory of f because f is continuous. Furthermore, x∗ satisfies the inequalities kx∗n − x∗ k < ξ

(7.28)

for all n because of (7.23) and the second inequality in (7.22). The inequalities (7.28) and (7.27) contradict the ξ-expansivity property, so the limit (7.25) must exist. The other limit follows similarly, which completes the proof. t u Note that only the direct shadowing part of bi-shadowing has been used in the above proof.

7.3.3 Semi-Hyperbolic Lipschitz Mappings From Theorems 7.13 and 6.30 we have the following corollary. Corollary 7.17. Let f be (s, h, δ)-semi-hyperbolic on a compact subset Y of X and x be a homoclinic trajectory of f contained in Y and define k(ε) by (7.21) and γ(ε) by γ(ε) =

1 min{β(s, h, δ), ε/(α(s, h)). 3

(7.29)

Then every mapping g ∈ C satisfying kg − f kC < γ(ε) is (ε, k)-chaotic on a neighbourhood of {x} for any positive integer k ≥ k(ε). From this corollary the chaotic behaviour of nonsmooth perturbations of a diffeomorphism with a hyperbolic homoclinic trajectory can be established, see [7,65]. In fact Theorems 7.13 and 6.30 can be used to consider homoclinic trajectories for much wider classes of systems, such as those with a Marotto snap-back repellor or generalizations thereof [89, 130, 140]. Example 7.18. Let 0 be a hyperbolic fixed point of a smooth mapping f on Rd and let W s and W u denote the local stable and unstable manifolds of f at 0. Suppose that there is a point x0 ∈ W u \ {0} and a positive integer m with f m (x0 ) = 0 and that the linear space Dx0 Txu0 is transversal to T0s , where

124


Dx is the derivative of f (m) at the point x and Txu and Txs are the tangent spaces to the stable and unstable manifolds. Since x0 ∈ W u \ {0}, there exist points x−n , n > 0, with limn→∞ x−n = 0 and f (x−n ) = x−n+1 . Let N be an arbitrary positive integer, write r(N ) =

kx−N k + kx−N +1 k 2

and U (N, ε) = Or(N ) (0)

[

Oε (x−N )

for arbitrary ε > 0. It can be shown that for suitable ε > 0 and integers N, M > 0 the mapping ( f (x), x ∈ Or(N ) (0), FN,M,ε = f N +M (x), x ∈ Oε (x−N ) is semi-hyperbolic on any compact subset of U (N, ε) [47]. Hence the mapping f itself is bi-shadowing and cyclically bi-shadowing on its homoclinic trajectory x = . . . , x−n , . . . , x−1 , x0 , f (x0 ), . . . , f m−1 (x0 ), 0, . . . , 0, . . . . Consequently, for every ε > 0 there exists a γ > 0 such that the chaos threshold of every small continuous perturbation g with kg − f kC < γ does not exceed ε.

7.3.4 Perturbations on Sets of Chain Recurrent Points Another mechanism for generating robust chaotic behaviour in nonsmooth dynamical systems involves its set of chain recurrent points. Let X is an open bounded subset of Rd , let Y be on open subset of X, and let f : X → X. A point x ∈ Y is called (ε, Y )-chain recurrent for f if there exists a finite ε-pseudo-trajectory {x0 , x1 , . . . , xL } of f in Y with x0 = xL = x, that is connecting x with itself. Let CR(f, ε, Y ) denote the set of all (ε, Y )-chain recurrent points of f . The set of Y -chain recurrent points of f is then defined as CR(f, Y ) = ∩ε>0 CR(f, ε, Y ). Note that if f is continuous and Y ⊆ X, then CR(f, Y ) is compact and f (CR(f, Y )) = CR(f, Y ). The following lemma, which is analogous to Smale’s Spectral Decomposition Theorem for Axiom A diffeomorphisms [65], is an important technical step in formulating and proving the main result of this section.

7.3 Chaos

125

Lemma 7.19. Let Y be an open set with Y ⊆ X and let f ∈ Lip(X, Rd ) be bi-shadowing and expansive on CR(f, Y ) with CR(f, Y ) ⊂ Y . Then the set CR(f, Y ) can be decomposed into a disjoint union of a finite number of closed sets Ω1 , . . . , ΩK such that f (Ωi ) = Ωi for i = 1, . . . , K. Moreover, for each i = 1, . . . , K there exists a bi-infinite trajectory z = {zn }n=∞ n=−∞ of f belonging to Ωi such that each of the two semi− 0 trajectories z + = {zn }∞ n=0 and z = {zn }n=−∞ are dense in Ωi ; (ii) each set Ωi can itself be decomposed into a disjoint union of a finite number of closed sets Ωij , j = 1, . . . , ni , which are cyclically permutated under f , such that f ni |Ωi1 is topologically mixing. That is, for any disjoint U , V relatively open in Ωi1 there exists an Ni,U,V such that (f ni )n (U ) ∩ V 6= ∅ for every n ≥ Ni,U,V ; (iii) the periodic points of f |CR(f,Y ) are dense in CR(f, Y ).

(i)

Proof of the first property. Points x, y ∈ CR(f, Y ) are called chain connected in CR(f, Y ) if for any ε > 0 there exist an ε-pseudo-trajectory x of f with x, y ∈ {x} ⊆ CR(f, Y ). The set CR(f, X) is partitioned into equivalence classes of points connected pairwise with each other and every such class E is a closed subset of CR(f, X) satisfying f (E ) = E . These equivalence classes are called the chain connected components of the restriction f |Y . Write E (x) for the chain connected component containing an element x ∈ CR(f, X). The proofs of the next lemma and its corollary are straightforward, and are omitted. Lemma 7.20 (cf. [47, Lemma 4]). For every x ∈ CR(f, Y ) and every ε > 0 there exists a q = q(x, ε) > 0 such that each q-pseudo-cycle x of f with x ∈ {x} ⊆ Y satisfies {x} ⊆ Oε (E (x)) Corollary 7.21. For every x ∈ CR(f, Y ) and q > 0 there exists a q-pseudocycle x of f with x ∈ {x} and {x} ⊆ E (x). A proof of the next lemma is given. Lemma 7.22. Let CR(f, Y ) ⊂ Y and f |Y be (s, h, δ)-semi-hyperbolic on CR(f, Y ) with respect to continuous mappings g : X → X. Let α, β and ξ be the corresponding bi-shadowing and expansivity parameters in Theorem 6.30 and suppose that x and x ˆ are members of distinct chain connected components E and Eb. Then kx − x ˆk ≥ min{β, ξ/α}. Proof. Assume the contrary. In particular, suppose that kx − x ˆk = q < r = min{β, ξ/α}. A contradiction to x and x ˆ belonging to distinct chain connected components is obtained by showing that for each ε > 0 there exists an ε-pseudo-cycle ˆ. For this it suffices to x∗ ⊆ Y which simultaneously contains both x and x construct two trajectories y and z satisfying, respectively,

126


y ⊆Y,

x ∈ {yn : n < 0}, x ˆ ∈ {yn : n > 0},

(7.30)

z ⊆Y,

x ˆ ∈ {zn : n < 0},

(7.31)

x ∈ {zn : n > 0}.

To construct the sequence y fix a positive number ε < r − q. By Corollary 7.21 there exist ε-pseudo-cycles x = {x0 , . . . , xm } ⊆ E (x) and ˆ = {ˆ x x0 , . . . , x ˆm x) satisfying x0 = x and x ˆ0 = x ˆ. Denote by u ˆ } ⊆ E (b ˆ the infinite periodic sequences defined, respectively, by uk = xk for and u k = 0, . . . , m and u ˆk = x ˆk for k = 0, . . . , m. ˆ Then consider the infinite sequence w defined by ( uk , k ≤ 0, (7.32) wk = u ˆk , k > 0. b are both ε-pseudo trajectories and ku0 − u Since u and u ˆ0 k = kx− x ˆk = q, the sequence w in (7.32) is a (q + ε)-pseudo-trajectory of f with w ⊆ CR(f, Y ). As f is bi-shadowing on CR(f, Y ) with the constants α, β and q + ε < r ≤ β, there exists a true trajectory y ∈ Y of f which is α(s, h)q close to w. It remains to prove that this trajectory satisfies the second and the third inclusions of (7.30). Because u−km = x, u ˆk m b for the positive integer k, ˆ = x it suffices to establish the limit relations lim kyn − un k = lim kyn − u bn k = 0,

n→−∞

n→∞

which can be done in the same way as for the limits in (7.24) above using the ξ-expansivity. The trajectory z is obtained similarly by bi-shadowing with respect to the ˆ defined by w pseudo-trajectory w ˆk = w−k for all integers k. t u The chain connected components of CR(f, Y ) are the required sets Ωi . Lemma 7.22 shows that there are only a finite number, while closedness and invariance follow from the definition. It remains to show that each set Ωi contains a dense trajectory. Fix i and consider a countable subset of points {x0 , . . . , xν , . . .} which is dense in Ωi . Fix ε > 0 sufficiently small. By definition, for each ν there exists ε/ν-pseudocycles xν = {xν0 , . . ., xνn(ν)−1 } with xν0 = xν . For each ν adjoin ν + 1 of the xν to form a pseudo-cycle y ν of length (ν + 1)n(ν). Then consider the bi-infinite sequence w of the form w = {. . . , y ν , . . . , y 1 , y 0 , y 1 , . . . , y ν , . . .} By definition this is a bi-infinite ε-pseudo-trajectory of f , so by the bi-shadowing property there exists a unique trajectory z in Ωi which is αε close to w. It is then straightforward to check that z has the necessary properties. t u Proof of the second property. The proof follows the usual manner, see [65] and the references therein, so will only be sketched here. Consider the set Ω1 . The essential step is to construct the subset Ω11 .

7.3 Chaos

127

For each ε > 0 let P (ε) be the set of positive integers p with the property that there exists a ε-pseudo-cycle x belonging to Ω1 of the period p. Define P = ∩ε>0 P (ε), which is non-empty because of cyclic shadowing, and denote by n1 the minimal common denominator of the integers in P . Let x be an arbitrary point in Ω1 . For each ε > 0 let A(ε) denote the set of all ε-pseudo-cycles containing the point x as their first element. Then denote by A∗ (ε) be totality of the elements of pseudo-cycles in A(ε) with indices of the form n1 k for k = 1, 2, . . .. Finally, define Ω11 = ∩A∗ε . This set has all of the required properties. t u Proof of the third property. This follows directly from Theorem 6.30. This completes the proof of Lemma 7.19.

t u t u

Using Lemma 7.19, Theorem 7.13 can be modified to establish the existence and robustness of chaotic behaviour under continuous perturbation of a semi-hyperbolic mapping on components of its chain recurrent set. Let Y be an open set with Y ⊆ X and let f ∈ Lip(X, Rd ) be bi-shadowing and expansive on CR(f, Y ) with CR(f, Y ) ⊂ Y and with spectral decomposition Ωij , j = 1, . . . , ni and i = 1, . . . , K. Denote γ(ε) =

1 min{β, ε/α} 3

where α, β are the bi-shadowing parameters. Property (ii) above implies the existence of a finite integer k(ε) defined by n o 1 1 k(ε) = max m : ∃v, w ∈ Ω11 with (f n1 )(m) Uγ(ε) (v) ∩ Uγ(ε) (w) = ∅ (7.33) where Uγ1 (x) = Uγ (x) ∩ Ω11 . Indeed, given ε > 0, consider a finite γ(ε)/3-net a = {a1 , . . . , aI } in the compact set Ω11 . By the topological mixing property, there then exists a positive integer N such that 1 1 (f n1 )n Uγ(ε)/3 (ai ) ∩ Uγ(ε)/3 (aj ) 6= ∅, i, j = 1, . . . , I, n > N. 1 On the other hand, each set of the form Uγ(ε) (x) for x ∈ Ω11 contains at least 1 one set of the form Uγ(ε)/3 (ai ), i = 1, . . . , I, so this N is a finite upper bound for k(ε) in (7.33).

Theorem 7.23. Let f, Ω11 be as n Lemma 7.19. Then every mapping g ∈ C satisfying kg − f kC < γ(ε) is (ε, k)-chaotic on a neighbourhood of Ω11 for any positive integer k ≥ k(ε). Theorems 7.13 and 7.23 have the following corollary. Corollary 7.24. Let CR(f, Y ) ⊂ Y and let f be (s, h, δ)-semi-hyperbolic on CR(f, Y ) with the spectral decomposition Ωij , j = 1, . . . , ni , and i = 1, . . . , K, where Ω11 contains more than one element. Define γ(ε) by (7.29) and k(ε) by

128


(7.33). Then every mapping g ∈ C satisfying kg −f kC < γ(ε) is (ε, k)-chaotic on a neighbourhood of Ω11 for any positive integer k > k(ε). In particular, the chaos threshold of every continuous mapping g satisfying kg − f kC < γ(ε) does not exceed ε. Example 7.25. Let f be the Smale horseshoe mapping on the square Q = [−1, 1] × [−1, 1] with compression factor 1/5 and expansion factor 5 [65] and let h be a Lipschitz mapping on Q with Lipschitz constant less than 2/3 and with khkC < 1/5. Then the mapping H = f +h is semi-hyperbolic on CR(H, Q) ⊂ Int Q [47]. Note that H need not be invertible on CR(H, Q). Corollary 7.24 is applicable here, so for each ε > 0 the chaos threshold of every sufficiently small continuous perturbation of f does not exceed ε.

Chapter 8

Applications

In this chapter various applications of semi-hyperbolicity and its consequences are considered, in particular in the context of delay differential equations, systems with hysteresis and the numerical approximation of chaotic attractors.

8.1 Semi-Hyperbolic Mappings in Banach Spaces It will be shown here that semi-hyperbolicity of a Lipschitz mapping on a given set implies bi-shadowing for a wide class of dynamical systems in infinite dimensional Banach spaces. Definitions of shadowing and bi-shadowing are given in the next section and that of semi-hyperbolicity for Lipschitz mappings in Section 8.1. The main results are stated in Subsection 8.1.2, an example of its application to delay equation is introduced in Section 8.4. Note that finite dimensional perturbations of infinite-dimensional semi-hyperbolic mappings were also considered in [46].

8.1.1 Bi-Shadowing: Completely Continuous Perturbations Let E be a Banach space with the norm k · k. Consider a mapping f : X → X where X is a subset of E. Definition 8.1. A dynamical system generated by a mapping f : X → X is said to be bi-shadowing with respect to completely continuous perturbations on a subset K of X with positive parameters α and β if for any given finite pseudo-trajectory x = {xn } ∈ Tr(f, K, γ) with 0 ≤ γ ≤ β and any completely continuous mapping ϕ : X → X satisfying

129

130

8 Applications

kϕ − f k∞ ≤ β − γ

(8.1)

there exists a trajectory y = {yn } ∈ Tr(ϕ, X) such that kxn − yn k ≤ α(γ + kϕ − f k∞ )

(8.2)

for all n for which y is defined. Definition 8.1 generalizes in a natural way the notion of bi-shadowing given in Definition 6.19 for mappings in finite-dimensional spaces. As is in finitedimensional case bi-shadowing with respect to completely continuous perturbations conceptualizes the robust relationship between observed dynamical behaviour of a dynamical system and its computer simulations, and can also be interpreted as a form of dynamical structural stability when restricted to specific classes of mappings, such as continuous mappings. A counterpart of bi-shadowing with respect to completely continuous perturbations for cycles and pseudo-cycles is also useful. Definition 8.2. A dynamical system generated by a mapping f : X → X is said to be cyclically bi-shadowing with respect to completely continuous perturbations on a subset K of X with positive parameters α and β if for any given pseudo-cycle x ∈ Per(f, K, γ) with 0 ≤ γ ≤ β and any completely continuous mapping ϕ : X → X satisfying (8.1) there exists a proper cycle y ∈ Per(ϕ, X) of period N equal to that of x such that (8.2) holds for n = 0, 1, . . . , N . Note that the cycle y in Definition 8.2 is required only to be in X rather than in the subset K. Finally, a generalization of the notion of semi-hyperbolicity for Lipschitz mappings acting in a Banach space is required. Definition 8.3. Let s = (λs , λu , µs , µu ) be a split and K a subset of an open set X ⊆ Rd . A Lipschitz mapping f : X → X is said to be s-semi-hyperbolic on K if there exist positive real numbers h, δ such that for each x ∈ K there exists a splitting E = Exs ⊕ Exu with corresponding projectors Pxs and Pxu satisfying Properties SH0(Lip)–SH2(Lip) from Definition 3.6: SH0(Lip): The space Exu is finite dimensional for all x and dim Exu = dim Eyu if x, y ∈ K with kf (x) − yk ≤ δ; SH1(Lip): supx∈K {kPxs k, kPxu k} ≤ h; SH2(Lip): The inclusion x+u+v ∈X (8.3) and the inequalities

8.1 Semi-Hyperbolic Mappings in Banach Spaces

131

kPys (f (x + u + v) − f (x + u ˜ + v)) k ≤ λs ku − u ˜k,

(8.4)

kPys kPyu kPyu


(8.5)


(8.6)

(f (x + u + v) − f (x + u + v˜)) k ≥ λu kv − v˜k

(8.7)

hold for all x, y ∈ K with kf (x) − yk ≤ δ and all u, u ˜ ∈ Exs and v, v˜ ∈ Exu such that kuk, k˜ uk, kvk, k˜ v k ≤ δ. Note that continuity in x of the splitting subspaces Exs , Exu or of the projectors Pxs , Pxu is not assumed here, nor is invariance of the splitting subspaces, as is the case in the definition of hyperbolicity of a diffeomorphism.

8.1.2 Main Results The main result of this section is that semi-hyperbolicity is sufficient to ensure bi-shadowing of a dynamical system generated by a Lipschitz mapping with respect to perturbed systems generated by completely continuous mappings. Theorem 8.4. Let f : X → X be a Lipschitz mapping which is s-semi-hyperbolic on a subset K of X with constants h, δ. Then it is bi-shadowing on K, with respect to completely continuous mappings ϕ : X → X with parameters α(s, h) = h

λu − λs + µs + µu (1 − λs )(λu − 1) − µs µu

(8.8)

and

(1 − λs )(λu − 1) − µs µu . (8.9) max{λu − 1 + µs , 1 − λs + µu } The proof of the theorem is not dissimilar to that of a finite-dimensional analogue, Theorem 6.21. Nevertheless, to make the presentation selfsufficient and to highlight some differences from the finite-dimensional case, a complete proof is given in the next section. Cyclic bi-shadowing is also a consequence of semi-hyperbolicity. β(s, h, δ) = δh−1

Theorem 8.5. Let f : X → X be a Lipschitz mapping which is s-semihyperbolic on a subset K of X with constants h, δ. Then it is cyclically bishadowing on K with parameters α(s, h) and β(s, h) given by (8.8) and (8.9), with respect to completely continuous mappings ϕ : X → X.

8.1.3 Proofs For a given split s = (λs , λu , µs , µu ) and semi-hyperbolicity constant h, define the matrix

132

8 Applications

M (s) =

λs µs µu /λu 1/λu

(8.10)

and a two-dimensional vector a = (a, b)T by a = (I − M (s))−1 h,

h = h(1, 1/λu )T .

(8.11)

Then the bi-shadowing constants (8.8) and (8.9) satisfy α(s, h) = a + b, β(s, h, δ) = δ min a−1 , b−1 .

(8.12)

where

As in the finite-dimensional case, the proof proceeds through a series of Lemmas. Denote by Bx (r) the closed ball centered at 0 of the radius r in the linear space Exu . For each x ∈ K and each z ∈ Rd satisfying kPxs zk ≤ δ define the finite-dimensional mapping Fx,z : Bx (δ) → Efu(x) by Fx,z (v) = Pfu(x) (f (x + Pxs z + v) − f (x + Pxs z)). Lemma 8.6. Let 0 ≤ r ≤ δ. Then Fx,z (Bx (r)) ⊇ Bf (x) (λu r).

(8.13)

Proof. Since equality of singleton sets occurs when r = 0, let r > 0. Respectively, let ∂Bx (r) and Bxo (r) denote the boundary and the interior of Bx (r). Clearly, Fx,z (0) = Pfu(x) (f (x + Pxs z) − f (x + Pxs z)) = 0 ∈ Bfo(x) (λu r); on the other hand, by the inequality (8.6) \ Fx,z (∂Bx (r)) Bfo(x) (λu r) = ∅.

(8.14)

(8.15)

By Condition SH0(Lip) of Definition 8.3, and the Principle of Domain Invariance (see, [5, p. 396]), (8.15) implies that \ ∂Fx,z (Bx (r)) Bfo(x) (λu r) = ∅. This, together with (8.14), implies (8.13) and the lemma is proved. This lemma and inequality (8.6) immediately give −1 Lemma 8.7. The operator Qx,z = Fx,z is defined and continuous on the ball Bf (x) (λu δ) and satisfies the estimate kQx,z (v)k ≤ λ−1 u kvk.

Denote by ZN the space of N + 1-tuples z = (z0 , z1 , . . . , zN ), zn ∈ E, n = 0, 1, . . . , N . The set ZN can be treated as the Banach space E N +1 with norm kzkE N +1 = max kzn k. 0≤n≤N


133

Let x = {x0 , x1 , . . . , xN } be a given γ-pseudo-trajectory of the system f . Let ϕ be a given completely continuous mapping. In what follows, suppose that the parameter β = γ + kf − ϕk∞ (8.16) satisfies β ≤ β(s, h, δ).

(8.17)

Define the operator H : ZN → ZN , which transforms z = (z0 , z1 , . . . , zN ) ∈ ZN into

H (z) = w = (w0 , w1 , . . . , wN ) ∈ ZN

defined by the relations Pxs0 w0 = 0 and Pxsn wn = Pxsn (ϕ(xn−1 + zn−1 ) − xn )

(8.18)

for n = 1, 2, . . . , N , and the relations PxuN wN = 0 and Pxun−1 wn−1 = Qxn−1 ,zn−1 (Pxun (−ϕ(xn−1 + zn−1 ) + f (xn−1 + zn−1 ) + xn − f (xn−1 + Pxsn−1 zn−1 ) + zn ))

(8.19)

for n = 1, 2, . . . , N . Consider the set S (β) = {z ∈ ZN : kPxsn zn k ≤ aβ, kPxun zn k ≤ bβ, n = 0, 1, . . . , N }. (8.20) By (8.12) and (8.17) aβ ≤ aβ(s, h, δ) ≤ δ,

bβ ≤ bβ(s, h, δ) ≤ δ

and so by (8.3) trajectories from S (β) belong to X. Lemma 8.8. The following statements are valid: The operator H is defined and completely continuous for z belonging to the set S (β). (ii) For any fixed point z = (z0 , z1 , . . . , zN ) ∈ S (β) of H , the sequence (i)

y = {x0 + z0 , x1 + z1 , . . . , xN + zN }

134

8 Applications

is a trajectory of the system ϕ. Proof. (i) Clearly, by (8.17) the right hand side of (8.18) is defined and completely continuously depends on z ∈ S (β). So we need only prove that for any n = 1, 2, . . . , N the right hand side of (8.19), which is finite-dimensional, is defined and continuous for z ∈ S (β). By Lemma 8.7 it is sufficient to establish that kPxun (−ϕ(xn−1 + zn−1 ) + f (xn−1 + zn−1 ) + xn − f (xn−1 + Pxsn−1 zn−1 ) + zn )k ≤ λu δ. Rewrite the last inequality in the form kJ1 + J2 + J3 k ≤ λu δ, where J1 = Pxun (−ϕ(xn−1 + zn−1 ) + f (xn−1 + zn−1 ) + xn − f (xn−1 )), J2 = J3 =

Pxun (f (xn−1 ) Pxun zn .

− f (xn−1 +

Pxsn−1 zn−1 )),

(8.21) (8.22) (8.23)

It suffices to find bounds for kJ1 k, kJ2 k and kJ3 k. For kJ1 k, note that by (8.16) kϕ − f k∞ ≤ β − γ, kxn − f (xn−1 )k ≤ γ, so by Condition SH1(Lip) of Definition 8.3 kJ1 k ≤ βh.

(8.24)

kJ2 k ≤ µu kPxsn−1 zn−1 k.

(8.25)

kJ3 k = kPxun zn k.

(8.26)

From the inequality (8.6),

Clearly, On the other hand, z ∈ S (β) implies that kPxsn−1 zn−1 k ≤ βa,

kPxun zn k ≤ βb.

(8.27)

From by (8.25)–(8.27) kJ1 + J2 + J3 k ≤ kJ1 k + kJ2 k + kJ3 k ≤ β(h + aµu + b) . But β(h + aµu + b) ≤ λu δ. To see this, from (8.10), (8.11)

(8.28)


135

λu − 1 + µs , (1 − λs )(λu − 1) − µs µu 1 − λs + µu b=h . (1 − λs )(λu − 1) − µs µu

a=h

and put h + aµu + b = λu b, rewriting (8.28) as βλu b ≤ λu δ

(8.29)

But then (8.29) follows from (8.12) and (i) is proved. (ii) It is sufficient to establish that xn + zn = ϕ(xn−1 + zn−1 ),

n = 1, 2, . . . , N.

(8.30)

Because z is a fixed point of H , equations (8.18) and (8.19) can be rewritten as Pxsn zn = Pxsn (ϕ(xn−1 + zn−1 ) − xn ), (8.31) and Pxun−1 zn−1 = Qxn−1 ,zn−1 (Pxun (−ϕ(xn−1 + zn−1 ) + f (xn−1 + zn−1 ) + xn − f (xn−1 + Pxsn−1 zn ) + zn )). (8.32) From (8.31) it follows that Pxsn (xn + zn ) = Pxsn ϕ(xn−1 + zn−1 ).

(8.33)

Apply the nonlinear, finite-dimensional operator Fxn−1 ,zn−1 = Q−1 xn−1 ,zn−1 to both sides of (8.32), to obtain Pxun f (xn−1 + zn−1 ) − f (xn−1 + Pxsn−1 zn−1 ) = Pxun ((xn + zn − ϕ(xn−1 + zn−1 )) +(f (xn−1 + zn−1 ) − f (xn−1 + P s xn−1 zn−1 ))) which, simplifying, gives 0 = Pxun (zn − ϕ(xn−1 + zn−1 ) + xn ). That is, Pxun (xn + zn ) = Pxun ϕ(xn−1 + zn−1 ) and (8.30) follows from (8.33) and (8.34). The lemma is proved. Lemma 8.9. The set S (β) is invariant for H . Proof. First, rewrite (8.18) in the form Pxsn wn = I1 + I2 ,

(8.34) t u

136

8 Applications

where I1 = Pxsn ((ϕ(xn−1 + zn−1 ) − f (xn−1 + zn−1 )) + (f (xn−1 ) − xn )) , I2 = Pxsn (f (xn−1 + zn−1 ) − f (xn−1 )) . Similarly, rewrite (8.19) in the form Pxun−1 wn−1 = Qxn−1 ,zn−1 (J1 + J2 + J3 ) where J1 , J2 , J3 are defined in (8.21). To estimate kI1 k note that by (8.16) kϕ − f k∞ ≤ β − γ,

kxn − f (xn−1 )k ≤ γ,

so by Condition SH1(Lip) of Definition 8.3 kI1 k ≤ hβ.

(8.35)

kI2 k ≤ λs kPxsn−1 zn−1 k + µs kPxun−1 zn−1 k.

(8.36)

By (8.4), (8.5)

Now for each z ∈ ZN define the pair of real nonnegative numbers ms (z) = max kPxsn zn k, 0≤n≤N

mu (z) = max kPxun zn k 0≤n≤N

and denote by m(z) the two-dimensional column vector with coordinates ms (z), mu (z). From the bounds (8.35), (8.36) and definition (8.20) of S (β), it follows that ms (H z) = ms (w) ≤ βh + βλs a + βµs b.

(8.37)

Similarly, from (8.24)–(8.26), the definition (8.20) of S (β) and Lemma 8.7, it follows that mu (H z) = mu (w) ≤ λ−1 u (βµu a + βb + βh).

(8.38)

The inequalities (8.37), (8.38) are equivalent to the coordinate-wise bound m(H z) = m(w) ≤ βM (s)a + βk,

z ∈ S (β).

(8.39)

In view of (8.11) βM (s)a + βk = β(M (s)(I − M (s))−1 + I)k = β(I − M (s))−1 k = βa. Consequently, (8.39) is equivalent to m(H z) = m(w) ≤ βa,

z ∈ S (β),

8.2 Nonautonomous Difference Equations

137

which means that the set S (β) is invariant for H .

t u

The proof of Theorem 8.4 is completed as follows. From (i) of Lemma 8.8, the operator H is completely continuous on the convex set S (β) and, in view of Lemma 8.9, H (S (β)) ⊆ S(β). So by the Schauder Fixed Point Theorem, H has a fixed point z = (z0 , z1 , . . . , zN ) ∈ S (β) and hence, by (ii) of Lemma 8.8, the sequence x∗ = {x0 + z0 , x1 + z1 , . . . , xN + zN } is a trajectory of the system ϕ. By definition (8.20) of the set S (β), kzn k ≤ kPxsn zn k + kPxun zn k ≤ (a + b)β,

n = 0, 1, . . . , N,

and thus by (8.12), (8.16), kzn k ≤ α(s, k)(γ + kϕ − f k∞ ),

n = 0, 1, . . . , N.

That is to say kxn − x∗n k ≤ α(s, k)(γ + kϕ − f k∞ ),

n = 0, 1, . . . , N

and Theorem 8.4 is proved.

t u

8.2 Nonautonomous Difference Equations Let P be a nonempty space and let θy : P → P be mapping, so that {θn : n ∈ Z+ } is a semi-group under composition. No assumption is made about the domain P , nor properties of the mapping p 7→ θp, but in applications P is often either a metric space (deterministic systems) or a measure space (random systems), with p 7→ θp being continuous in the former and measurable in the latter. In addition, let X = {(Xp , k · kp ) : p ∈ P } be a family of Banach spaces (possibly subspaces of a common space) and let f = {(f (p, ·) : p ∈ P } be a family of mappings f (p, ·) : Xp → Xθp ,

p ∈ P,

which are at least continuous in x. Consider the nonautonomous difference equation xn+1 = f (θn p, xn ),

(8.40)

138

8 Applications

on X with xn ∈ Xθn p for each n ∈ Z+ . This generates a discrete-time nonautonomous semi-dynamical on X with the autonomous semi-dynamical system on P generated by θ as its driving system and the associated cocycle mapping on X defined by φ(0, p, x) := {x}

and φ(n, p, x) := f (θn p, ·) ◦ · · · ◦ f (p, x)

for all

n ≥ 1,

where φ(n, p, ·) : Xp → Xθn p , see [70, 71]. A simple example is a sequence of mappings fn on Rd with P = Z, f (n, ·) = fn (·) and θ the left shift on Z+ . A nontrivial example arises when the sequence of mappings fn on Rd is chosen periodically or, perhaps less regularly, from a finite family of mappings {g1 , . . . , gr }, that is with fn = gin where the in ∈ {1, . . . , r}. The nonautonomous difference equation xn+1 = gin (xn )

(8.41) +

on Rd can be rewritten in the form (8.40) with the set P = {1, . . . , r}Z of infinite sequences p = {in : n ∈ Z+ } with in ∈ {1, . . . , r} and the left shift operator θ defined on P by θ(in ) = in+1 . The space P here may be given the structure of a compact metric space with metric d (p, p0 ) =

∞ X

(r + 1)−|n| |in − i0n | .

(8.42)

n=0

An important application is a variable time-step method for an autonomous differential equation x˙ = f (x), such as the Euler method xn+1 = xn + hn f (xn ), where the variable time steps hn > 0, in practice, come from a finite set, so gin (x) = x + hn f (x) in (8.41) and the sequence space metric is as in (8.42). More generally, a variable parameter q ∈ Q may be involved in a difference equation xn+1 = f (xn , qn ), where qn ∈ Q. For example, let f (x, q) =

|x| + q 2 , 1+q

x ∈ R, Q = [0.5, 1]. +

Here, the sequence space is P = [0.5, 1]Z of infinite sequences p = {qn : n ∈ Z+ }, qn ∈ Q, which is a compact metric space with the metric d(p, p0 ) =

∞ X n=0

2−|n| |qn − qn0 |.


139

A simple example indicating the usefulness of different spaces Xp is given in the next section.

8.2.1 Semi-Hyperbolicity As in the autonomous case, a 4-tuple of real numbers called a split measures the expansion, contraction and other rates in the definition of a nonautonomous semi-hyperbolic system (8.40). Let f = {(f (p, ·) : p ∈ P } a family of mappings with f (p, ·) : Xp → Xθp for p ∈ P and let K = {Kp : p ∈ P } be a family of closed and bounded subsets with Kp ⊂ Xp and f (p, Kp ) ∩ Kθp 6= ∅ for each p ∈ P . Definition 8.10. A splitting Xp = Exs (p) ⊕ Exu (p) of a family of Banach spaces X = {(Xp , k · kp ) : p ∈ P } with projectors Pxs (p) : Xp → Exs (p) and Pxu (p) : Xp → Exu (p) for each x ∈ Xp such that Pxs (p)Xp = Exs (p),

Pxs (p)Exu (p) = 0,

Pxu (p)Xp = Exu (p),

Pxu (p)Exs (p) = 0,

is said to be uniform on the family K = {Kp : p ∈ P } of closed and bounded subsets with respect to the family of mappings f = {(f (p, ·) : p ∈ P } on X with f (p, Kp ) ∩ Kθp 6= ∅ for each p ∈ P if there exist positive real numbers δ, h such that SH0: dim Eyu (θp) = dim Exu (p) for all x ∈ Kp and y ∈ Kθp such that kf (p, x) − y)kθp ≤ δ for every p ∈ P ; SH1: supp∈P supx∈Kp {kPxu (p)kp , kPxs (p)kp } ≤ h for every p ∈ P . The continuity in x of the splitting subspaces Exs (p), Exu (p) or of the projectors Pxs (p), Pxu (p) is not assumed in the above definition, nor is the invariance of the subsets Kp ∈ K = {Kp : p ∈ P } with respect to the corresponding mappings f (p, ·) ∈ f = {(f (p, ·) : p ∈ P }. That is, neither Kθp = f (p, Kp ) nor Kθp ⊆ f (p, Kp ) has to hold for any p ∈ P . This allows considerably more flexibility in applications, see for example [43, 46]. Definition 8.11. A family of mappings f = {f (p, ·) : p ∈ P } on a family of Banach spaces X = {Xp : p ∈ P } is said to be uniformly s-semi-hyperbolic with a split s = (λs , λu , µs , µu ) on a family K = {Kp : p ∈ P } of nonempty closed and bounded subsets of X such that f (p, Kp ) ∩ Kθp 6= ∅ for all p ∈ P , if there exists a uniform splitting Xp = Exs (p) ⊕ Exu (p) of X with projectors Pxs (p) and Pxu (p) for each x ∈ Kp and p ∈ P (that is satisfying conditions SH0 and SH1) such that SH2:

for each p ∈ P the inclusion x + u + v ∈ Xp and the inequalities

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8 Applications

kPys (θp) [f (p, x + u + v) − f (p, x + u ˜ + v)] kθp ≤ λs ku − u ˜kp , kPys (θp) [f (p, x + u + v) − f (p, x + u + v˜)] kθp ≤ µs kv − v˜kp , kPyu (θp) [f (p, x + u + v) − f (p, x + u ˜ + v)] kθp ≤ µu ku − u ˜kp , kPyu (θp) [f (p, x + u + v) − f (p, x + u + v˜)] kθp ≥ λu kv − v˜kp . hold for all x ∈ Kp and y ∈ Kθp with kf (p, x) − y)kθp ≤ δ and all u, u ˜ ∈ Exs (p) and v, v˜ ∈ Exu (p) with kukp , k˜ ukp , kv|p , k˜ v kp ≤ δ A simple example can be constructed in terms of a sequence of linear mappings fn (x) = An x in Rd with hyperbolic matrices An for which the unstable linear spaces all have the same dimension. This can be put into formalism with P = Z and θ the left shift on Z+ with all Xn = Rd . If the unstable subspaces are each mapped onto their successor, a split with µs = µu = 0 can be used. To see why it may be convenient to distinguish the elements Xp of X, consider a real symmetric 2 × 2 hyperbolic matrix A0 and a 2 × 2 rotation matrix R by π/2 and consider a nonautonomous linear difference equation with matrices An = Rn A0 for n ∈ Z+ , so P = Z+ . Suppose that the space R2 has a fixed coordinate system and define Xn := Rn R2 , so Xn+1 = RXn . Note that the stable and unstable manifolds would be mapped one into the other if the spaces were not also rotated. Here a split with µs = µu = 0 is also appropriate. This example is, of course, trivial — one could avoid the rotations altogether and consider simply the autonomous linear difference equation with matrix A0 . A less trivial example will be given in Sect. 8.2.3. As above, let K = {Kp : p ∈ P } be a family of nonempty closed and bounded subsets with Kp ⊂ Xp for each p ∈ P such that f (p, Kp ) ∩ Kθp 6= ∅, p ∈ P , for a family of mappings f = {f (p, ·) : p ∈ P } with f (p, ·) : Xp → Xθp , p ∈ P. Definition 8.12. A trajectory of a family of mappings f on X is a sequence x = {xn } with xn ∈ Xθn p satisfying xn+1 = f (θn p, xn ) . A γ-pseudo-trajectory of a family of mappings f on X is a sequence y = {yn } with yn ∈ Xθn p satisfying kyn+1 − f (θn p, yn )kθn+1 ≤ γ,

γ > 0,

for n = −n− , . . . , 0, . . . , n+ where n± ≤ ∞. The qualifier finite may be used when n± < ∞ and infinite otherwise. Given a finite or infinite interval I of integers Z, let TrI (f, K, γ) denote the totality of corresponding γ-pseudo-trajectories of f in K. Since a true trajectory can be regarded as a γ-pseudo-trajectory for any γ ≥ 0, in particular with γ = 0, denote the corresponding set of true trajectories by TrI (f, K, 0)


141

or simply by TrI (f, K). Obviously TrI (f, K) ⊂ TrI (f, K, γ), with strict inclusion since there are γ-pseudo-trajectories which are not trajectories. Let g = {g(p, ·) : p ∈ P } with g(p, ·) : Xp → Xθp for each p ∈ P . The semi-norm kg − fk∞ = sup sup kg(p, x) − f (p, x)kθp p∈P x∈Xp

gives a notion of distance between families g and f on X. Definition 8.13. A nonautonomous semi-dynamical system generated by a family of Lipschitz mappings f = {f (p, ·) : p ∈ P } with f (p, ·) : Xp → Xθp , such that f (p, ·) : Xp → Xθp for a family K = {Kp : p ∈ P } of nonempty closed and bounded subsets with Kp ⊂ Xp and f (p, Kp ) ∩ Kθp 6= ∅, p ∈ P , is said to be bi-shadowing with respect to completely continuous perturbations on K with positive parameters α and β, if for any given finite pseudo-trajectory x = {xn } ∈ TrI (f, K, γ) with 0 ≤ γ ≤ β and any family of completely continuous mappings g = {g(p, ·) : p ∈ P } with g(p, ·) : Xp → Xθp , p ∈ P , satisfying kg − fk∞ ≤ β − γ, there exists a trajectory y = {yn } ∈ TrI (g, X) such that kxn − yn kp ≤ α(γ + kg − fk∞ ) = αβ for all n for which y is defined. If the spaces Xp are finite-dimensional of like dimension, then the requirement that the sets Kp are closed and bounded is equivalent to compactness. The condition that the perturbed mappings g are completely continuous, appearing in the definition of bi-shadowing is then equivalent to continuity.

8.2.2 Semi-Hyperbolicity Implies Bi-Shadowing The main result of this section is that semi-hyperbolicity is sufficient to ensure bi-shadowing of a nonautonomous semi-dynamical system generated by a family f of Lipschitz mapping with respect to a class of perturbed systems generated by families g of continuous mappings. As with the corresponding result for autonomous systems, which it generalizes, it also provides explicit values of the shadowing parameters. Theorem 8.14. Let K = {Kp : p ∈ P } be a family of closed and bounded subsets of X = {Xp : p ∈ P } for a family f = {f (p, ·) : p ∈ P } of Lipschitz mappings with f (p, ·) : Xp → Xθp , p ∈ P , which is semi-hyperbolic on K with a uniform split s and constants h, δ. Then, the nonautonomous semidynamcial system generated by f is bi-shadowing on K with respect to any

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8 Applications

family g = {g(p, ·) : p ∈ P } of completely continuous mappings g(p, ·) : Xp → Xθp , p ∈ P , with parameters α = α(s, h) and β = β(s, h, δ) defined by α(s, h) := h and β(s, h, δ) := δh−1

λu − λs + µs + µu (1 − λs )(λu − 1) − µs µu

(1 − λs )(λu − 1) − µs µu . max{λu − 1 + µs , 1 − λs + µu }

Moreover, if each set Kp is compact, then the bi-shadowing holds with respect to trajectories on infinite intervals. The proof follows very closely those given in [43, 47] for the autonomous case, noting only that all estimates hold uniformly in p ∈ P , and will not be repeated here. A particular strength of the result is that it does not require smooth pertubations and is thus applicable to, for example, perturbations due to hysteresis effects [46]. It also provides explicit error bounds. An important application is when a variable time-step numerical method such as the Euler method is applied to an autonomous differential equation. Due to the discreteness of computer arithmetic, which causes round-off error, an approximation of the Euler mapping is in fact computed [41, 43]. The bi-shadowing result ensures that the computed dynamics reflects that of the Euler method.

8.2.3 Twisted Horseshoe Mapping The twisted horseshoe mapping f = (f1 , f2 ) on I 2 = [0, 1] × [0, 1] defined by ( 2x, 0 ≤ x ≤ 12 , x y 1 f2 (x, y) = + + f1 (x, y) = 1 2 10 4 2 − 2x, 2 < x ≤ 1, has been investigated by Guckenheimer, Oster and Ipaktchi [57]. It has a fixed point 2 35 (¯ x, y¯) = , , 3 54 1 . All the conditions of which is a saddle point with eigenvalues −2 and 10 a theorem in [71] which generalizes Marotto’s snap-back repeller result to saddle points are thus satisfied, so the twisted horseshoe mapping behaves chaotically in a slight generalization of the sense of Li and Yorke (see [68]). In fact, the twisted-horseshoe mapping also satisfies the Sharkovsky cycle co-existence ordering, due to a generalization in [70] of Sharkovsky’s theorem to triangular shaped mappings like the twisted horseshoe mapping.


143

The autonomous difference equation generated by the twisted horseshoe mapping can be interpreted as a nonautonomous difference equation xn+1 =

pn 1 xn + + 10 2 4

(8.43)

driven by the semi-dynamical system on [0, 1] ( 2pn , 0 ≤ pn ≤ 12 , pn+1 = 2 − 2pn , 21 < pn ≤ 1.

(8.44)

Note that any trajectory of the driving system can be extended backwards indefinitely, though typically not uniquely. Let p = {pn : n ∈ Z} be such an entire trajectory and hold it fixed. Then the explicit solution of (8.43) given by n−1 n−1 1 X j+1−n 1 X j+1−n xn = 10−n+n0 xn0 + 10 pj + 10 . 2 j=n 4 j=n 0

0

Let n fixed and let the xn0 take values in a fixed bounded subset of R. Then, taking the pullback limit as n0 → −∞ (see [72, 73]), gives x ¯n =

n−1 n−1 1 X 1 X 10j+1−n pj + 10j+1−n , 2 j=−∞ 4 j=−∞

that is, x ¯n =

n−1 1 X 10 10j+1−n pj + . 2 j=−∞ 36

In particular, for two different initial values xn0 and x0n0 , with the same driving sequence p, it is easily shown that |xn − x0n | = |xn0 − x0n0 |10−n+n0 → 0

as

n → ∞,

and hence, with x0n ≡ x ¯n , |xn − x ¯n | → 0

as

n → ∞.

Thus all solutions of (8.43) with different starting points converge to the same limit if they correspond to the same driving sequence. Define x ¯(p) := x ¯0 and let θ be the shift operator on the space of sequences Z+ ¯n = x ¯(θn p) and the singleton subsets Ap := {¯ x(p)} P = [0, 1] . Then x for the pullback attractor (which is also forwards attracting) of the nonautonomous dynamical system generated by the twisted horseshoe system. Let Xp be the space R with the origin shifted to x ¯(p). With the change of coordinates zn := xn − x ¯(p) the mapping (8.43) simplifies to

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8 Applications

zn+1 =

1 zn , 10

with zn ∈ Xθn p and zn+1 ∈ Xθn+1 p . Semi-hyperbolicity on Kp ≡ Xp for the split s = (10−1 , 0, 0, 0) and trivial unstable subspaces follows immediately. The bi-shadowing theorem 8.14 is applicable to this NDS with continuous perturbations of the linear mapping (8.43). The bi-shadowing theorem 8.14 is also applicable to the twisted horseshoe mapping in both variables on the product space R2 with P = Z+ . This allows for shadowing with perturbations of both the linear mapping (8.43) and the tent-mapping (8.44).

8.3 Chaotic Attractors under Discretization The solutions φ(t; x0 ) with initial values φ(0; x0 ) = x0 of an ordinary differential equation dx = f (x) (8.45) dt on Rd generate a continuous-time dynamical system on Rd , which is a flow {φ(t : ·)} of diffeomorphisms on Rd if f is smooth enough. It is well-known (cf. [74,134]) that if this system has a compact attractor A0 , then the discretetime dynamical system generated by a one-step numerical scheme xn+1 = Fh (xn ) := xn + hf (h; xn )

(8.46)

with constant time-step h > 0, such as an Euler or Runge–Kutta scheme, also has a compact attractor Ah such that dist(Ah , A0 ) → 0

as

h → 0+,

(8.47)

where dist is the Hausdorff semi-distance of nonempty compact subsets of Rd defined by dist(A, B) = max dist(a, B) = max min ka − bk. a∈A

a∈A b∈B

Convergence (8.47) is often described as the upper semi-continuity of the numerical attractor Ah at h = 0. In general, a numerical attractor Ah need not be lower semi-continuity under time-discretization at h = 0, that is with dist(A0 , Ah ) → 0

as

h → 0+,

(8.48)

unless additional assumptions are made about the dynamics inside the attractor A0 . Here it will be assumed that A0 is a hyperbolic invariant set

8.3 Chaotic Attractors under Discretization

145

for the flow generated by the differential equation (8.45) containing a point x ¯0 ∈ A0 for which the ω-limit set ω + (¯ x0 ) = A0 , which is typical of many chaotic attractors, where ω + (¯ x0 ) =

\ [

{φ(s; x ¯0 }.

(8.49)

T ≥0 s≥T

The main technical tool to be used in the proof is the bi-shadowing property satisfied by the time-1 map φ(1 : ·) of the continuous-time system on the hyperbolic set A0 . Recall that a one-step numerical scheme (8.46) with time-step h > 0 is of pth order if for every bounded subset D of Rd there exists a constant CD such that its one-step discretization error satisfies the inequality kφ(h; x0 ) − Fh (x0 )k ≤ CD hp+1 for all x0 ∈ D and h > 0. There then exists a constant γ > 0 such that the global discretization error satisfies the inequality kφ(nh; x0 ) − Fhn (x0 )k ≤ CD eγnh hp

(8.50)

for all x0 ∈ D, h > 0 and n = 0, 1, . . . , ND,h such that φ(t; x0 ) ∈ D for 0 ≤ t ≤ hND,h . Theorem 8.15. Suppose that the mapping f : Rd → Rd in the differential equation (8.45) is p + 1 times continuously differentiable and that the onestep numerical scheme Fh : Rd → Rd (8.46) is of pth order. If the differential equation (8.45) has a hyperbolic attractor A0 which contains a dense recurrent trajectory, then the numerical scheme (8.46) has an attractor Ah which is continuous in h at h = 0, that is H (Ah , A0 ) = max {dist(Ah , A0 ), dist(A0 , Ah )} → 0

as

h→0+.

Proof. By the assumed existence of the attractor A0 there is an open bounded subset X of Rd which is positively invariant for the flow φ generated by the differential equation (8.45) and attention can be restricted to this set without loss of generality. The results of [58, 74, 134] then establish the existence of a numerical attractor Ah of the numerical scheme (8.46) for sufficiently small h > 0 which are upper semi-continuous in h at h = 0 in the sense of convergence (8.47). Hence it remains only to show the lower semi-continuity of the Ah at h = 0. Write Φ(x) := φ(1; x) for the time-1 mapping of the differential equation (8.45), so Φ : X → X . Note that Φ is a diffeomorphism on Φ(X ) and that the original attractor A0 ⊂ Φ(X ) is also an attractor for the discretetime dynamical system generated by Φ, and hence A0 is invariant under Φ. Moreover, the diffeomorphism Φ inherits the hyperbolicity of the flow φ on A0 . Hence by Theorem 6.19 there exist positive constants α and β such that

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8 Applications

the diffeomorphism Φ is bi-shadowing with parameters α and β on A0 with respect to the space of continuous functions C (X ). It can be assumed without loss of generality that mapping Fh in the numerical scheme (8.46) satisfies Fh (X ) ⊆ X for sufficiently small h, in particular for h ≤ 1. The Fhn ∈ C (X ) for n = 1, 2, . . . and by (8.50) sup kφ(nh; x) − Fhn (x)k ≤ CX eγnh hp

x∈X

n = 1, 2, . . .. Suppose that CX eγ hp ≤ β and that there is an integer Nh such that h = 1/Nh , that is Nh h = 1. Then

N N (8.51)

Φ − Fh h := sup Φ(x) − Fh h (x) ≤ CX eγ hp ≤ β. ∞

x∈X

Now let x ¯0 ∈ A0 be any point such that ω + (¯ x0 ) = A0 and define x ¯0 (t) = φ(t, x ¯0 ) for each t ∈ [0, 1]. For each such t consider the trajectory {Φn (¯ x0 (t))}n∈N of the mapping Φ. By the bi-shadowing property of Φ in A0 with δ = 0 there then exists a y¯0 (t) ∈ X such that

n

x0 (t)) − FhnNh (¯ y0 (t)) ≤ α Φ − FhNh ≤ αCX eγ hp (8.52)

Φ (¯ ∞

for all n = 0, 1, . . .. In particular, for n = 0 we have k¯ x0 (t) − y¯0 (t)k ≤ αCX eγ hp , that is

y¯0 (t) ∈ CαCX eγ hp ({¯ x0 (t)}) ⊂ CαCX eγ hp (A0 )

where Cρ (S) is the closed ball of radius ρ about a subset S of Rd . Since the points y¯0 (t) for 0 ≤ t ≤ 1 belong to a common bounded subset of Rd , by the absorbing property of the (maximal) numerical attractor Ah for each ε > 0 there exists an integer N (ε, h) such that yj (t) := Fhj (¯ y0 (t)) ∈ Cε/2 (Ah )

for all j ≥ N (ε, h), 0 ≤ t ≤ 1.

But, from (8.52), kΦn (¯ x0 (t)) − ynNh (t)k ≤ αCX eγ hp , that is Φn (¯ x0 (t)) ∈ CαCX eγ hp ({ynNh (t)})

for all n ≥ 0, 0 ≤ t ≤ 1.

Hence Φn (¯ x0 (t)) ∈ CαCX eγ hp +ε/2 (Ah ) for all n ≥ N (ε, h)/Nh and 0 ≤ t ≤ 1, or equivalently [ {Φn (¯ x0 (t))} ⊆ CαCX eγ hp +ε/2 (Ah ) . (8.53) n≥N (ε,h)/Nh , 0≤t≤1

Now the group property of the flow φ gives

8.3 Chaotic Attractors under Discretization

147

Φn (¯ x0 (t)) = φ(n, x ¯0 (t)) = φ(n, φ(t, x ¯0 (t))) = φ(n + t, x ¯0 ), so

[

[

{Φn (¯ x0 (t))} =

and hence from (8.53) [

{φ(s, x ¯0 )}

s≥N (ε,h)/Nh

n≥N (ε,h)/Nh , 0≤t≤1

{φ(s, x ¯0 )} ⊆ CαCX eγ hp +ε/2 (Ah ) .

s≥N (ε,h)/Nh

But ω + (¯ x0 ) = A0 , so by 8.49 A0 ⊆

[

{φ(s, x ¯0 )} ⊆ CαCX eγ hp +ε/2 (Ah )

s≥N (ε,h)/Nh

or equivalently H ∗ (A0 , Ah ) ≤ αCX eγ hp + ε/2. Restricting h further to 0 < h < h0 (ε) so that αCX eγ hp ≤ ε/2 also holds and so H ∗ (A0 , Ah ) ≤ ε (8.54) for all 0 < h < h0 (ε) with h−1 an integer. Finally, the case that h−1 is not an integer needs to be considered. Let Nh be the integer such that hNh < 1 < h(Nh + 1). Then kFhNh (x) − Φ(x)k = kFhNh (x) − φ(1; x)k ≤ kFhNh (x) − φ(hNh ; x)k + kφ(hNh ; x) − φ(1; x)k

Z 1

≤ CX eγhNh hp + f (φ(s; x)) ds

hNh

γ

p

≤ CX e h + M (1 − hNh ) ≤ CX eγ hp + M h ≤ Kh where M = maxx∈X kf (x)k and K = 2 max{CX eγ , M }. Repeating the above argument with kFhNh − Φk∞ ≤ Kh instead of the global discretization estimate (8.51), that is with CX eγ hp replaced by Kh. With an appropriate modification to h0 (ε), the inequality (8.54) then holds for all h sufficiently small. Convergence (8.48), that is the lower semi-continuity of the numerical attractor at h = 0, then holds. This completes the proof of Theorem 8.15. t u

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8 Applications

8.4 Delay Differential Equations Consider the linear delay equation x0 (t) = Ax(t) + Bx(t − h).

(8.55)

Here x(t) ∈ Rd , A and B are real d-matrices and h is a positive constant. We shall call this equation hyperbolic if det(wI − A − ewB ) = 0

(8.56)

does not have a purely imaginative solution w = ip. To each solution w of this equation there corresponds a solution of the delay equation (8.55) of the form ewt a,

−∞ < t < ∞

(8.57)

where a is an eigenvector of the matrix wI − A − ewB with the eigenvalue w. We will also consider nonlinear delay equations of the form y 0 (t) = Ay(t) + By(t − h) + F (y(t), y(t − h)).

(8.58)

Here F (y, v) is a continuous Rd -valued function, which is locally Lipschitz in y and A,B are as before. Denote by L(F ) the set of all continuous function y(t), t ≥ −h satisfying the equation (8.58) for t > 0. In particular, L(0) denotes the set of all continuous function x(t), t ≥ −h satisfying the equation (8.55) for t > 0. Theorem 8.16. Let the equation (8.55) be hyperbolic. Then there exists a constant γ > 0 with the following properties. (i)

For each x(t) ∈ L(0) and for each uniformly bounded F (x, u) there exists a continuous function y(t) ∈ L(F ), satisfying the inequality |y(t) − x(t)| < γ sup |F (y, v)|,

t ≥ −h.

(8.59)

y,v

(ii) Let F (y, v) be a uniformly bounded and y(t) ∈ L(F ). Then there exists a function x(t) ∈ L(0) satisfying (8.59). This demonstrates robustness of solutions of a hyperbolic delay equation with respect to arbitrary continuous perturbations of small amplitude. In particular, any nonlinear perturbation (8.58) of a linear equation (8.55) has bounded at t → ∞ solutions which shadow a given bounded at t → ∞ solution of the linear equation. The main steps in the proof of this theorem are summarized as follows. Step 1. For each continuous function ξ(s), s ∈ [−h, 0] the equation (8.58) has a unique solution y(t; ξ, F ), t ≥ −h which is continuous and satisfies

8.4 Delay Differential Equations

149

y(s; ξ, F ) = ξ(s),

s ∈ [−h, 0],

because F (y, v) is supposed continuous, uniformly bounded and satisfy local Lipschitz condition in y. Introduce the shift operator SF for equation (8.58) by (SF ξ)(τ ) = y(h − τ ; ξ, F ), −h ≤ τ ≤ 0. The operator SF is completely continuous as an operator in the space C([−h, 0], Rd ). In particular, denote by S the shift operator for the linear equation (8.55). The operator S is a linear completely continuous operator in C. Step 2. Let ξ(τ ), τ ∈ [−h, 0] be an eigenfunction of the complexification of the operator S with a complex eigenvalue w. Then by the definition ξ(s) satisfies the equation wξ 0 (τ ) = wAξ(τ ) + Bξ(τ ),

−h ≤ τ ≤ 0.

Thus the set of nonzero eigenvalues of the linear operator S coincides with the set of complex number z = ehw where w is a solution of the equation (8.56) (The corresponding complex eigenfunction are restrictions of functions (8.57) on [−h, 0].) Since S is completely continuous, the spectrum of S consists of zero and all complex numbers ewh where w is a solution of the equation (8.56). Step 3. By the previous step and the hyperbolicity of the linear equation (8.55) the spectrumSσ(S) of the linear operator S consists of two disjoint parts σ(S) = σ s (S) σ u (S), such that σ s is located strictly inside the unit disc of a the complex plane and σ u is located strictly outside the unit disc. By the Decomposition Theorem [127, p. 421], it means that the space C can be decomposed into a direct sum C = Es ⊕ Eu so that both E s and E u are invariant for S with σ(S|E s ) = σ s ,

σ(S|E u ) = σ u .

Further, since S is completely continuous, the subspace E u is finite-dimensional. Note that the parallel projection P s of C onto E s in the direction of E u can be written in an explicit form as Z 1 (S − zI)−1 dz. (8.60) Ps = − 2πi |z|=1 Step 4. Introduce an auxiliary norm k · ks onto the subspace E s by

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8 Applications

kxks =

∞ X

kS n xkC .

n=0

Clearly this norm is equivalent to the norm k · k and the restriction of the operator S onto E s contracts in this norm with some constant λs < 1. Analogously, introduce an auxiliary norm k · ku onto the subspace E u by kxku =

∞ X

kS −n xkC .

n=0

This norm is also equivalent to the C-norm and the restriction of the operator S onto E u expands in this norm with some constant λu > 1. Introduce in C an auxiliary norm k · k∗ by kξk∗ = max{kP s ξks , kP u ξku } where P s is defined by (8.60) and P u = I − P s . Denote by s the split (λs , λu , 0, 0). By construction, the linear operator S is s-semi-hyperbolic with constants h, δ where k = max{kP s k∗ , kP u k∗ } and δ is an arbitrary positive number. Step 5. In Step 1 the shift operator SF of the nonlinear equation (8.58) is completely continuous. This operator also satisfies the estimate kSF ξ − Sξk∗ < γ1 sup |F (y, v)|,

t ≥ −h.

y,v

for some positive γ1 . Thus Theorem 8.4 is applicable and, taking into account the equivalence of norms k · kC and k · k∗ , as a corollary to that theorem it follows that: Corollary 8.17. There exist a constant γ > 0 with the following properties. (i)

For each trajectory η = η0 , η1 , . . .

(8.61)

of the shift operator S there exists a trajectory F η F = η(0) , η1F , . . .

(8.62)

kηn − ηnF kC ≤ γ sup |F (y, v)|,

(8.63)

of the operator SF with y,v

(ii) For each trajectory (8.62) of the shift operator SF there exists a trajectory (8.61) of the operator S satisfying (8.63). Theorem 8.16 then follows.

t u

8.5 Systems with Hysteresis

151

Note that the construction of the last section can be carried out also for some systems described by parabolic equations. Also note that some hysteresis perturbations, like Prandtl, Besseling and Ishlinskii models in plasticity or Preisach, Giltay and Madelung models in magnetism [22, 77, 137] can be taken into account both in analysis of delay and parabolic equations.

8.5 Systems with Hysteresis Consider a smooth mapping f : Rd → Rd . The dynamical system generated by a difference equation of the form xn = f (xn−1 ),

n = 1, 2, . . . ,

(8.64)

is often used in technical, physical or mechanical applications, where f usually occurs via a Poincaré section. Throughout, this will be referred to as the system f . Realistically, a system (8.64) can describe the actual underlying system only approximately. Thus an important mathematical problem is the robustness of the system to perturbations. Classical results in this direction state that a C r dynamical system preserves some of its structural properties under a small smooth perturbation [56, 111, 131]. However, there are some kinds of nonsmooth perturbations which are very important. In this section we analyze a specific class of perturbations which arise in systems with weak hysteresis nonlinearities. An important feature of such models is that hysteresis nonlinearities are treated as continuous but nonsmooth dynamical systems W , often with an infinite dimensional set Ω = Ω(W ) of internal states ω. This includes such nonlinearities as play, stop, the Besseling–Ishlinskii and Preisach–Giltay models and so on. Further details may be found in [77, 91]. In such situations the natural description of state space of a perturbed system (8.64) is Q = Rd × Ω. So it is more realistic to describe the dynamics of the perturbed system W by relations of the form (xn , ωn ) = W (xn−1 , ωn−1 ) = (ϕ(xn−1 , ωn−1 ), ψ(xn−1 , ωn−1 )). Here ϕ : Rd × Ω → Rd and ψ : Rd × Ω → Ω are continuous mappings. Some concrete examples of systems which arise in the theory of hysteresis nonlinearities are given in Sects. 8.5.1 and 8.5.2. We are concerned with the relationship between the trajectories of a smooth system f and those of systems W which are close to f in some sense. An appropriate measure of the distance between the two types of system was introduced and investigated in Sect. 6.2.3. It is important to note that, without extra assumptions, the system (8.64) is not structurally stable in general. A natural, additional assumption is hyperbolicity. In these circumstances, estimates of the distance between trajectories of f and its perturbation W

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8 Applications

should not depend explicitly upon the time interval over which the trajectories are considered. Instead, it is preferable that any estimate should be uniform so long as the trajectories remain in the region in question. This is the principal question that we address in this section.

8.5.1 Transducer Stop Recall that the nonlinearity stop with threshold value h or transducer stop [77, p. 23–24] is a system Uh with the state space [−h, h], scalar inputs u(t) and outputs ω(t). For a smooth input u(t), t ≥ 0, and initial state ω0 ∈ [−h, h] the corresponding output ω(t) = (Uh [ω0 ]u)(t),

t ≥ 0,

is defined as a unique absolutely continuous solution of the problem ω 0 = q(ω, u0 (t)), where

ω(0) = ω0

  min{u, 0}, ω ≥ h, q(ω, u) = u, |ω| < h,   max{u, 0}, ω ≤ −h.

Consider the system described by the equations x0 = G(x, ω),

ω(t) = (Uh [ω0 ](c, x))(t).

(8.65)

Here x ∈ Rd ; h > 0 and ω ∈ [−h, h] are parameters, c is a fixed vector from Rd and Uh is the stop nonlinearity with threshold value h. Equations of such type arise as description of mechanical systems with elastic-plastic Prager elements, technical systems with plays or stops and many control systems. Suppose that the function G satisfies a global Lipschitz condition. Then the equation (8.65) has a unique solution for any initial condition x(0) = x0 and each initial state ω0 of the hysteresis nonlinearity Uh . Let the shift operator Sh (x0 , ω0 ) denote the image of the initial value (x0 , ω0 ) after unit time along the trajectories of the system (8.65). Suppose that F (x) = G(x, 0) is a smooth function, satisfying F (0) = 0 and the matrix DF0 does not have eigenvalues with zero real part. Similarly, let S0 (x0 ) be the image of the initial value x0 after unit time along the trajectories of equation x0 = F (x).

(8.66)

8.5 Systems with Hysteresis

153

The mappings W (x, ω) = Sh (x, ω) and f (x) = S0 (x) generate dynamical systems W and f respectively, where the state space of the system W is the product Rd × [−h, h] Clearly, the system f is semi-hyperbolic in some open ball centered at the origin. From Theorem 6.15 it follows immediately Theorem 8.18. There exist α > 0 and h0 > 0 with the following property: for any trajectory x(t) ∈ B, 0 ≤ t < t∗ ≤ ∞, of the equation (8.66) and any h ≤ h0 there exists a trajectory (xh (t), ωh (t)), 0 ≤ t < t∗ , of (8.65) satisfying |x(t) − xh (t)| ≤ αh,

0 ≤ t < t∗ .

Corollary 8.19. There exist α > 0 and h0 > 0 with the following property: for any x0 ∈ B belonging to the stable manifold of the equation (8.66) there exists a trajectory (xh (t), ωh (t)), t ≥ 0, of (8.65) satisfying |x0 (t) − xh (t)| ≤ αh,

t > 0.

This result can be treated as a kind of ‘the stable manifold robustness theorem’ with respect to hysteresis perturbations of a system. Analogues of Theorem 8.18 are valid for equations with such nonlinearities as play or generalized play, with multi-dimensional plays and stops, with Mises and Treska models [77], and so on.

8.5.2 Ishlinskii and Besseling Systems Let Uh be the stop nonlinearity with threshold h, as in Sect. 8.5.1. Consider h as a parameter, 0 ≤ h ≤ ∞, and let µ be a Borel measure on [0, ∞] satisfying Z ∞ h dµ(h) < ∞. 0

Denote by H the totality of continuous functions z(h), h ≥ 0, satisfying |z(h)| ≤ h. Now introduce a system Wµ , with scalar inputs and outputs and with state space H , as follows. For a given smooth input u(t), t ≥ 0, and an initial state z0 ∈ H , the corresponding output z(t) = (Wµ [z0 ]u)(t), t ≥ 0, is defined as Z ∞ z(t) = (U [z0 (h)]u)(t) dµ(h). 0

A model of this type includes fundamental mechanical models such as the Ishlinskii and Besseling systems [77, p. 342–346]. It might be thought of as describing a continuum of linked transducers. Suppose that the function G is globally Lipschitz, as in previous section. Consider the system described by equations

154

8 Applications

x0 = G(x, z),

z(t) = (Wµ [z0 ](c, x))(t).

(8.67)

This extends the system (8.65). Again, (8.67) has a unique solution x(t), t ≥ 0, for each initial condition x(0) = x0 . Define the corresponding shift operator Sµ (x0 ). From Theorem 6.15 it follows that Theorem 8.20. There exist α > 0 and ε0 > 0 with the following property: for any trajectory x(t), 0 ≤ t < t∗ ≤ ∞, of the equation (8.66), for any measure µ satisfying Z ∞ h dµ(h) ≤ ε0 r(µ) = 0

and any z(h) ∈ H , there exists a trajectory (xµ (t), zµ (t)), 0 ≤ t < t∗ , of (8.67) satisfying |x(t) − xµ (t)| ≤ αr(µ),

0 ≤ t < t∗ .

Analogues of Theorem 8.20 are valid for models such as the multi-dimensional Ishlinskii system, the Preisach–Giltay model [77] and its multi-dimensional analogue [76].

Chapter 9

Extensions and Further Applications

Two extensions of semi-hyperbolicity and their applications are considered in this chapter. The first extension, split-hyperbolicity [120], is a modification of the semi-hyperbolicity concept that is applicable to analysis of mappings defined in metric spaces; it appeared also useful in rigorous computer aided analysis of chaotic behaviour. The second extension, topological hyperbolicity [120], is a tool to prove shadowing for various classes of dynamical systems. Split-hyperbolicity is briefly discussed in the next section, and a simplified version of the topological hyperbolicity concept is discussed in Sect. 9.2. In the last three sections some recent applications of split-hyperbolicity and topological in various subject areas are briefly sketched.

9.1 Split-Hyperbolicity 9.1.1 Split-Hyperbolicity in Product-Spaces Let J ∈ Z be a set, possibly infinite, of consecutive integers. Let Mns , Mnu where n ∈ J be complete metric spaces with metrics ρsn , ρun . Suppose that the non-empty balls in Mns and in Mnu are connected, that is they cannot be represented as a disjoint union of two nonempty sets each of which is both relatively open and relatively closed. Elements from the Cartesian product Mn = Mns × Mnu are treated as pairs x = (xs , xu ). The spaces Mn are endowed with the usual metric ρn (x, y) := max {ρsn (xs , y s ), ρun (xu , y u )} . Let f denote the bi-infinite sequence {fi }i∈Z of continuous maps, which may be partially defined, from Mn to Mn+1 : fn (xs , xu ) = (fns (xs , xu ), fnu (xs , xu ))

155

156

9 Extensions and Further Applications

s u where fns : Mns × Mnu → Mn+1 and fnu : Mns × Mnu → Mn+1 . Let the sequence x = {xn }n∈J of the elements xn ∈ Mn be fixed, such that the images fn (xn ) are defined for all n ∈ J. Denote by Bns [r] the closed r-ball in Mns centered at xsn ∈ Mns ; the balls Bnu [r] for xun ∈ Mnu , r > 0, are defined in a similar way. Let δ s , δ u be some positive constants. Denote

Un := Bns [δ s ] × Bnu [δ u ]. Let Dn be the set of those y ∈ Un which satisfy fn (y) ∈ Un+1 . Recall that a four-tuple s = (λs , λu , µs , µu ) of non-negative real numbers is called split if λs < 1 < λu

and ∆(s) := (1 − λs )(λu − 1) − µs µu > 0.

Definition 9.1. The sequence f of the maps fn is said to be split (s-) hyperbolic in the (δ s , δ u )-neighbourhood of the sequence x if it satisfies the following three conditions: C0: Dn is closed for all n ∈ J, and, for each boundary point y of Dn , either y belongs to the boundary of Un or f (y) belongs to the boundary of Un+1 (whenever n + 1 ∈ J). C1: For all n, n + 1 ∈ J and all y, z ∈ Dn the following inequalities hold ρsn+1 (fns (y), fns (z)) ≤ λs ρsn (y s , z s ) + µs ρun (y u , z u ), ρun+1 (fnu (y), fnu (z)) ≥ −µu ρsn (y s , z s ) + λu ρun (y u , z u ). u [δ u ] for C2: The map w 7→ fnu (v, w) is open as a map from Bnu [δ u ] to Bn+1 s s u each v ∈ Bn [δ ], in the sense that the image fn (v, U ) of an open subset U u of Bnu [δ u ] is relatively open in Bn+1 [δ u ] (whenever n, n + 1 ∈ J).

Conditions C0 and C2 are technical and intended for use in some complicated situations. The following simple lemma is stated without proof. Lemma 9.2. Let fn be defined on Un and let Mnu = Rdu . Then conditions C0 and C2 from the definition of split-hyperbolicity hold. For a given sequence x = {xn }n∈J and a given f the discrepancy D(x; f ) is defined by the formula: D(x; f ) := max ρn+1 (fn (xn ), xn+1 ). n,n+1∈J

Introduce the numbers as :=

λu − 1 + µs , ∆(s)

au :=

1 − λs + µu . ∆(s)

One of the main tools in the investigation of split-hyperbolic systems is the following Shadowing Theorem taken from [120].

9.1 Split-Hyperbolicity

157

Theorem 9.3 (Shadowing Theorem). Let the sequence f be split-hyper¯ = {¯ bolic in the (δ s , δ u )-neighbourhood of the sequence x xn }n∈J and let the discrepancy D(¯ x; f ) satisfy the inequality s u δ δ u , ,δ . D(¯ x; f ) < min as au Then there exists a trajectory x = {xn }n∈J of f satisfying ρsn (xsn , x ¯sn ) ≤ as D(¯ x; f )

and

ρun (xun , x ¯un ) ≤ au D(¯ x, f ).

In the case when J is the set of all integers the trajectory x is unique.

9.1.2 An Application to Hysteresis In this section, following [117], it is shown the split-hyperbolicity can be used to analyse systems with hysteresis. Let f (t, x) : R+ × Rd be T -periodic in time t and Lipschitz continuous in x: |f (t, x) − f (t, y)| < `f |x − y|. Consider an ordinary differential equation x0 = f (t, x), with x ∈ Rd . Suppose that this equation has a T -periodic solution with x(0) = xT . Let F be a shift operator along the trajectories for the time T . Thus the operator F has a fixed point xT . It will be assumed that the linearization of F in a small neighbourhood of xT has no eigenvalues with absolute values equal to 1. Then it is possible to change coordinates to split the space into two invariant subspaces Rd = Rds × Rdu corresponding to eigenvalues inside and outside the unit circle, respectively. Now consider an extended system that involves the output of a hysteresis nonlinearity: x0 = f (t, x) + εg(x, z(t)),

z(t) = (Γ [z0 ]Lx)(t).

(9.1)

Here L : Rd → Rm is a linear map with the Euclidean norm estimate kLk ≤ l; Γ [z0 ] is an operator which transforms functions u : R+ → Rm ,

R+ = [0, +∞),

to functions z : R+ → Z , where Z is a complete metric space equipped with a metric ρz ; the argument z0 represents initial memory; and, finally, g : Rd × Z → Rd is a global Lipschitz continuous function. As usual, the

158


notation (Γ [z0 ]u)(t) refers to the value of the function z = Γ [z0 ]u at the time t. Let Wt1,1 be the Banach space of absolutely continuous functions u : [0, t] → Rm equipped with the standard norm Z kukW 1,1 = |u(0)| + T

and define

t

|u0 (s)| ds

0

1,1 Wloc = {u : R+ → Rm , u|[0,t] ∈ Wt1,1 }.

1,1 Definition 9.4. A nonlinearity Γ : Wloc × Z → C(R+ ; Rm ) is called a normal nonlinearity [23]) with threshold h > 0 if it satisfies the Volterra property:

u(s) = v(s),

0 ≤ s ≤ t implies

(Γ [z0 ]u)(t) = (Γ [z0 ]v)(t)

for all

t ≥ 0,

the semigroup property: (Γ [(Γ [z0 ]u)(t1 )]v)(t2 − t1 ) ≡ (Γ [z0 ]u)(t2 )

where

v(t) = u(t − t1 ),

the Lipschitz condition N1 and the contraction property N2 below: N1: There exists a constant γu > 0 such that for every z0 ∈ Z , every t ≥ s ≥ 0 and every u, v ∈ Wt1,1 the inequality ρz ((Γ [z0 ]u)(s), (Γ [z0 ]v)(s)) ≤ γu ku − vkW 1,1 holds; T N2: There exists a continuous and bounded function q : R+ → R+ with q(α) < 1 for α > h such that ρz ((Γ [z0 ]u)(t), (Γ [z1 ]u)(t)) ≤ q(osct u)ρz (z0 , z1 ) holds for all t ≥ 0, all z0 , z1 ∈ Z and all u ∈ Wt1,1 . Here osct u is defined as follows osct u := sup |u(τ ) − u(σ)|. 0≤τ,σ≤t

Although the definition seems to be complicated, numerous hysteresis nonlinearities are in fact normal nonlinearities. Examples are the so-called stop or von Mises nonlinearity [77, 79], and certain modifications of the Preisach nonlinearity [125]. Theorem 9.5. For sufficiently small ε it is possible to introduce metrics ρs in Rds × Z and ρu in Rdu such that the shift operator G along trajectories of (9.1) is split-hyperbolic in U × Z , where U is a small neighbourhood of xT . Corollary 9.6. There exists ε1 such that for all 0 < ε < ε1 , the shift operator G has a unique fixed point in U × Z .

9.1 Split-Hyperbolicity

159

9.1.3 Split-Hyperbolicity and Strong Compatibility A rectification of Theorem 9.5, which has been used in computer aided analysis of chaotic behaviour, is formulated here, see [117] and Sect. 9.3.1 below. For any positive integer m, let Σ(m) = Σ({1, . . . , m}) denote the totality of all bi-infinite sequences ∞

ω = {ωi }i=−∞

with ωi ∈ {1, . . . , m}

for i = 0, ±1, ±2, . . . ,

and let σ = σm denote the (left) shift on Σ(m) given by 0 σm (ω) = ω 0 = (. . . , ω−1 , ω00 , ω10 , . . .)

where ωi0 = ωi+1 . In addition, let A = (ai,j ),

i, j = 1, . . . , m,

be a square m × m-matrix, the components of which are either zeros or ones, and introduce the set ΣA = ω ∈ Σ(m) : aωi ,ωi+1 = 1, i = 0, ±1, ±2, . . . . The set ΣA is shift invariant and the restriction σA of σm to ΣA is a topological Markov chain with the matrix A. Let f : M → M be a continuous map in a complete metric space M with the metric ρ. A trajectory of f (or, to be more precise, of the dynamical system generated by f ) is a sequence x = {xi }∞ i=−i− satisfying xi+1 = f (xi ),

i = −i− , . . . , 0, 1, 2, . . . ,

where 0 ≤ i− < ∞ (note that i− = i− (x) depends on the particular trajectory x). Let σf be the left shift map naturally defined on the set Tr(f ) of bi-infinite trajectories of f . It is convenient to endow the sets ΣA and Tr(f ) with the usual metrics: d(ω, ω 0 ) =

∞ X |ωi − ωi0 | , 2|i| i=−∞

d(x, x0 ) =

∞ X ρ(xi , x0i ) . 2|i| i=−∞

Definition 9.7. Let X = (X1 , . . . , Xm ) be a finite family of compact connected subsets of M . A continuous map f is said to be strongly (X , σA )-

160


compatible if for any ω ∈ ΣA there exists a unique trajectory x = ϕ(ω) ∈ Tr(f ) satisfying xi ∈ Xωi . The following assertion is well known. Lemma 9.8. Let f be strongly (X , σA )-compatible. Then the map ϕ has the following properties (i) A shift of ω ∈ ΣA induces a shift of the trajectory ϕ(ω): ϕσA = σf ϕ; (ii) if ω ∈ ΣA is p-periodic, then the trajectory x = ϕ(ω) is also p-periodic; (iii) the map ϕ is continuous. Theorem 9.9. Suppose that the nonempty balls in the metric spaces Mis , Miu are connected, that is cannot be represented as a disjoint union of two nonempty sets each of which is both relatively open and relatively closed. u s u s Suppose that gi,j = h−1 j f hi is s-hyperbolic on Bi [δ , δ ], Bj [δ , δ ] whenever ai,j = 1. Suppose also that s u δ δ u , , δ ρj (gi,j (˜ yi ), y˜j ) < min as au whenever ai,j = 1. Then f is strongly (X , σA )-compatible, where X = {Xi = hi (Bi [δ u , δ s ]) : i = 1, . . . , m} . This follows from Theorem 9.3. Various properties of split-hyperbolicity were investigated in [120].

9.2 Topological Hyperbolicity Topological hyperbolicity is a tool that can be used in the computer-aided analysis of long periodic trajectories and chaotic behaviour. It has been applied for analysis various interesting phenomena such as piece-wise systems [118, 143], hysteresis in the trade cycle [10, 96], oscillators with hysteresis [22], complicated behaviour in a truncated Lang–Kobayashi model [126], chaotic Wave Patterns in KdVB-type equation [36], emerging avian influenza virus [107] and Section 9.4 below. Note also recent applications to the investigation of chaos in singularly perturbed systems, see [119, 121] and Section 9.5 below, are not computer aided. Other toolboxes have been developed for computer-aided analysis of chaotic behavior: we mention the Tacker’s method [135, 136], and Michaikov-Mrozek method [97, 98] (see also some further developments in [54, 99]) and Zgliczy´ nski method [141, 142]. Each of the aforementioned methods have some advantages and disadvantages. In particular, the semi-hyperbolicity approach seems to be more robust with respect to higher-dimensional and hysteresis type perturbations, and it may be more convenient in analysis of multi-rate systems.

9.2 Topological Hyperbolicity

161

Informally, a parallelogram P is said to be topologically hyperbolic if the image of P under a continuous mapping F , F (P ), intersects P in such a way that they form a distorted ‘cross-shape’ with one another. A topologically hyperbolic parallelogram always contains a fixed point of the mapping F . The mapping F ‘expands’ along the unstable (dashed) direction and ‘contracts’ along the stable (solid) direction, see Fig. 9.1. Note that a ‘stretching’ behaviour is observed in one direction (the unstable direction) and a ‘compression’ is observed in the other (stable) direction. If P and its image, F (P ) intersect in this manner, then, from the two dimensional version of topological degree theory [78] or [38], there exists a fixed point of the mapping F in the intersection of P and F (P ). Next, suppose that there is a chain of parallelograms P1 , . . . , Pq , where the image of the preceding parallelogram in the chain intersects the next parallelogram in the manner outlined above. Figure 9.2 shows three such parallelograms, P1 , P2 and P3 , and their images under the mapping F . In this Figure, the image of P1 , F (P1 ), crosses P2 , F (P2 ) crosses P3 , and F (P3 ) crosses P1 . Note that the set F (P1 ) intersects P1 in a ‘cross’ shape, in addition to intersecting P2 . Therefore, there exist periodic orbits with any given period that are greater than or equal to 3. This is an example of a 9-periodic orbit. Then the Product Theorem [78] guarantees that there exists a periodic trajectory with minimal period, in this case, with period 3, the elements of which belong to the corresponding intersections. This assertion is also true for a chain of parallelograms of arbitrary length q > 1. Note that in Fig. 9.2, the parallelogram P1 is also in a cross-type position with respect to its image F (P1 ). Applying the Product Theorem [78] once more, we can guarantee that there exist periodic orbits with any given period that are greater than, or equal to 3. In Fig. 9.2, there exists a 9-periodic orbit, the elements of which belong to the intersections of the parallelograms and their images. This theorem holds for a chain of parallelograms of arbitrary length q > 1.

Fig. 9.1 A topologically hyperbolic parallelogram P and its image F (P ).

More rigorously, topological hyperbolicity may be defined as follows. Fix two positive integers du , ds such that du + ds = d. The indices ‘u’ and ‘s’

162


F (P2 )

F (P1 ) P2

P3

F (P3 ) P1 Fig. 9.2 A topologically hyperbolic chain of three parallelograms.

signify ‘unstable’ and ‘stable’, respectively. Let V and W be bounded, open and convex product-sets V = V (u) × V (s) ⊂ Rdu × Rdu ,

W = W (u) × W (s) ⊂ Rdu × Rds ,

satisfying the inclusions 0 ∈ V , 0 ∈ W , and let f : V → Rdu × Rds be a continuous mapping. It is convenient to treat f as the pair (f (u) , f (s) ), where f (u) : V → Rdu and f (s) : V → Rds . Definition 9.10. The mapping f is (V, W )-hyperbolic, if (s) \ (u) W = ∅, f (u) ∂V (u) × V \ (u) f (V ) W × (Rds \ W (s) ) = ∅

(9.2) (9.3)

and deg(f (u) , V (u) ) 6= 0.

(9.4)

Condition (9.2) means geometrically that the image of the ‘unstable’ (s)

boundary, ∂V (u) × V of V , does not intersect the infinite cylinder C = (u) ds W × R . Relationship (9.3) means that the image of the entire set f (V ) (u)

can intersect the cylinder C only through its central fragment W × W (s) . Thus, condition (9.2) means that the mapping ‘expands in a rather weak sense along the first, unstable ‘u’-coordinate in the Cartesian product Rdu × Rds , whereas condition (9.3) describes a type of ‘contraction’ along the second,

9.3 Semiconductor Lasers Dynamics

163

stable ‘s’-coordinate. Condition (9.4) means that the rotation of the vector field f (u) at V (u) is non-zero, that is there exists a fixed point in the set V (u) . Establishing the (V, W )-hyperbolicity of a particular group of sets provides information on the existence of long periodic trajectories of a particular mapping. In addition, (V, W )-hyperbolicity may be used to establish the existence of chaotic trajectories of such a mapping. Let X = (X0 , . . . , Xm−1 ) be a finite family of compact connected subsets of Rd and A = (ai,j ), i, j = 1, . . . , m. Definition 9.11. A continuous mapping f is called (X , σA )-compatible if there exists a mapping ϕ : ΣA → Tr(f ) which satisfies the following requirements: R1: the trajectory x = ϕ(ω) satisfies xi ∈ Xωi for each ω ∈ ΣA and all integers i; R2: ϕσA = σf ϕ: a shift of ω ∈ ΣA induces a shift of the trajectory ϕ(ω); R3: if ω ∈ ΣA is p-periodic, then the trajectory x = ϕ(ω) is also p-periodic. This definition is similar, but significantly weaker than the definition of strong compatibility 9.7: neither the uniqueness of the mapping ϕ nor its continuity are assumed, so ϕ need not be a semi-conjugacy [65, p. 68]. On the other hand, the subshift σA is a factor [65] of a restriction of the system f to S some set S ⊂ Xi , providing that f is (X , σA )-compatible. The (X , σA )compatible mappings have some features of chaotic behaviour if A has some and if sufficiently many subfamilies of X have empty intersections. Theorem 9.12. Let A be a square m × m-matrix whose entries are either zeros or ones, hi : Rdu × Rds → Rd be homeomorphisms, and Vi be bounded, open and convex product sets. Suppose that gi,j = h−1 j f hi is (Vi , Vj )-hyperbolic whenever ai,j = 1. Then f is (X , σA )-compatible for X = {hi (Vi ), i = 1, . . . , m}.

9.3 Semiconductor Lasers Dynamics External cavity semiconductor lasers present many interesting features for both technological applications and fundamental non-linear science. Their dynamics has been the subject of numerous studies over the last twenty years. Motivations for these studies vary from the need for stable tunable laser sources, for laser cooling or multiplexing, to the general understanding of their complex stability and chaotic behaviour. In [117] the split-hyperbolicity concept was applied to study the dynamics of the truncated Lang–Kobayashi equations that describe the behaviour of semiconductor lasers with feedback. For particular values of the parameters it was rigorously proved that the system exhibits chaotic behaviour in the Smale sense: some power of a suitable first return map is topologically conjugate to the left shift operator in the set

164


of symbolic sequences. The proof is computer assisted, with all errors estimated and taken into account. The main result from [117] is briefly described below.

9.3.1 Truncated Lang–Kobayashi equations A typical experiment is usually described by a set of delay differential equations introduced by Lang & Kobayashi [81]: E˙ = κ (1 + iα) (N − 1) E + γe−iϕ0 E (t − τ ) , N˙ = −γq N − J + |E|2 N .

(9.5)

Here E is the complex amplitude of the electric field, N is the carrier density, J is pumping current, κ is the field decay rate, 1/γq is the spontaneous time scale, α is the linewidth enhancement factor, γ represents the feedback level, ϕ0 is the phase of the feedback if the laser emits at the solitary laser frequency and τ is the external cavity round trip time. A typical setup is shown on Fig. 9.3. Numerical simulations of these equations have successfully re100%

Laser feedback

94% semi−transparent mirror

Fig. 9.3 Laser with a semi-transparent mirror

produced many experimental observations, such as mode hopping between external cavity modes [101] and a period doubling route to chaos [83]. However there are few analytical results since delay equations are nonlocal. This model was recently reduced to a 3-D dynamical system describing the temporal evolution of the laser power P = |E|2 , carrier density N and phase difference η(t) = ϕ (t) − ϕ (t − τ ). This was achieved by assuming P (t − τ ) = P (t) together with the approximation ϕ˙ = η/τ + η/2. ˙ This expression remains valid when the phase fluctuates on a time scale much shorter than the re-injection time τ . Under these approximations the Lang– Kobayashi equations (9.5) reduce to:


165

P˙ = 2 (κ (N − 1) + γ cos (η + ϕ0 )) P, N˙ = −γq (N − J + P N ) , 1 η˙ = − η + 2κα (N − 1) − 2γ sin (η + ϕ0 ) . τs This model was successfully used to describe low frequency fluctuations commonly observed in semiconductor lasers with optical feedback, but its behaviour for a low feedback level (see Fig. 9.4) has not yet been investigated.

Fig. 9.4 γq = 0.03, γ = 0.06.

9.3.2 Poincar´ e Map To improve some estimates crucial for the numerical computations and to somehow center the graph of trajectories while keeping all parameters as rational numbers it is convenient to introduce new scaled variables: x(1) = ln P,

x(2) = 5N − 97/20,

x(3) = 5(η + 2)/16.

Choosing numerical values for the parameters in line with [126] the system under investigation can be rewritten as

166


3 2 3 16 dx(1) = − + x(2) + cos 1 − x(3) , dt 50 5 25 5 (2) dx 609 3 (2) 3 x(1) 97 (2) = − x − e +x , dt 2000 100 100 20 7 3 (2) 1 (3) 3 16 (3) dx(3) =− + x − x + sin 1 − x dt 160 8 50 80 5

(9.6)

and in vector notation as x˙ = F (x)

where

x = x(1) , x(2) , x(3)

for the system (9.6). For the elements of the plane x(3) = 0 the notation x = (x(1) , x(2) ) will be used as a synonym for x = (x(1) , x(2) , 0). Note that the condition of transverse ˙ 6= 0, is broken only for the set of W defined by intersection with W , x(3) x(2) =

7 3 (2) − sin(1) = xT . 60 10 (2)

Denote by S the half-plane x(3) = 0, x(2) > xT . Since S is, by definition, transversal to trajectories of the system, the Poincaré map Φ, which is defined as a (partial) map from S to itself, can be obtained by following trajectories from one intersection with S to the next. Let R denote the union set of two rectangles R1 , R2 ⊂ S given by R1 = {(0.32, 0.43) + (−0.3675, 0.3255)α + (0.039, 0.039)β : α, β ≤ 1}; R2 = {(−0.04, 0.83) + (0.075, 0)α + (0, 0.1)β : α, β ≤ 1}. Theorem 9.13. The Poincaré map Φ is defined for all y ∈ R and Φ(R) ⊂ R.

(a) Rectangle R1 and its im- (b) Rectangle R2 and its im- (c) Previous age age bined Fig. 9.5 Geometry of the Poincar´ e map on the set R = R1

S

R2

figures

com-


167

The geometry of the map Φ on R is illustrated by the numerically constructed Figs. 9.5(a), 9.5(b) and 9.5(c). On the naive level, Fig. 9.5(c) provides a ‘proof’ for the proposition above. However, the detailed justification of this figure is cumbersome. It requires a rigorous proof of continuity of the Poincaré map Φ as well as estimates of the accuracy of discrete computer arithmetic and estimates of the precision of the numerical method. A rigorous computer assisted proof of Theorem 9.13 was given in [126].

9.3.3 Main Theorem Consider the set P consisting of twenty parallelograms Pi , i = 1, . . . , 20 given by Pi = {xi + αpi + βqi : |α|, |β| ≤ 0.00002}. (1) (2) Here the coordinates of the center points xi = xi , xi and of the vectors (1) (2) (1) (2) pi = pi , pi , qi = qi , qi , are given in Table 9.1.

Table 9.1 The centre points xi , and the vectors pi , qi i

xi

pi

qi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

(0.019587, 0.66699) (0.020515 , 0.66915) (0.020479, 0.66906) (0.020479, 0.66907) (0.020482, 0.66907) (0.020475, 0.66907) (0.020492, 0.66906) (0.020453, 0.66910) (0.020545, 0.66900) (0.020328, 0.66923) (0.020842, 0.66869) (0.019627, 0.66996) (0.022516, 0.66695) (0.032140 , 0.67407) (0.019587, 0.65706) (-0.0048159, 0.69655) (0.092305, 0.59933) (-0.074542, 0.79970) (0.44510, 0.30273) (0.042707 , 0.72057)

(0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (0.69432 , -0.71967) (-0.75835 , 0.78856) (0.79844 , -0.84560) (-0.92879 , 0.94270) (0.84009 , -0.95290) (-1.2827 , 1.1753) (0.29435 , -0.73198) (-1.0545 , 0.86816) (-0.58329 , 0.65993)

(0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (0.39613, 0.91820) (-0.36270, -0.85146) (0.35310, 0.80491) (-0.29803, -0.73433) (0.34887, 0.74206) (-0.13304, -0.67436) (0.72214, 1.3124) (-1.0288, -0.51327) (-0.38881, -0.89935)

Figure 9.6 graphs these parallelograms together with R1 , R2 .

168


Fig. 9.6 Rectangles R1 , R2 and parallelograms Pi .

Introduce a 20 × 20 matrix A = (ai,j ), i, j = 1, . . . , 20 defined by a20,1 = a3,3 = 1, ai,i+1 = 1 for all 1 ≤ i ≤ 19, ai,j = 0 if pair (i, j) is none of above. Theorem 9.14. The Poincaré map Φ is strongly (P, σA ) compatible. S ∗ By definition the union set Pi has 15 connected components U1∗ , . . . , U15 (see Fig. 9.6). Since Φ is an homeomorphism, Theorem 9.14 implies the following corollary. Corollary 9.15. The restriction of Φ38 to a suitable Φ-invariant Cantor set S K ⊂ Pi is topologically conjugate to the left shift σ15 . Moreover, the inclusion Φ38i (ϕ(ω)0 ) ∈ Uωi , ω ∈ Σ(15) is satisfied and the the topological entropy is bounded below by E top (Φ) ≥ ln(15)/38 > 0.07. Proof. This follows from the fact that the matrix A is 38-transitive.

9.4 Avian Influenza in a Seabird Colony Novel pathogens pose a significant threat to the health status of man, of domestic animals and of wildlife. Therefore, it is important to know how such pathogens spread in immunologically naive host populations. A model of the long-term dynamics of a novel bird flu pathogen H5N1 in a mixed population of marine birds, which takes into account seasonality effects that affect multiple processes in a seabird population is discussed here. Such seasonal variations may considerably affect the transmission of a pathogen in such a population. In particular, a seasonally perturbed Susceptible-InfectedRecovered (SIR) model is considered. Such systems are known to exhibit

9.4 Avian Influenza in a Seabird Colony

169

complex dynamics as the amplitude of the seasonal perturbation term is increased. Furthermore, it has been long observed that chaotic solutions may appear in such a model: computer simulations of a seasonally perturbed SIR system suggest the existence of chaos in the model. A rigorous proof of the existence of chaos was given in [107] based on the concept of topological hyperbolicity. The main result from [107] is briefly described here.

9.4.1 Basic Model An SIR (Susceptible-Infected-Recovered) model is used to describe the spread of H5N1 in a seabird colony. According to the classic assumption the population is divided into three classes of epidemiological significance, namely the susceptible birds S(t), the infected bird I(t) and recovered (and immune) birds R(t). The infection is transmitted via direct contacts between the susceptible and the infected birds. frequency-dependent bi-linear (or massaction) transmission is assumed in which the number of contacts between the susceptibles (S) and the infectives (I) is independent of the population size. After an instance of infection, the infected bird immediately becomes infectious; that is this model disregards a latent period. This is a reasonable assumption because the latent period of influenza is comparatively short. The infected birds recover and become immune to the disease at a rate γ. The size of a colony N (t) is limited only by the availability of the breading spots, and the number of these are constant. Space is scarce, and a vacant nesting spots are immediately occupied by a candidate. Thus, for a colony the reqruitment exactly balances the mortality disregarding whether it is inflicted by the disease (at a rate ωI ) or by natural causes (at a rate ω). Under these assumptions, the SIR model is given by the following system of differential equations (cf. [106]): S˙ = ω(S + I + R) + ωI I − βSI − ωS I˙ = βSI − (γ + ω + ωI )I

(9.7)

R˙ = γI − ωR. The number of adults in a colony is limited by the available nesting spots, and hence is constant. However, the total colony population varies in time according to the birds seasonal breading pattern: chicks appear increasing the colony population, and then, after some time, the chicks leave the colony. For example, chicks of the common guillemot (Uria aalge) leave a colony after about 21 days after the birth, to return to the colony after 5 years to breed. The breading seasonal pattern of the birds is known, as well as is known an average number of the chicks per an adult bird. Therefore it can be safely assumed that the colony size is a given function of time, N (t). The population

170


is composed of three classes, that is S(t) + I(t) + R(t) = N (t), and hence one of the three equations of system (9.7) may be omitted. Usually, it is the equation for R(t) that is omitted. However, any of the other two equations may be as well omitted instead, and omitting either of these two, rather than one for R(t), has some advantages. Here, the equation for S(t) is omitted. This choice has an advantages that, for instance, the resulting two-dimensional system I˙ = (β(N − I − R) − γ − ωI − ω)I, R˙ = γI − ωR is equivalent for SIR and SIRS models, including the models with the vertical transmission. This system always has an infection-free equilibrium state that is conveniently located in the origin. Furthermore, it is readily seen that the positive quadrant is a positive invariant set of this system, and hence any phase trajectory initiated in this quadrant indefinitely remains there. Finally, when the system parameters are constant, this system is globally asymptotically stable [105]. That is, this system always has a globally attractive equilibrium state. depending on the basic reproduction number R0 = β/(γ +ω+ωI ), this equilibrium state being either positive (endemic) or infection-free. Seasonality may be introduced into a SIR model by incorporating a periodical forcing term into the system. For example, the most common approach of introducing seasonality is periodically perturbing the transmission rate β(t). However, as mentioned above, the seasonality in a seabird colony is mostly associated with the seasonal breading pattern of the birds, and respectively with seasonally-varying population size. It is reasonable to assume, therefore, that in this case the seasonally perturbed quantity is the total population N (t) of the colony. That is, it is assumed that N (t) = S(t) + I(t) + R(t) is a given 1-periodic function, such that N (t + 1) = N (t) holds. This leads to the following system of non-autonomous differential equations: I˙ = (Q(t) − βI − βR)I, R˙ = γI − ωR,

(9.8)

where Q(t) is a given 1-periodic function defined by the equation Q(t) = Q0 (1 + ε(t)). Here the constant Q0 = βN − (γ + ω + ωI ) is the mean value of the function Q(t), and |ε(t)| < Q0 . The seasonal perturbation ε(t) may be assumed to be a sinusoidal or step function. The parameter Q0 incorporates the effects of the transmission rate β, the recovery rate γ,


171

the death rates ω, ωI and, in particular, the population size N . Below it is assumed that Q(t) = Q0 (1 − Q1 sin 2πt).

9.4.2 Numerical Evidence of Chaotic Trajectories The effect of varying the amplitude Q1 of Q(t) = Q0 (1 − Q1 sin 2πt) on the long-term dynamics of the virus in a population has been investigated via numerical experiments. The numerical values used for these experiments were chosen to realistically simulate the dynamics of H5N1 virus in a seabird colony (see [33] for relevant details). The output of the translation operator for the time of order 1 along a particular trajectory of system (9.8) provides further evidence of chaotic behaviour. The output of the translation operator is the solution of system (9.8) shifted by time t = 1 years. This is achieved by iterating the map from a given initial condition, recording the solution at the end of 1 year and then repeating the iteration from this point. Therefore, the translation operator for system (9.8) is a discrete mapping, which outputs the values of the infected and recovered populations at the end of each year. (Note that it is not a Poincaré mapping.) Let Φ denote the time-1 translation operator along trajectories of the seasonally perturbed system with the parameters given in Table 9.2. Table 9.2 Parameter values for the mapping Φ along numerical trajectories of system (9.8). Parameter Symbol Parameter value Total population N 10000 Basic reproductive number R0 10 Transmission rate β 0.0924 Recovery rate γ 365/4 Natural death rate ω 1/10 Disease-associated death rate ωI 3/4 Mean value of Q(t) Q0 831.928 Amplitude Q1 0.025

The output of Φ is plotted in Fig. 9.7(a), after discarding the initial transient values of this. It is noteworthy that this set closely resembles the famous strange attractor of the Hénon mapping [59] f (x, y) = (1 − 1.4x2 + y, 0.3x),

172


shown in Fig. 9.7(b) for comparison. The existence of the set in Fig. 9.7(a) will provide the foundation for the proof of the existence of chaotic trajectories of system (9.8). From this Figure, it is also clear that the infected population persists, albeit at very low levels.

(a)

(b)

Fig. 9.7 (a) A trajectory of Φ. (b) A trajectory of H´ enon mapping.

To prove that the translation operator Φ is a chaotic mapping, and hence, prove the existence of chaotic trajectories of system (9.8), the topological hyperbolicity method was employed.

9.4.3 Main Assertion The topological hyperbolicity method will be used to prove that the translation operator Φ is a chaotic mapping and, hence, to prove the existence of chaotic trajectories of system (9.8). Theorem 9.16. The system (9.8) has periodic orbits with all minimal periods greater than 15. Proof. The formal definition of topological hyperbolicity will be used here to prove the assertion rigorously. Firstly, define the parallelograms Pi oriented around the points xi = (Ii , Ri ) of the quasi-periodic orbit of period 15, (u)

(s)

Pi = {xi + κ1 ai pi + κ2 ai qi }, where |κ1 |, |κ2 | ≤ 1. Numerical values for the points xi = (Ii , Ri ) of the (u) (s) quasi-periodic orbit, sizes of parallelograms ai and ai and vectors pi and qi , which define the orientations of parallelograms Pi are given in Table 9.3. The procedure for locating parallelograms was the following. The first step in identifying suitable topologically hyperbolic parallelograms is to find a quasi-periodic orbit of length q of the mapping Φ, the


173

translation operator of the time of order 1 along trajectories of system (9.8). The parallelograms are defined so that they are oriented around the points of the quasi-periodic orbit. The orbit should have elements xi , i = 1, . . . , q, in a small vicinity of a fixed point of Φ and have elements that are far away from this fixed point. The fixed point may be found using the broken orbits method [122]. However, due to the stiffness of system (9.8), this method may not be precise enough for identifying a suitable quasi-periodic point close to the fixed point of Φ. Instead, an alternative procedure may be used: a small rectangle around the fixed point may be identified, the iterated mapping Φq may be defined and a loop may be used to search for a suitable quasi-periodic point xq such that |xq − Φq (xq )| 1. By using smaller rectangles, this point was refined by choosing the point with the smallest norm. Once a suitable quasi-periodic point is found, a linearization of the iterated mapping Φq about each point xi , i = 1, . . . , q of the quasi-periodic orbit may be obtained. This gives information about the local dynamics of the system in the neighbourhood of each point xi , and therefore allows the orientations of the parallelograms to be determined. The solution of the linearized system at xi is a 2 × 2 matrix. The eigenvectors of this matrix were used as first approximations of the orientations of the parallelograms. However, the sizes (u) (s) (ai , ai ) of the parallelograms had to be adjusted until the desired results were obtained. This is a delicate operation involving much trial and error and experiment. If this procedure is successfully applied, then the mapping Φ is topologically hyperbolic. (u)

Table 9.3 The points xi = (Ii , Ri ), sizes of parallelograms (ai

(s)

(u)

, ai ) and vectors xi

,

(s) xi .

i

xi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

(37.113, 9307.1) (36.6294, 9309.91) (39.0891, 9300.52) (33.9295, 9319.77) (45.0663, 9277.67) (24.5481, 9354.05) (73.2449, 9150.18) (5.48759, 9429.4) (25.3733, 8601.2) (0.05103, 9431.51) (0.66918, 8535.46) (0.05242, 9339.88) (90.2127, 9153.63) (31.1815, 9321.67) (38.515, 9304.28)

(u)

(ai

(s)

, ai )

(3.75, 6) (8, 8) (12, 5) (4, 15) (25, 12) (5, 10) (4, 20) (2.5, 0.3) (2, 10) (0.9, 0.8) (1, 2.5) (0.06, 0.7) (1, 6) (5.5, 5) (5, 4)

pi

qi

(-0.193582, 0.981084) (-0.263871, 0.964558) (-0.25983, 0.965654) (-0.049546, -0.998772) (-0.257906, 0.96617) (0.287037, -0.957919) (-0.000440847, 1.) (0.234088, -0.972215) (0.521902, -0.853006) (-0.000324325, -1.) (-0.000228771, -1.) (0.0466627, 0.998911) (-0.051018, 0.998698) (-0.342414, 0.939549) (0.291732, -0.9565)

(0.381538, -0.924353) (0.384224, -0.92324) (0.375556, -0.9268) (-0.394077, 0.919077) (0.355469, -0.934688) (-0.435555, 0.900162) (0.254311, -0.967123) (-0.987824, 0.155575) (0.0888201, 0.996048) (-0.006101, -0.999981) (-0.020363, -0.999793) (0.0058613, 0.999983) (0.273534, -0.961862) (0.396575, -0.918002) (-0.378975,0.925407)

(u)

(s)

Denote the product sets in R2 by Vi = {(ai , ai ) ∈ R2 }, where i = 1, . . . , 15. These are mapped into the IR plane by the functions

174


hi : R2 → R2 ,

i = 1, . . . , 15,

as follows: hi : R2 → R2 ,

(u)

(s)

(u)

(s)

(ai , ai ) 7→ xi + ai pi + ai qi .

It is easy to see that each hi is a homeomorphism and thus, the mappings h−1 j Φhi are meaningful. Note also that Pi = hi (Vi ). Lemma 9.17. The mapping h−1 j Φhi is (Vi , Vj )-hyperbolic for all j = i + 1 mod 15. Furthermore, the mapping h−1 1 Φh1 is (V1 , V1 )-hyperbolic, (V2 , V1 )hyperbolic, (V3 , V1 )-hyperbolic, and (V3 , V2 )-hyperbolic. Introduce a 15 × 15 matrix A = (ai,j ) i, j = 1, . . . , 20 by a1,1 = a2,1 = a3,1 = a15,1 = 1, ai,i+1 = 1 for all 1 ≤ i ≤ 14, ai,j = 0 if pair (i, j) is none of above. R Theorem 9.18. The Poincaré map Φ is (P, σA ) compatible.

Clearly, some pairs of parallegrams Pi , Pj do not intersect. Therefore, Theorem 9.18 implies Theorem 9.16. t u A lower bound on the topological entropy E top (Φ) of the mapping Φ, can also be calculated. The topological entropy is greater than or equal to the natural logarithm of the maximal eigenvalue λmax of the k-transitive matrix A, namely E top (Φ) ≥ ln(λmax ) = 0.693168.

9.4.4 Existence of Chaotic Behaviour Let U = {U1 , . . . , Um }, m > 1, be a family of disjoint subsets of Rd and let us denote the set of one-sided sequences ω = ω0 , ω1 , . . . by Σ R (m). Sequences in Ω R (m) will Sm be used to prescribe the order in which sets Ui are to be visited. For x ∈ i=1 Ui define I(x) to be the number i satisfying x ∈ Ui . A small modification of Definition 7.11 will be used below. Definition 9.19. A mapping S f is called (U , k)-chaotic, if there exists a compact f -invariant set S ⊂ i Ui with the following properties: R P1: for any ω ∈ Σm there exists x ∈ S such that f ik (x) ∈ Uωi for i ≥ 1; R P2: for any q-periodic sequence ω ∈ Σm there exists a q-periodic point ik x ∈ S with f (x) ∈ Uωi ;

9.5 Chaotic Canard-Type Trajectories

175

P3: for each η > 0 there exists an uncountable subset S(η) of S, such that the simultaneous relationships lim sup |I(f ik (x)) − I(f ik (y))| ≥ 1, i→∞

lim inf |f ik (x) − f ik (y)| < η i→∞

hold for all x, y ∈ S(η), x 6= y. Similarly to Definition 7.11, this definition describes rigorously the important attributes of chaotic behaviour of a mapping f : Rd → Rd . These are the sensitive dependence on initial conditions, an abundance of periodic trajectories and an irregular mixing effect, that is the existence of a finite number of disjoint sets which can be visited by trajectories of f in any prescribed order. This definition of chaos can be applied to the family U of connected components of the set of 15 parallelograms that are defined above. By inspection, the family U has 9 elements. Therefore, the following theorem was established: Theorem 9.20. The mapping Φ is (U , 26)-chaotic.

9.5 Chaotic Canard-Type Trajectories In this section topological methods are used to in the analyse canard-type periodic and chaotic trajectories following [121]. Consider the slow-fast system x˙ = X(x, y, ε),

εy˙ = Y (x, y, ε),

(9.9)

where x ∈ R2 ,

y ∈ R1 ,

and ε > 0 is a small parameter. The subset S = (x, y, z) ∈ R2 × R1 : Y (x, y, 0) = 0

(9.10)

of the phase space is called a slow surface of the system (9.9): on this surface the derivative y˙ of the fast variable is zero, the small parameter ε vanishes. The part of S where Yy0 (x, y, 0) < 0 > 0 is called attractive (repulsive, respectively). The subsurface L ⊂ S which separates attractive and repulsive parts of S is called a turning subsurface. Trajectories which at first pass along, and close to, an attractive part of S and then continue for a while along the repulsive part of S are called canards or duck-trajectories [15, 100].

176


Topological hyperbolicity will be used to prove the existence of and to locate with a given accuracy chaotic canards of system (9.9). The canards which are found in this way are topologically robust: they vary only slightly if the right hand side of the system is disturbed. Assumption 9.21 Suppose that the function X is continuously differentiable and that the function Y is twice continuously differentiable. A point (xc , yc ) is called a critical point of the system (9.9), if it satisfies the equations hX(xc , yc , 0), Yx0 (xc , yc , 0)i = 0,

(9.11)

Y (xc , yc , 0) = 0,

(9.12)

Yy (xc , yc , 0) = 0.

(9.13)

This is a system of three equations with three variables, so in general case it is expected to have solutions. The existence of critical points is important for the phenomenon of canard type solutions, because every canard, which first goes along the stable slow integral manifold for t < 0 and then along the unstable slow integral manifold for t > 0, must pass through a small vicinity of a critical point. Thus, attention will focus on critical points and the behaviour of (9.9) in a vicinity of such a point. A critical point is called non-degenerate, if the following inequalities hold: X(xc , yc , 0) 6= 0,

Yx0 (xc , yc , 0) 6= 0,

00 Yyy (xc , yc , 0) 6= 0.

Note that non-degeneracy is stable with respect to small perturbations of the right-hand side of (9.9). Only non-degenerate critical points are considered here. Without loss of generality it can be assumed that the critical point is situated at the origin: xc = yc = 0. (9.14) Consider the auxiliary system x˙ = X(x, y, 0),

h(x, ˙ y), ˙ Yx0 (x, y)i = 0.

If the initial point (x0 , y0 ) lies on the slow surface S0 = (x, y) ∈ R3 : Y (x, y, 0) = 0

(9.15)

(9.16)

of the system (9.9), then (9.15) is equivalent to x˙ = X(x, y, 0),

(x, y) ∈ S0 .

(9.17)

The system (9.17) is important because it describes the singular limits of the solutions of (9.9) which lie on the slow surface. Equations (9.15) can also be rewritten in the form:

9.5 Chaotic Canard-Type Trajectories

x˙ = X(x, y, 0),

177

yY ˙ y (x, y, 0) = −hX(x, y, 0), Yx (x, y, 0)i,

(9.18)

which, by (9.13), has a singularity at the origin. Therefore, the existence and uniqueness of a solution of (9.18) which starts at the origin requires an additional assumption and will be discussed in detail later. To describe the dynamics of the system (9.9) near the origin, a special coordinate system (x(1) , x(2) , y) is introduced in the three-dimensional space of pairs (x, y). Choose x(1) to be co-directed with the gradient Y 0 ((0, 0), 0, 0), and x(2) to be orthogonal to x(1) and y. In the coordinate system (x(1) , x(2) , y) the gradient of Y (x(1) , x(2) , y, 0) at the origin takes the form Y 0 (0, 0, 0, 0) = (ξ, 0, 0),

ξ > 0,

(9.19)

and the equation (9.9) takes the form x˙ (1) = X (1) (x(1) , x(2) , y, 0), x˙ (2) = X (2) (x(1) , x(2) , y, 0), y˙ = Y (x(1) , x(2) , y, 0). Equations (9.19) and (9.11) imply X (1) (0, 0, 0, 0) = 0. The non-degeneracy of the origin guarantees the inequalities X (2) (0, 0, 0, 0) > 0,

00 Yyy (0, 0, 0, 0) = ζ > 0,

by changing, if necessary, the directions of the x(2) and y axes. The existence of canards and uniqueness of solutions of (9.18) is guaranteed by the following assumption: (1)

00 2Xx(2) (0, 0, 0, 0)Yyy (0, 0, 0, 0) − Xy(1) (0, 0, 0, 0)Yx00(2) y (0, 0, 0, 0) < 0,

Xy(1) (0, 0, 0, 0) > 0. Lemma 9.22. There exist Ta < 0 < Tr such that system (9.15) with the initial condition x(0) = y(0) = 0, has the unique solution w∗ (t) = (x∗ (t), y ∗ (t)),

Ta < t < T r ,

and the inequalities Yy (x∗ (t), y ∗ (t)) > 0,

0 < t < Tr ,

Yy (x∗ (t), y ∗ (t)) < 0,

Ta < t < 0

178


hold. In other words, the half-trajectory (x∗ (t), y ∗ (t)), Ta < t < 0 lives on the attractive part of the slow surface (9.16), while the half-trajectory (x∗ (t), y ∗ (t)), 0 < t < Tr lives on the repulsive part. Lemma 9.22 implies a strict restriction on the possible location of canards of system (9.9) that passing near the origin: such canards should follow closely the solution w∗ (t) for a certain interval ta < 0 < tr . The above argument shows that a canard should have segments of fast motion from a small neighbourhood of some point of the repulsive part of w∗ (t) to a small neighbourhood of the attractive part of w∗ (t); this fast motion is, consequently, almost vertical (that is almost parallel to the y axis). More precisely, if there is a limit of a canard as ε → 0, then the limiting curve has necessarily vertical segments connecting the repulsive and attractive parts of w∗ (t). Assumption 9.23 Let the trajectories Γa and Γr intersect K ≥ 2 times, that is, there exist τi and σi , i = 1, . . . , K, such that x∗ (τi ) = x∗ (σi ) = x∗i , with Ta < τi < 0 < σi < Tr , and τi 6= τj for i 6= j. It is also required that the curves Γa and Γr do not self-intersect and that Y (x∗i , y) < 0,

y ∈ [y ∗ (τi ), y ∗ (σi )],

i = 1, . . . , K.

Assumption 9.24 All the intersections are transversal, i = 1, . . . , K. Theorem 9.25. There exist disjoint sets Pi ⊂ Π, the plan y = 0,Swith Pi 3 x∗ (τi ), and ε0 > 0 such that for any ε < ε0 the Poincaré map Φ : i Pi → Π of system (9.9) is (U, 1)-chaotic, where U = {P1 , . . . , PK }. S Let S ⊂ i Pi , i = 1, . . . , K, be the compact Φ-invariant set from Definition 9.19; its existence is guaranteed by Theorem 9.25. Denote by ES the topological entropy of the Poincaré map Φ with respect to the compact set S, see [65, p. 109]. Corollary 9.26. Under conditions of Theorem 9.25, for sufficiently small ε the following inequality holds: ES ≥ log K. This corollary follows from the {Pi }-chaoticity of Φ and the definition of topological entropy. See [39] and references therein for another approach to understanding chaotic canards. As an example consider the system

9.6 Other Applications

179

x˙ (1) = −ax(2) + y/3, x˙ (2) = x(1) + 1, εy˙ = x(1) + y 2 + x(2) y. The curves Γa and Γr intersect transversally on the plane (x(1) , x(2) ), see Fig. 9.8.

Fig. 9.8 Curves Γa (solid) and Γr (dashed) for a = 3.

According to the theorem above this system has a chaotic canard for a small ε.

9.6 Other Applications 9.6.1 Kaldor Model of the Business Cycle The Kaldor model of the trade cycle [63] is one of the earliest nonlinear models in macroeconomics. Kaldor was among the foremost economists in the pre-war period and based his work on Keynes’ General Theory [67]. According to Keynes, an economy should reach an equilibrium when the levels of savings and investments are equal, however such equilibria are not observed in the real world, and Kaldor’s model attempted to explain why. In Kaldor’s model investment and savings depend on the economic activity through a nonlinear relationship. He explained this by noting that investment opportunities saturate at high activity (i.e., all of the easy investments are taken, leaving higher-cost, higher-risk investments), and at low activity the level of investment will also level out due to the low levels of expected income. This leads to a sigmoidal dependance on activity. From the mathematical perspective Kaldor model of the business cycle is a discrete time nonlinear business cycle model comprised of the following

180


difference equations for income Y and capital stock K: Yt+1 = Yt + α(It − St )

(9.20)

Kt+1 = (1 − δ)Kt + It

(9.21)

with the following parameters; •

• •

•

α is the speed of adjustment (α > 0) i.e. it measures the entrepreneurs reaction to the demand excess. A small value of α means a prudent or careful reaction from the entrepreneur. A high value of α, (α > 1) means a rash reaction from the entrepreneur. δ is the depreciation rate of the capital stock (the rate at which the value of stock is reduced). The dynamic variables Yt and Kt represent the income level in period t and the capital stock in period t, respectively. It represents the investment demand in period t and St represents savings in period t. Savings are assumed to be proportional to the current level of income: St = s Yt , where s, 0 < s < 1, represents the propensity to save.

The main reason for more recent interest in the Kaldor model is the discovery of the presence of rich and complicated dynamics in its behaviour. The existence of multiple attractors, and the presence of global attractors, was proved in [17]. Here they also mention the apparent existence of chaotic behaviour in certain regions of the parameter space. This behaviour had been noted in the literature before and since, however no formal proof of the existence of chaos in the Kaldor mapping was presented until recently, by Lisa Cronin in her M.Sc. Dissertation. In [96] the Kaldor model of the business cycle is modified by the incorporation of a Preisach nonlinearity. The paper then presents a rigorous proof of chaotic behaviour, including the existence of periodic orbits of all minimal periods p > 57. The corresponding control problems were considered in [10] using the split-hyperbolicity technique.

9.6.2 Periodically Forced KdVB Equations The paper [36], which was moitivated by experimental observations presented in [28], is concerned with the existence of chaotic behaviour of second order periodically forced ordinary differential equations. These equations are a reduced form of a periodically forced Korteweg-de Vries–Burgers’ (KdVB) and extended KdVB (eKdVB) equations when viscous damping is included, and which are given by ∂u ∂u ∂u ∂3u ∂2u ∂u +∆ + au2 + bu + d 3 + ν 2 = R0 (x) . ∂t ∂x ∂x ∂x ∂x ∂x

(9.22)


181

Here ∆, a, b, d and ν are real parameters, R(x) is a periodic function and R0 is the derivative of R. When a = 0, (9.22) reduces to the forced KdVB equation, which is a well known model for weakly nonlinear dispersive waves with viscous damping. When b = 0, (9.22) reduces to the forced eKdVB equation. The ordinary differential equation ∆u0 + au2 u0 + buu0 + du000 + νu00 = R0 (x) ,

(9.23)

where u = u(x) describes the steady, time independent profiles for the eKdVB equation. In [36] the existence of profiles which are close to any shuffling of two basic profiles is proved, and hence the existence of spatially chaotic and recurrent solutions. The proofs are based on the topological hyperbolicity technique.

9.6.3 Piecewise Linear Oscillator Consider a semi-linear equation of the type x00 + α2 x = b(t) + f (x).

(9.24)

Here f is a scalar continuous bounded nonlinearity and b is 2π-periodic. The behaviour of this equation is a classical object of studies in the theory of dynamical systems. Nevertheless a lot of important questions are still open and just some recent developments in this area are mentioned here. Consider the following equation: √ (9.25) x ¨ + x = sin( 2t) + s(x) , where

 1  −1 if x ≤ − 5 , s(x) = 5x if − 15 < x < 51 ,   1 if x ≥ 15 .

This equation has been already considered in the engineering literature as a model of the saturation effects. Bifurcation phenomena and boundedness of solutions have been studied respectively in [86] and [108]. Denote by √ F the shift operator along the trajectories of equation (9.25) for the time 2π, i.e., the period of the forcing term. In other words, for any√pair (x0√, x00 ) ∈ R2 the value F (x0 , x00 ) is a two-dimensional vector (x( 2π), x0 ( 2π)), where x(t) denotes the solution of the initial value problem x(0) = x0 , x0 (0) = x00 for equation (9.25). Fig. 9.9 shows three typical long trajectories of (9.25): •

an outer invariant curve bounding all trajectories inside itself;

182

• •


two filled black circles representing a so called ‘chain of islands’, see [85]; a cloud of circles, which is a long trajectory of F identifying an area of instability; the structure of this area is similar to the classical Aubry– Mather instability area for twist maps, see [65].

Fig. 9.9 Typical trajectories of (9.25).

In general, an understanding of the dynamics of Aubry–Mather instability area is not complete. It is known that there are signs of homoclinic behaviour, see for example [65]. Nevertheless, the area in this case is not a chaotic attractor in a traditional sense: for example, it seems that the box counting dimension of a typical trajectory is fractional, while the correlation dimension is not. The existence of chaotic behaviour in the Smale sense inside the instability area was proved in [118], i.e., that a suitable restriction of some iteration F m of the mapping F is topologically conjugated to the left shift operator in the set of symbolic sequences. A typical result from [118] is as follows: √ Theorem 9.27. The point 0, − 62 is a transversal homoclinic fixed point of F . The results follow from the application of topological hyperbolicity technique and the computer-assisted verification of a set of inequalities. The beauty of this work resides in its simplicity: even though the construction


183

of the proof is computer-assisted, the set of inequalities is really small and it could be verified by hand.

9.6.4 Oscillations of a Ferromagnetic Pendulum Consider an iron pendulum oscillating in an external magnetic field, see Fig. 9.10.

Fig. 9.10 An iron pendulum in a magnetic field

Such oscillators are common in nature and in industry: a satellite orbiting the Earth, or swings in a playground (in both cases because the Earth is a huge magnet), numerous cases in microelectronics etc. Without a magnetic field, the pendulum dynamics are described (in the linear approximation) by the equation x00 + ax0 + x = sin(ωt), which can be integrated explicitly. The situation when the magnetic field is present is, however, much more complex. The iron substance of the pendulum will be magnetized and demagnetized. This process is called ferromagnetic hysteresis. The magnetization will interact with the external magnetic field, and thus the equation will have the form x00 + ax0 + x = sin(ωt) + y(t),

y(t) = Γ x(t),

(9.26)

where the nonlinearity Γ x(t) describes the interaction between the external magnetic field and the magnetized pendulum itself, and will be described more fully later. This term, Γ x(t), depends on the pendulum position as well as on the pendulum magnetization. The pendulum’s current magnetization depends in turn on the whole previous history of the pendulum motion. Thus the actual equation is of a differential-operator type.

184


Strange attractors of fractional dimension in the system (9.26) were discovered in [22] inand a rigorous computer-aided proof of chaotic behaviour, based on topological hyperbolicity, was presented there.

Appendix A

Semi-Hyperbolicity: Estimations Marcin Mazur, Jacek Tabor, Piotr Ko´ scielniak

In this appendix we look at the relation between the semi-hyperbolicity and hyperbolicity conditions from the point of view of possible numerical applications. We provide some estimations that can help in strict numerical verification of the hyperbolicity condition via the semi-hyperbolicity one. Due to its rather numerical character and frequent use of Banach algebra tools, we present a mainly self-sufficient approach, which partially differs from that presented in Chaps. 4 and 5.

A.1 Linear Case The aim of this section is to recall and generalize some results of Sect. 4.3.1 and to present comparison between hyperbolic and semi-hyperbolic linear operators. In the following sections we will generalize this approach to obtain some numerical estimations of hyperbolicity constants (in the nonlinear case) with the use of semi-hyperbolicity. Let E be a Banach space. Let us begin with the following Lemma A.1. Let S, T : E → E be commuting bounded linear operators. If ST is invertible, then T is invertible. Proof. Let U be the inverse operator to ST . Then (U S)T = U (ST ) = I and T (SU ) = (ST )U = I, which means that T has a left and right inverse, so T is invertible. t u Proposition A.2. Let E = E1 ⊕ E2 and T21 : E1 → E2 , T12 : E2 → E1 be given bounded linear operators. Define the operator T : E → E by the following matrix formula IE1 T12 T := . T21 IE2 If kT12 T21 k < 1 and kT21 T12 k < 1, then T is invertible. 185

186

A Semi-Hyperbolicity: Estimations

Proof. Let S :=

IE1 −T12 . −T21 IE2

One can easily check that IE1 − T12 T21 0 ST = T S = . 0 IE2 − T21 T12 Then, using Lemma A.1, we conclude that T is invertible (note, that IE1 − T12 T21 and IE2 − T21 T12 are invertible). t u From now on, let A : E → E be a bounded linear operator. The following theorem was used in Sect. 4. We now present its proof that comes from [93]. Theorem 4.28. If the operator A is semi-hyperbolic with respect to a split s = (λs , λu , µs , µu ) and a norm k · k, then A is spectrally hyperbolic. Proof. Take a semi-hyperbolic splitting E = E s ⊕ E u and corresponding projections P s , P u . Put A11 := P s A|E s ,

A12 := P s A|E u ,

A21 := P u A|E s ,

A22 := P u A|E u .

To show that A is hyperbolic is enough to check that the operator λIE S − A11 −A12 λIE − A := −A21 IE u − A22 is invertible for any λ ∈ C, |λ| = 1. Clearly, λIE S − A11 and λIE u − A22 are invertible and (λIE s − A11 )−1 = λ−1 (λIE u − A22 )−1 =

∞ X

(λ−1 A11 )n ,

n=0 ∞ X n (λA−1 −A−1 22 ) . 22 n=0

Thus the following estimates hold kλIE s − A11 )−1 k ≤

∞ X

kA11 kn ≤

n=0

k(λIE u − A22 )−1 k ≤ kA−1 22 k

∞ X

1 , 1 − λs

n kA−1 22 k ≤

n=0

Since λIE − A = ST , where λIE s − A11 0 S := 0 IE u − A22

1 . λu − 1

A.1 Linear Case

187

is an invertible operator and IE s −(λIE s − A11 )−1 A12 T := , −(λIE u − A22 )−1 A21 IE u to obtain the conclusion we need to check the invertibility of T . But due to the following estimates µs µu < 1, (1 − λs )(λu − 1) µs µu − A11 )−1 A12 k ≤ < 1, (1 − λs )(λu − 1)

k(λIE s − A11 )−1 A12 (λIE u − A22 )−1 A21 k ≤ k(λIE u − A22 )−1 A21 (λIE s

this is an immediate consequence of Proposition A.2.

t u

Now we show how to easily obtain a strengthened version of Theorem 4.28. Theorem A.3. If the operator A is semi-hyperbolic with respect to a split s = (λs , λu , µs , µu ) and a norm k·k, then the spectrum of A does not intersect the ring R(λ∗s , λ∗u ) := {λ ∈ C : λ∗s < |λ| < λ∗u }, where

p (λu − λs )2 − 4µs µu λs + λu = − < 1, 2 2 p (A.1) 2 (λu − λs ) − 4µs µu λs + λu ∗ λu = + > 1. 2 2 Before proceeding to the proof let us mention that the estimations obtained in (A.1) are sharp. To observe this, consider the linear operator A : R2 → R2 given by λs µs A= . µu λu λ∗s

Obviously, A is semi-hyperbolic with respect to the the split s from Theorem A.3, but one can easily check that the eigenvalues of A are exactly λ∗s and λ∗u . Consequently, there is no larger (in the sense of inclusion) ring then R(λ∗s , λ∗u ) which does not intersect the spectrum of A. Proof. Take a semi-hyperbolic splitting E = E s ⊕ E u , corresponding projections P s and P u . Put Aλ =

1 A λ

for any

λ ∈ C \ {0}.

Obviously, if λ ∈ σ(A) then Aλ is not hyperbolic. Since kP s Aλ |E s k = and, analogously,

kP s A|E s k λs ≤ |λ| |λ|

188


kP s Aλ |E u k ≤

µs , |λ|

kP u Aλ |E s k ≤

µu |λ|

and

k(P u Aλ |E u )−1 k ≤

|λ| , λu

we conclude that the operator Aλ is semi-hyperbolic if the following inequalities hold λs < |λ| < λu

and

µs µu < (|λ| − λs )(λu − |λ|).

Solving the above system we obtain λ ∈ R(λ∗s , λ∗u ). The proof is completed by the following sequence of implications λ ∈ R(λ∗s , λ∗u ) ⇒ Aλ is semi-hyperbolic ⇒ Aλ is spectrally hyperbolic ⇒ λ ∈ / σ(A). t u At the end of this section we would like to fix our attention on the inverse problem. Thus we would like to ask the following question Problem A.4. Suppose that the spectrum of the operator A does not intersect the ring R(λ∗s , λ∗u ). Does there exist an equivalent norm on E such that for arbitrary λs ∈ [0, λ∗s ] the operator A is semi-hyperbolic with respect to a split s = (λs , λu , µs , µu ), for some λu , µs , µu for which (A.1) holds? Roughly speaking we ask if we can ‘decrease’ the value of λ∗s by ‘putting’ it into the not invariant splitting (according to the semi-hyperbolicity condition). Of course, one can also ask the dual question for λu ∈ [λ∗u , ∞) but since this is an analogue, we consider only the case of λs . Let us first show that under no additional assumptions the answer for the above question is negative. Example A.5. Consider the mapping A : x 7→ x/2,

x ∈ R.

Then the only spectral value of A is 1/2, hence σ(A) ∩ R(1/2, ∞) = ∅. However, one can easily notice that for every λs < 1/2 and λu , µs , µu such that (A.1) holds, the operator A does not satisfy semi-hyperbolicity conditions with respect to the split s = (λs , λu , µs , µu ) and any norm on R. Nonetheless, we show that in some cases, the inverse holds.

A.1 Linear Case

189

Example A.6. Let λ∗s ∈ [0, 1), λ∗u ∈ (1, ∞). Consider the mapping A : R2 → R2 given in the matrix form by ∗ λ 0 A := s ∗ . 0 λu Then

σ(A) = {λ∗s , λ∗u } ⊂ C \ R(λ∗s , λ∗u ).

To show that in this case the answer to Problem A.4 is positive, let us fix b1 , b2 ∈ R such that b1 · b2 ∈ [0, λ∗s /λ∗u ]. Consider the operator B : R2 → R2 , defined, in the matrix form, by 0 b1 B= . b2 0 Put E s = (I − B)(R × {0})

and

E u = (I − B)({0} × R),

where I denotes the identity operator. It is easy to see that the projections corresponding to the splitting E s ⊕ E u are given by 1 1 b1 s , P = 1 − b1 b2 −b2 −b1 b2 1 −b1 b2 −b1 Pu = . b2 1 1 − b1 b2 Then, choosing a norm k · k on R2 such that kes k = keu k = 1, where es = [1, −b2 ]T ∈ E s and eu = [−b1 , 1]T ∈ E u , we obtain ∗ λs − λ∗u b1 b2 s ke k = kP Ae k = 1 − b1 b2 ∗ λs b2 − λ∗u b2 u u s ke k = kP Ae k = 1 − b1 b2 ∗ λu b1 − λ∗s b1 s s u ke k = kP Ae k = 1 − b1 b2 s

s

λ∗

1 − λu∗ b1 b2 λ∗s − λ∗u b1 b2 s = λ∗ ≤ λ∗s < 1, 1 − b1 b2 1 − b1 b2 s λ∗u − λ∗s |b2 |, 1 − b1 b2 λ∗u − λ∗s |b1 |, 1 − b1 b2

∗ λ∗ ∗ ∗ 1 − λ∗s b1 b2 λu − λ∗s b1 b2 u λ − λ b b 1 2 u s u ke k = = λ∗ ≥ λ∗u > 1. kP Ae k = 1 − b1 b2 1 − b1 b2 1 − b1 b2 u u

u

Thus, putting λs =

λ∗s − λ∗u b1 b2 , 1 − b1 b2

λu =

λ∗u − λ∗s b1 b2 , 1 − b1 b2

190


and µs =

λ∗u − λ∗s |b1 |, 1 − b1 b2

µu =

λ∗u − λ∗s |b2 |, 1 − b1 b2

one can easily check, by direct computations, that (A.1) holds and that λs ∈ [0, λ∗s ],

µs µu < (1 − λs )(λu − 1).

We have obtained even more then we wanted — by modifying the values of b1 and b2 we can not only realize any value λs ∈ [0, λ∗s ], but we control also one of the constants µs , µu . The above example leads us to the following Conjecture A.7. Let A be a hyperbolic operator such that the spaces E s and E u are isomorphic. Then the answer to Problem A.4 is positive.

A.2 Functional Calculus and its Consequences Since we will need some basic results from the operator functional calculus, for the convenience of the reader and to establish notation we present them in this section. For more information on this subject we refer the reader to [64]. By Sr = {λ ∈ C : |λ| = r} we denote the positively oriented circle with the radius r. Assume that σ(A) ∩ R(λ∗s , λ∗u ) = ∅ for some constants λ∗s < 1 and λ∗u > 1. Hence, by Theorem 4.28, A is metrically hyperbolic in some equivalent norm. We put Z 1 (λI − A)−1 dλ, P∗u = I − P∗s for r ∈ (λ∗s , λ∗u ). (A.2) P∗s = 2πi Sr Then P∗s and P∗u does not depend on the choice of r ∈ (λ∗s , λ∗u ). Moreover, P∗s and P∗u are the projections corresponding to the unique splitting from the metric definition of hyperbolicity (we have E = E∗s ⊕E∗u for E∗s,u = P∗s,u (E)). Moreover, we have the following formulas for the iterations of A on spaces E∗s and E∗u Z Ak x s = λk (λI − A)−1 xs dλ for xs ∈ E∗s , k ≥ 0, (A.3) Sr Z (A|E∗u )−k xu = − λ−k (λI − A)−1 xu dλ for xu ∈ E∗s , k ≥ 1. (A.4) Sr

A.2 Functional Calculus and its Consequences

191

Our aim in this section it to give exact estimations of respective iterates of A on subspaces E∗s and E∗u . Theorem A.8. Let A be a semi-hyperbolic operator with respect to a split s = (λs , λu , µs , µu ), a constant h and a norm k · k, λ∗s , λ∗u be given by (A.1) and r ∈ (λ∗s , λ∗u ) be arbitrary. We put λu − λs + µs + µu Cr := h. (r − λs )(λu − r) − µs µu Then k(A|E∗s )k k ≤ Cr · rk ,

k(A|E∗u )−k k ≤ Cr · r−k

k ∈ N, k ≥ 1.

for

Clearly, if we are interested in the estimation of k(A|E∗s )k k we should take r ∈ (λ∗s , 1), and r ∈ (1, λ∗u ) if we need to estimate k(A|E∗u )−k k. Theorem A.8 is a direct consequence of (A.3), (A.4) and the following proposition: Proposition A.9. Let A be a semi-hyperbolic operator with respect to a split s = (λs , λu , µs , µu ), a constant h and a norm k · k. (Hence, by Theorem A.3, σ(A) ∩ R(λ∗s , λ∗u ) = ∅ where λ∗s and λ∗u are given by (A.1).) Then k(λI − A)−1 k ≤

λu − λs + µs + µu h (|λ| − λs )(λu − |λ|) − µs µu

for

λ ∈ R(λ∗s , λ∗u ).

Proof. Take a semi-hyperbolic splitting E = E s ⊕ E u , corresponding projections P s and P u . Choose x, y ∈ E, λ ∈ R(λ∗s , λ∗u ) such that (λI − A)x = y. Then λx = y + Ax and hence λP s x = P s y + P s AP s x + P s AP u x, P u x = P u y + P u AP s x + P u AP u x. Since U = P u A|E u : E u → E u is an invertible operator with kU −1 k ≤ (λu )−1 , we obtain P u x = U −1 (λP u x − P u y − P u AP s x). It follows that |λ|kP s xk ≤ hkyk + λs kP s xk + µs kP u xk, λu kP u xk ≤ hkyk + |λ|kP u xk + µu kP s xk and, in consequence,

192


(|λ| − λs )kP s xk − µs kP u xk ≤ hkyk, (λu − |λ|)kP u xk − µu kP s xk ≤ hkyk. Then µu (|λ| − λs )kP s xk − µs µu kP u xk ≤ µu hkyk, (|λ| − λs )(λu − |λ|)kP u xk − µu (|λ| − λs )kP s xk ≤ (|λ| − λs )hkyk, and hence kP u xk ≤

µu + |λ| − λs hkyk. (|λ| − λs )(λu − |λ|) − µs µu

kP s xk ≤

µs + λu − |λ| hkyk. (|λ| − λs )(λu − |λ|) − µs µu

Analogously,

Let us note that the above calculations are based on the following estimates λs < |λ| < λu

and

µs µu < (|λ| − λs )(λu − |λ|),

which can be easily verified as in the proof of Proposition A.3. Thus we have k(λI − A)−1 yk = kxk ≤ kP s xk + kP u xk ≤

λu − λs + µs + µu hkyk. (|λ| − λs )(λu − |λ|) − µs µu t u

As a direct consequence of Proposition A.9 and (A.2) we obtain the following Corollary A.10. Let A be a semi-hyperbolic operator with respect to a split s = (λs , λu , µs , µu ), a constant h and a norm k · k. Then kP∗s k ≤ L,

kP∗u k ≤ L + 1,

where P∗s , P∗u are given by (A.2) and L=

λu − λs + µs + µu h. (1 − λs )(λu − 1) − µs µu

A.3 General Case The results of this section are strictly related to ones obtained in Chap. 5. Generally, we show that any diffeomorphism of a compact Riemannian manifold, which is semi-hyperbolic on some compact invariant set is, in fact,

A.3 General Case

193

hyperbolic on this set. We consider here a variable norm version of semihyperbolicity condition (see Definition 3.4). Nevertheless, one can easily adapt all the proofs to the other concepts of semi-hyperbolicity that were presented in Sect. 3.1. Let f : M → M be a diffeomorphism of a compact Riemannian manifold M . For x ∈ M consider the Banach space Ex ⊂ l∞ (Z, T M ) consisting of all bounded sequences v = {vn } of vectors vn ∈ Tf n (x) M ⊂ T M , endowed with the supremum norm kvk∞ x = sup kvn kx,n . n∈Z

Here and later on k · kx,n denotes, for each n ∈ Z, an underlying Riemannian norm on Tf n (x) M . We define the bounded operator Ax on Ex by the following formula (Ax v)n+1 = T ff n (x) vn

for

v ∈ Ex .

From the above definition it directly follows that we can identify the operator Ax with Ay in the case of x and y lying on the same trajectory of f . (Compare the above notation with that used in Chap. 4.) Now we are ready to present the main result of this appendix (cf. [95]). Theorem A.11. Assume K is a compact f -invariant subset of M , s = (λs , λu , µs , µu ) is a split and h is a real positive number. Let k · kx be a Riemannian norm on Tx M for x ∈ K. If f is (s, h)-semi-hyperbolic for f according to the norm k · kx , then for every x ∈ K the operator Ax is semi-hyperbolic with respect the split s, the constant h and the norm k · k∞ x . (ii) Assume that for every x ∈ K the operator Ax is semi-hyperbolic with ∗ ∗ respect the split s, the constant h and the norm k · k∞ x . Let λs , λu be ∗ ∗ given by (A.1) and γs , γu be arbitrary reals such that

(i)

λ∗s < γs∗ < 1 < γu∗ < λ∗u . Put C ∗ = max

(λu − λs + µs + µu )h (λu − λs + µs + µu )h , (γs∗ − λs )(λu − γs∗ ) − µs µu (γu∗ − λs )(λu − γu∗ ) − µs µu

.

Then there exists a splitting Tx M = Exs ⊕ Exu varying continuously in x ∈ K, such that T ff n (x) (Efs n (x) ) = Efs n+1 (x) ,

T ff n (x) (Efun (x) ) = Efun+1 (x) ,

kT ffkn (x) v s kx,n+k ≤ C ∗ (γs∗ )k kv s kx,n , u ∗ ∗ −k kT ff−k kv u kx,n n (x) v kx,n−k ≤ C (γu )

for all x ∈ K, n, k ∈ Z, k > 0, v s ∈ Efs n (x) and v u ∈ Efun (x) .

(A.5) (A.6)

194


Before we proceed with the proof let us quote an important direct consequence of Theorem A.11. Theorem A.12. Assume that the diffeomorphism f is semi-hyperbolic on a compact f -invariant subset K of the manifold M . Then the set K is hyperbolic for f . Proof of Theorem A.11. (i) Take x ∈ K and, for each n ∈ Z, projections s u Px,n and Px,n corresponding to a given semi-hyperbolic splitting Tf n (x) M = s u Ex,n ⊕ Ex,n . Then it is easy to see that the operators Pxs and Pxu , defined by Y Y s u Pxs = Px,n and Pxu = Px,n , n∈Z

n∈Z

are bounded projections inducing a splitting of the space Ex that makes Ax a semi-hyperbolic operator with respect to s, h and k · k∞ x . (ii) Fix γs∗ ∈ (λ∗s , 1), γu∗ ∈ (1, λ∗u ) and take x ∈ K. For the operator Ax consider a hyperbolic splitting Ex = Exs ⊕ Exu , corresponding to projections Pxs and Pxu . Then by Corollary A.10 and Theorem A.8 we obtain that ∗ ∗ k k(Ax |Exs )k k∞ x ≤ C (γs ) ,

∗ ∗ −k k(Ax |Exu )−k k∞ x ≤ C (γu )

for all

k > 0,

u ∞ max{kPxs k∞ x , kPx kx } ≤ L + 1.

where L= C ∗ = max

λu − λs + µs + µu h, (1 − λs )(λu − 1) − µs µu

(λu − λs + µs + µu )h (λu − λs + µs + µu )h , ∗ ∗ ∗ (γs − λs )(λu − γs ) − µs µu (γu − λs )(λu − γu∗ ) − µs µu

.

For each n ∈ Z put Ex,n = Tf n (x) M

and

Ax,n = T ff n (x)

and note that T ffkn (x) = T ff n+k−1 · · · T ff n (x) = Ax,n+k−1 · · · Ax,n , −1 −1 −1 −1 T ff−k n (x) = T ff n−k+1 · · · T ff n (x) = Ax,n−k · · · Ax,n−1 . s u Consider the spaces Ex,n , Ex,n ⊂ Ex,n defined as follows s Ex,n = {v ∈ Ex,n : sup kAx,n+k−1 · · · Ax,n vkx,n+k < ∞}, k≥0

u Ex,n

−1 = {v ∈ Ex,n : sup kA−1 x,n−k · · · Ax,n−1 vkx,n−k < ∞}. k≥0

Obviously, since Ax,n is invertible, s s Ax,n (Ex,n ) = Ex,n+1

and

u u Ax,n (Ex,n ) = Ex,n+1 .

(A.7)

A.3 General Case

195

s u n Firstly note that Pxn (Exs ) ⊂ Ex,n and Pxn (Exu ) ⊂ QEx,n , where Px denotes the canonical projection on the n-th coordinate in n∈Z Ex,n . Indeed, taking sequences vs ∈ Exs and vu ∈ Exu it is easily seen that for k ∈ N, k ≥ 1

kAx,n+k−1 · · · Ax,n vns kx,n+k ≤ kAxk vs k∞ ≤ C ∗ (γs∗ )k kvs k∞ ≤ C ∗ kvs k∞ x x x , −1 u −k u ∞ ∗ ∗ −k ∗ u ∞ kA−1 kvu k∞ x ≤ C kv kx . x,n−k · · · Ax,n−1 vn kx,n−k ≤ kAx v kx ≤ C (γu ) s u Now we show that, for each n ∈ Z, the spaces Ex,n and Ex,n form splitting of the space Ex,n . s u Indeed, if v ∈ Ex,n ∩ Ex,n then −1 −1 v = (. . . , A−1 x,n−2 Ax,n−1 v, Ax,n−1 v, vn = v, Ax,n v, Ax,n+1 Ax,n v, . . .) ∈ Ex

and since Ax is hyperbolic and Ax v = v we obtain v = 0. The equality s u Ex,n = Ex,n + Ex,n follows immediately from the following observation v = vn = (Pxs v + Pxu v)n = (Pxs v)n + (Pxu v)n s u ∈ Pxn (Exs ) + Pxn (Exu ) ⊂ Ex,n + Ex,n

for any v ∈ Ex with vn = v ∈ Ex,n , e.g., v = (. . . , 0, vn = v, 0, . . .). u s . Note that ⊕ Ex,n This finishes the proof of the condition Ex,n = Ex,n (A.5) is a trivial consequence of (A.7). Finally, we also obtain that Y Y s u Pxs = Px,n , and Pxu = Px,n , n∈Z

n∈Z

u s denote the projections corresponding to the splitting and Px,n where Px,n u s Ex,n = Ex,n ⊕ Ex,n , as the result of the earlier remarks and the following sequence of implications that hold for each n ∈ Z s u v ∈ Ex ⇒ vn = (Pxs v)n + (Pxu v)n ∈ Ex,n + Ex,n s ⇒ Pn,x vn = (Pxs v)n

u and Pn,x vn = (Pxu v)n .

In particular, if v = (. . . , 0, vn = v, 0, . . .) then s Pxs v = (. . . , 0, Px,n v, 0, . . .)

u and Pxu v = (. . . , 0, Px,n v, 0, . . .).

s u Now take v s ∈ Ex,n , v u ∈ Ex,n , corresponding sequences s vs = (. . . , 0, vns = v s , 0, . . .) = (. . . , 0, Px,n v s , 0, . . .) = Pxs vs ∈ Exs , u vu = (. . . , 0, vnu = v u , 0, . . .) = (. . . , 0, Px,n v u , 0, . . .) = Pxu vu ∈ Exu ,

and note that the following estimates hold for all k > 0

196


∗ ∗ k s ∞ kAx,n+k−1 · · · Ax,n v s kx,n+k ≤ kAxk vs k∞ x ≤ C (γs ) kv kx

= C ∗ (γs∗ )k kv s kx,n , −1 u −k u ∞ ∗ ∗ −k kA−1 kvu k∞ x x,n−k · · · Ax,n−1 v kx,n−k ≤ kAx v kx ≤ C (γu )

= C ∗ (γu∗ )−k kv u kx,n , which completes the proof of the condition (A.6). Since the uniform boundedness of projections guarantees continuity of a hyperbolic splitting (see [110, s u Sect. 2.2]), it is enough to verify that the projections Px,n and Px,n are uniformly bounded, that is sup x∈K,n∈Z

s u max{kPx,n kx,n , kPx,n kx,n } < ∞.

Indeed, for v ∈ Ex,n and v = (. . . , 0, vn = v, 0, . . .) we have s Pxs v = (. . . , 0, Px,n v, 0, . . .)

u and Pxu v = (. . . , 0, Px,n v, 0, . . .)

and then we obtain the following estimates s s ∞ ∞ s ∞ kPn,x vkx,n = kPxs vk∞ x ≤ kPx kx kvkx = kPx kx kvkx,n ≤ (L + 1)kvkx,n , u u ∞ ∞ u ∞ kPn,x vkx,n = kPxu vk∞ x ≤ kPx kx kvkx = kPx kx kvkx,n ≤ (L + 1)kvkx,n .

But L depends neither on x ∈ K nor on n ∈ Z, which makes the proof complete. t u

Appendix B

Set-Valued Semi-Hyperbolic Mappings Janosch Rieger

The concept of semi-hyperbolicity can be extended to set-valued dynamics generated by a set-valued mapping, see, e.g., [114–116]. As this topic is a subject of ongoing research, we can only provide a very limited presentation of this generalization. We will introduce a strong and a weak set-valued notion of semi-hyperbolicity and discuss the issues of expansivity and shadowing.

B.1 Definition Let C (Rd ) be the collection of all nonempty compact subsets of Rd . For A, B ∈ C (Rd ), we define the Hausdorff semi-distance and the Hausdorff distance by dist(A, B) := sup dist(a, B) := sup inf ka − bk a∈A

a∈A b∈B

and distH (A, B) := max{dist(A, B), dist(B, A)}, respectively. When equipped with the Hausdorff distance, C (Rd ) is a complete metric space. A set-valued mapping F : Rd → C (Rd ) is a function which assigns a nonempty convex and compact subset to any point in Rd . The Hausdorff metric allows us to specify notions of continuity for F . We call such a mapping continuous at x ∈ Rd if limx˜→x distH (F (x), F (˜ x)) = 0, and we say that F is continuous on an open subset X ⊂ Rd if it is continuous at any point x ∈ X. Moreover, a set-valued mapping F is called Lipschitz continuous with Lipschitz constant L ≥ 0 on X if distH (F (x), F (˜ x)) ≤ Lkx − x ˜k

for any

x, x ˜ ∈ X.

197

198

B Set-Valued Semi-Hyperbolic Mappings

Consider a set-valued mapping F : X → C (X), where X is an open subset of Rd and C (X) denotes the subcollection of sets from C (Rd ) which are contained in X. The mapping F generates two types of discrete time set-valued dynamical systems by successive iteration. a)

A sequence {xn }n∈I in X is an individual trajectory of the system on a finite or infinite interval I ⊂ Z if xn+1 ∈ F (xn )

b)

holds whenever n, n + 1 ∈ I. A sequence {Xn }n∈I in C (X) is a set trajectory of the system on a finite or infinite interval I ⊂ Z if Xn+1 = F (Xn ) holds whenever n, n + 1 ∈ I. As usual, F (Xn ) := ∪x∈Xn F (x).

Both points of view have their merits, as we will see below.

B.1.1 Strong semi-hyperbolicity The notion of strong semi-hyperbolicity is defined in terms of conditions on the full image of the mapping F . In [115], a shadowing and an inverse shadowing theorem are provided which require convexity of the images, but are less restrictive with respect to the regularity of F . Definition B.1. Let s = (λs , λu , µs , µu ) be a split and K a compact subset of an open set X ⊂ Rd . A set-valued mapping F : X → C (X) is called strongly s-semi-hyperbolic on K if it can be represented as F (x) = f (x) + G(x),

(B.1)

where 1) 2) 3)

f : X → X is a Lipschitz continuous single-valued function; there exists a splitting Rd = Exs ⊕ Exu on K with projectors Pxs and Pxu for each x ∈ K; there exists a norm k · k on Rd and positive constants δ and h such that (i) dim Exu = dim Eyu for all x, y ∈ K with dist(y, F (x)) ≤ δ; (ii) supx∈K {kPxs k, kPxu k} ≤ h; (iii) the inclusion x+u+v ∈X and the inequalities

B.1 Definition

199

kPys (f (x + u + v) − f (x + u ˜ + v))k ≤ λs ku − u ˜k,

(B.2)

kPys (f (x kPyu (f (x kPyu (f (x

+ u + v) − f (x + u + v˜))k ≤ µs kv − v˜k,

(B.3)

+ u + v) − f (x + u ˜ + v))k ≤ µu ku − u ˜k,

(B.4)

+ u + v) − f (x + u + v˜))k ≥ λu kv − v˜k

(B.5)

hold for all x, y ∈ K with dist(y, F (x)) ≤ δ and all u, u ˜ ∈ Exs and u v, v˜ ∈ Ex such that kuk, k˜ uk, kvk, k˜ v k ≤ δ; 4)

G : X → C (X) is a Lipschitz continuous set-valued mapping with Lipschitz constant L > 0. satisfying 10 · Ldh < ν(s), where ν(s) =

(B.6)

(1 − λs )(λu − 1) − µs µu λu − λs + µs + µu

is the robustness constant associated with the split s. This notion of semi-hyperbolicity requires a certain degree of uniformity of the splitting. If the images of G are large, conditions (B.2) to (B.5) must be satisfied for y varying in a large region. At the same time, the restricion (B.6) imposed on the Lipschitz constant of G ensures that the semi-hyperbolicity of f dominates the behaviour of the set-valued mapping F . The factor 10 · d originates from a geometric principle, the so-called Parametrization Theorem, see Lemma B.4. For set-valued mappings with convex and compact values, it is possible to define strong semi-hyperbolicity and obtain strong shadowing results without this factor (see [115]), because in that case, continuous local parametrizations can be obtained from minimal-norm-type selections. In [55], spatial discretization effects near compact invariant sets are modelled by inflated right hand sides. If the original mapping is semi-hyperbolic, the resulting set-valued mapping is strongly semi-hyperbolic if either the inflation and the variation of the splitting are small enough, see Example B.2. The saddle dynamics analyzed in [34] are generated by a strongly semi-hyperbolic set-valued mapping. The focus of this paper is the characterization of analogs of the stable and unstable manifolds. Example B.2. Let X = Rd and let F (x) = f (x) + Bg1 (x) (g2 (x)), where f : Rd → Rd is Lf -Lipschitz and s-semi-hyperbolic on some compact set K and g1 : Rd → R+ and g2 : Rd → Rd are L1 - and L2 -Lipschitz. Assume that the projectors Pxs and Pxu satisfy kPys − Py˜s kop , kPyu − Py˜u kop ≤ LP ky − y˜k, where LP > 0 and k · kop is the operator norm induced by the vector norm according to which f is semi-hyperbolic. Let us check under which conditions on P s , P u , g1 , and g2 , the mapping F is strongly semi-hyperbolic on K with respect to the original splitting and

200


a modified split. For any x ∈ K and y ∈ Rd with dist(y, F (x)) ≤ δ, there exists some y˜ ∈ Rd such that ky − y˜k ≤ g1 (x) + kg2 (x)k and k˜ y − f (x)k ≤ δ, and hence, kPys (f (x + u + v) − f (x + u ˜ + v))k = k(Py˜s + (Pys − Py˜s ))(f (x + u + v) − f (x + u ˜ + v))k ≤ kPy˜s (f (x + u + v) − f (x + u ˜ + v))k + kPys − Py˜s kop · kf (x + u + v) − f (x + u ˜ + v)k ≤ λs ku − u ˜k + LP ky − y˜kLf ku − u ˜k ≤ λs + LP Lf (g1 (x) + kg2 (x)k) ku − u ˜k, ˜-semi-hyperbolic with respect to the split etc., and thus, F is strongly s ˜ = (λs + κ, λu − κ, µs + κ, µu + κ), s provided that κ = LP Lf sup (g1 (x) + kg2 (x)k) < ν(s)

(B.7)

x∈K

and the Lipschitz constant L = L1 + L2 of the mapping x 7→ Bg1 (x) (g2 (x)) satisfies 10 · Ldh < ν(˜ s). (B.8) By the very definition of semi-hyperbolicity, the Lipschitz constant Lf of f is larger than one. Consequently, there is a tradeoff between the tolerable roughness of the splitting measured in terms of the Lipschitz constant LP and the maximal sizes of the the radii of the images g1 and the perturbation g2 expressed by inequality (B.7).

B.1.2 Weak semi-hyperbolicity The notion of weak hyperbolicity is based on a representation via selections. A single-valued function f : X → X is called a selection of a set-valued mapping F : X → C(X) if f (x) ∈ F (x) for all x ∈ X. We will say that F is weakly semi-hyperbolic if it is parametrized by a family of hyperbolic selections subject to the same split and splitting. This hyperbolicity condition is more general than the notion of strong semi-hyperbolicity, but less graphic and more difficult to check. For a more detailed presentation of this issue, we refer to [116]. Definition B.3. Let s = (λs , λu , µs , µu ) be a split and let K be a compact subset of an open set X ⊆ Rd . The set-valued mapping F : X → C (X) is called weakly s-semi-hyperbolic on K if there exist

B.1 Definition

1) 2)

201

a splitting Rd = Exs ⊕ Exu on K with projectors Pxs and Pxu for each x ∈ K, a norm k · k on Rd , and positive constants δ and h; a parametrization f : X × U → X of F such that (i) U is an arbitrary index set; (ii) F (x) = f (x, U ) := ∪u∈U f (x, u) for any x ∈ X; (iii) for any u ∈ U , the function f (·, u) is Lipschitz and s-semihyperbolic with respect to the split s, the above splitting, the norm k · k and δ and h in the sense of Definition 3.6. A strongly semi-hyperbolic mapping is weakly semi-hyperbolic.

Lemma B.4. Let F : X → C (X) be a set-valued mapping which is strongly ˜-semi-hyperbolic s-semi-hyperbolic on a compact set K. Then F is weakly s with respect to the same splitting and the split ˜ = (λs + κ, λu − κ, µs + κ, µu + κ), s where κ = 10 · Ldh. Proof. If F is strongly s-semi-hyperbolic, then F (x) = f (x) + G(x), where f is s-semi-hyperbolic and G is L-Lipschitz. According to the Parametrization Theorem (see [12, Th. 9.7.2]), there exists a parametrization g : X × B1 (0) → X of G such that G(x) = g(x, B1 (0)) for all x ∈ X and g is Lipschitz in the first argument with Lipschitz constant 10 · Ld. Hence, we can parametrize F (x) = f (x) + g(x, B1 (0)) for all x ∈ X, and the parametrizing selections f (·) + g(·, z), z ∈ B1 (0), satisfy kPys [f (x + u + v) + g(x + u + v, z)] − [f (x + u ˜ + v) + g(x + u ˜ + v, z)] k ≤ kPys (f (x + u + v) − f (x + u ˜ + v))k + kPys k · kg(x + u + v, z) − g(x + u ˜ + v, z)k ≤ (λs + 10 · Ldh)ku − u ˜k etc. for all x, y ∈ K with ky − f (x) − g(x, z)k ≤ δ, which shows that the ˜-semi-hyperbolic, because κ < ν(s). selections are s t u Example B.5. As in Example B.2, consider the set-valued mapping F (x) = f (x) + Bg1 (x) (g2 (x)), where f : Rd → Rd is Lf -Lipschitz and s-semihyperbolic on some compact set K and g1 : Rd → R+ and g2 : Rd → Rd are L1 - and L2 -Lipschitz. Defining g(x, z) := g2 (x) + g1 (x)z, z ∈ B1 (0), we obtain a parametrization F (x) = f (x) + g(x, B1 (0)) such that g(·, z) is L-Lipschitz in the first argument with L = L1 + L2 . Fix any z ∈ B1 (0). For any y with dist(y, F (x)) ≤ δ, there exists some y˜ such that ky − y˜k ≤ g1 (x) + kg2 (x)k and k˜ y − f (x)k ≤ δ, and consequently,

202


kPys [f (x + u + v) + g(x + u + v, z)] − [f (x + u ˜ + v) + g(x + u ˜ + v, z)] k ≤ kPys f (x + u + v) − f (x + u ˜ + v) k + kPys g(x + u + v, z) − g(x + u ˜ + v, z)k s s s ≤ k(Py˜ + (Py − Py˜ )) f (x + u + v) − f (x + u ˜ + v) k + hLku − u ˜k ≤ λs ku − u ˜k + LP ky − y˜kLf ku − u ˜k + hLku − u ˜k ≤ λs + LP Lf (g1 (x) + kg2 (x)k) + hL ku − u ˜k ˜-semi-hyperbolic with respect etc., so that f (·)+g(·, z) and consequently F is s to the split ˜ = (λs + κ, λu − κ, µs + κ, µu + κ), s if κ = LP Lf sup (g1 (x) + kg2 (x)k) + hL < ν(s).

(B.9)

x∈K

The counterpart of the heavy restriction (B.8) required for strong semihyperbolicity is the additional term hL in (B.9). The factor 10 · d originating from the Parametrization Theorem is obsolete, because F has a simple structure and possesses a particularly mild parametrization.

B.2 Expansivity Let us adapt Definition 2.8 to set trajectories. Fix any norm k · k on Rd . For any subset K ⊆ X and k with 0 ≤ k ≤ ∞, let STr±k (F, K) denote the collection of all set trajectories X = {X−k , · · · , X0 , · · · , Xk } of the mapping F that are entirely contained in the set K, and for any ˜ ∈ STr±k (F, K) define X, X ˜ = max distH (Xn , X ñ) ρk (X, X) −k≤n≤k

if k < ∞ and

˜ = max distH (Xn , X ˜ n ). ρ∞ (X, X) n∈Z

Definition B.6. A set-valued dynamical system generated by a mapping F : X → C (X) is said to be ξ-expansive on X if for any two infinite set ˜ the inequality trajectories X and X, ˜ ≤ξ ρ∞ (X, X) ˜0. implies that X0 = X

B.2 Expansivity

203

Lemma B.7. Let F : K → C (K) be a continuous ξ-expansive mapping. Then for every 0 < ε and θ < ξ there exists a positive integer ℵ(ε, θ) such ˜ > θ holds for all X, X ˜ ∈ STr±k (F, K) with distH (X0 , X ˜0) ≥ ε that ρk (X, X) and k ≥ ℵ(ε, θ). In order to prove this lemma, we need a set-valued analog of the BolzanoWeierstrass Theorem. Lemma B.8. Let K ⊆ Rd be compact and let {Kn }n∈N be a sequence of compact sets such that Kn ⊆ K. Then there exists a subsequence {Knj }j∈N of {Kn }n∈N and a compact set K ∗ ⊆ K such that distH (Knj , K ∗ ) → 0 as j → ∞. (0)

(0)

Proof. Define a grid ∆0 := {x1 , . . . , xl(0) } := K ∩ 20 Z and a system of sets (0)

−1 S0 := {∪m ): j=1 Q(xij , 2

ij ∈ {1, . . . , l(0)}, ij 6= ij 0

for

j 6= j 0 , 1 ≤ m ≤ l(0)},

Qd where Q(x, r) := i=1 [xi − r, xi + r] is the cube with radius r centered at x. Since S0 has finitely many elements, there exists some S0∗ ∈ S0 such that for infinitely many Kn , S0∗ is the minimal element of S0 (with respect to inclusion) containing Kn . We denote this subsequence by Kn0 and set K1∗ := K10 . (1) (1) Now we define a finer grid ∆1 := {x1 , . . . , xl(1) } := S0∗ ∩ 2−1 Z on S0∗ and a system of sets (1)

−2 ): S1 := {∪m j=1 Q(xij , 2

ij ∈ {1, . . . , l(1)}, ij 6= ij 0

for j 6= j 0 , 1 ≤ n ≤ l(m)}.

Since S1 has finitely many elements, there exists some S1∗ ∈ S1 such that for infinitely many Kn0 , S1∗ is the minimal element of S1 containing Kn0 . We denote this subsequence by Kn00 and set K2∗ := K200 . Iterating this procedure, we obtain a subsequence Kn∗ of the original Kn and a nested sequence Sn∗ of nonempty compact sets such that Kn∗ ⊆ Sn∗ . It is well-known that the intersection K ∗ := ∩n∈N Sn∗ is a nonempty compact set. We have dist(Sn∗ , K ∗ ) → 0 as n → ∞, because otherwise there exist ε > 0 and a sequence of elements xnj ∈ Sn∗j such that dist(xnj , K ∗ ) > ε. Since K is compact, we can extract a subsequence (without changing notation) such that limj→∞ xnj =: x ¯ ∈ K. But then x ¯ ∈ Sn∗ for every n ∈ N, and the ∗ ∗ resulting x ¯ ∈ K = ∩n∈N Sn is a contradiction. Consequently, dist(Kn∗ , K ∗ ) ≤ dist(Kn∗ , Sn∗ ) + dist(Sn∗ , K ∗ ) → 0

as n → ∞.

On the other hand, we have dist(Sn∗ , Kn∗ ) ≤ 2−n−1 by construction. Hence,

204


dist(K ∗ , Kn∗ ) ≤ dist(K ∗ , Sn∗ ) + dist(Sn∗ , Kn∗ ) → 0

as

n → ∞,

which proves that limn→∞ distH (Kn∗ , K ∗ ) = 0.

t u

Proof of Lemma B.7. Suppose the contrary. Then for any k ≥ 0, there exist ˜ (k) ∈ STr±k (F, K) satisfying set trajectories X(k) , X (k) ˜ (k) distH (X0 , X 0 )≥ε

˜ (k) ) ≤ θ < ξ. and ρk (X(k) , X (k )

(k)

By Theorem B.8, the sequence {X0 }k∈N contains a subsequence {X0 j }j∈N which converges to some nonempty compact set X0∗ with respect to the Hausdorff distance. (k) Now we apply this argument to Xn , n = ±1, ±2, . . ., taking subsequences of the subsequences generated in the previous steps, in order to establish the existence of a limit sequence X∗ := {Xn∗ }n∈Z . Note that by construction, we can assume that on any finite interval I ⊆ Z, the limits Xn∗ , n ∈ I, are generated by the same subsequence. ∗ Let us check that X∗ ∈ STr±∞ (F, K). For arbitrary x∗n+1 ∈ Xn+1 , there (kj ) (kj ) (kj ) ∗ exists a sequence xn+1 ∈ Xn+1 with limj→∞ xn+1 = xn+1 . By definition, (k )

(k )

(k )

(k )

j xn+1 ∈ F (xn j ) for some xn j ∈ Xn j . By compactness of K, and since the ∗ are generated by the same subsequence, we can once more limits Xn∗ and Xn+1 (k ) extract a subsequence (without changing notation) so that limj→∞ xn j = x∗n for some x∗n ∈ Xn∗ . Hence

dist(x∗n+1 , F (x∗n )) (k )

(k )

j j ∗ j) ≤ |x∗n+1 − xn+1 | + dist(xn+1 , F (xn(kj ) )) + dist(F (x(k n ), F (xn )) → 0

as j → ∞. Since the images of F are compact, x∗n+1 ∈ F (x∗n ) and thus ∗ Xn+1 ⊆ F (Xn∗ ). On the other hand, let x∗n ∈ Xn∗ and x∗n+1 ∈ F (x∗n ) be given. Again, there (k ) (k ) (k ) exists a sequence xn j ∈ Xn j such that limj→∞ xn j = x∗n . But then, ∗ dist(x∗n+1 , Xn+1 ) ∗ j) ≤ dist(x∗n+1 , F (x∗n )) + dist(F (x∗n ), F (xn(kj ) )) + dist(F (x(k n ), Xn+1 ) → 0 ∗ ∗ , we have x∗n+1 ∈ Xn+1 and thus F (Xn∗ ) ⊆ as j → ∞. By compactness of Xn+1 ∗ Xn+1 . ˜ (k) obtaining a limiting set We can repeat the whole construction with X ∗ ∗ ∗ ˜ ˜ trajectory X . But then X and X are set trajectories such that

˜ 0∗ ) ≥ ε and distH (X0∗ , X

˜ n∗ ) ≤ θ < ξ, distH (Xn∗ , X

which contradicts the initial assumption that F is expansive.

t u

B.3 Bi-Shadowing

205

It is still unclear at present how semi-hyperbolicity is related to expansivity in the set-valued context. The following lemma only guarantees the existence of individual trajectories which satisfy the classical expansivity condition, but not expansivity in the sense of Definition B.6. Lemma B.9. Let K ⊆ Rd be a compact set. If F : K → C (K) is weakly s-semi-hyperbolic and K is invariant under F , then the generated set-valued dynamical system is exponentially expansive in the following sense: There exist an exponent r > 1 and constants ξ, c > 0 such that for any finite trajectory x = {x−n− , . . . , x0 , . . . , xn+ } and any y0 ∈ K, there exists some trajectory y = {y−n− , . . . , y0 , . . . , yn+ } such that at least one of the following inequalities holds: kxn − yn k ≥ crn kx0 − y0 k,

n = 1, 2, . . . , n+

(B.10)

n = −1, −2, . . . , −n− .

(B.11)

or kxn − yn k ≥ cr−n kx0 − y0 k, Proof. By Definition B.3, xn+1 ∈ F (xn ) = f (xn , U ),

n ∈ [−n− , n+ − 1],

and thus there exists a sequence {un } ⊆ U such that xn+1 = f (xn , un ),

n ∈ [−n− , n+ − 1].

Since K is invariant under F , it is also invariant under the sequence of mappings {f (·, un )}. In particular, there exists a sequence {yn } satisfying yn+1 = f (yn , un ),

n ∈ [−n− , n+ − 1].

As the selections f (·, un ) of F are semi-hyperbolic according to the same split and splitting, the results from Chapter 6 remain valid for the sequence {f (·, un )}. In particular, Theorem 6.5 applies and guarantees that either estimate B.10 or B.11 holds. t u

B.3 Bi-Shadowing The setting of weak semi-hyperbolicity and individual trajectories allows to prove fairly general shadowing and inverse shadowing theorems. Since strongly semi-hyperbolic mappings are weakly semi-hyperbolic (see Lemma

206


B.4), a similar result can be shown for set-trajectories. A complete and selfcontained coverage of the topic can be found in [116]. Definition B.10. A γ-pseudo-trajectory of a set-valued dynamical system generated by a mapping F : X → C (X) is a sequence y = {yn } ⊆ X on some finite or infinite interval I ∈ Z satisfying dist(yn+1 , F (yn )) ≤ γ whenever n, n + 1 ∈ I. Let Tr(F, K, γ) denote the set of all infinite γ-pseudo-trajectories remaining in K, and let Tr(F, K) = Tr(F, K, 0) denote the set of all individual trajectories remaining in K. Definition B.11. An individual trajectory x = {xn } ⊆ X is said to εshadow a γ-pseudo-trajectory y = {yn } defined on some finite or infinite interval I ⊆ Z if kxn − yn k ≤ ε, n ∈ I. Let us formulate a bi-shadowing result similar to Theorem 6.27. If we do not want to impose restrictive conditions on the geometry of the setvalued mappings, we have to measure their distance in terms of the Hausdorff distance between their relevant selections rather than the Hausdorff distance of their images. Definition B.12. Let F, Φ : X → C (X) be set-valued mappings with parametrizations F (x) = f (x, U ) and Φ(x) = φ(x, V ) for all x ∈ X, where U and V are arbitrary fixed index sets. The distance between F and Φ with respect to f and φ is defined by distf,φ (F, Φ) := max{sup inf kφ(·, v)−f (·, u)k∞ , sup inf kφ(·, v)−f (·, u)k∞ }. u∈U v∈V

v∈V u∈U

Theorem B.13. A set-valued dynamical system generated by a weakly ssemi-hyperbolic mapping F : X → C (X) is bi-shadowing on a compact subset K of X with positive parameters α = α(s, h) and β = β(s, h, δ) specified in (6.44) and (6.45) in the following sense: Let x = {xn } ∈ Tr(F, K, γ) be any given infinite pseudo-trajectory with 0 ≤ γ ≤ β and let Φ : X → C (Rd ) be any mapping that can be parametrized by a function φ : X × V → X such that (i) (ii) (iii) (iv)

V is an arbitrary index set; Φ(x) = φ(x, V ) := ∪v∈V φ(x, v) for any x ∈ X; φ(·, v) is continuous for every v ∈ V ; distf,φ (F, Φ) ≤ β − γ.

Then there exists a trajectory y = {yn } ∈ Tr(Φ, K) such that kxn − yn k ≤ α(γ + distf,φ (F, Φ)),

n ∈ Z.

B.3 Bi-Shadowing

207

Proof. An individual trajectory of the set-valued system is defined by xn+1 ∈ F (xn ) = f (xn , U ),

n ∈ Z.

If we restrict our attention to this trajectory, we may select a sequence {un } ⊆ U such that xn+1 = f (xn , un ), n ∈ Z. (B.12) By definition, all selections f (·, un ) of F are semi-hyperbolic according to the same split and splitting, so that the results developed in Chapter 6 remain valid for the non-autonomous dynamical system (B.12). Moreover, we have distf,φ (F, Φ) ≤ β − γ by assumption, so that for any n ∈ N we find some vn ∈ V such that kf (·, un ) − φ(·, vn )k∞ ≤ β − γ. Now a repetition of the chain of arguments proving Theorem 6.27 yields the existence of a sequence {yn } satisfying yn+1 = φ(yn , vn ) ∈ Φ(yn ),

n ∈ Z,

which is the desired trajectory of the set-valued mapping Φ.

t u

It was shown in [114, Remark 3] that even under stronger assumptions, the trajectory {yn } of Φ cannot be expected to be unique. Combining Lemma B.4 and Theorem B.13 yields a shadowing theorem for strongly semi-hyperbolic mappings. Theorem B.14. Consider a set-valued dynamical system generated by a mapping F (x) = f (x) + G(x), x ∈ X, which is strongly s-semi-hyperbolic on a compact subset K of X. Then F is ˜ and positive parameters α = α(˜s, h) bi-shadowing on K with a modified split s and β = β(˜ s, h, δ) specified in Theorem 6.21 in the following sense: Let x = {Xn } ∈ STr(F, K, γ) be any given infinite γ-pseudo-trajectory with 0 ≤ γ ≤ β and let Φ : X → C (Rd ) be any mapping that can be parametrized by a function φ : X × V → X such that (i) (ii) (iii) (iv)

V is an arbitrary index set; Φ(x) = φ(x, V ) := ∪v∈V φ(x, v) for any x ∈ X; φ(·, v) is continuous for every v ∈ V ; distf,φ (F, Φ) ≤ β − γ.

Then there exists a set trajectory y = {Yn } ∈ STr(Φ, K) such that distH (Xn , Yn ) ≤ α(γ + distf,φ (F, Φ)),

n ∈ Z.

The following lemma is useful for the proof of Theorem B.14. Lemma B.15. Let X be an open subset of Rd and let K ⊆ X be compact. Let γ ≥ 0 and let x(k) ∈ Tr(F, K, γ), k ∈ N, be a sequence of γ-pseudo(k) trajectories. If there exists some accumulation point x ∈ X of {xn0 : k ∈ N}

208


for some n0 ∈ Z, then there exists some x ∈ Tr(F, K, γ) such that xn0 = x (k) and xn is an accumulation point of {xn : k ∈ N} for all n ∈ Z. Proof. The construction of the limiting trajectory is the same as the one in the proof of Lemma B.7. Since K is closed, all limits are in K, and the distances are not affected by taking limits. t u Proof of Theorem B.14. Since x ∈ STr(F, K, γ), it is clear that there exists a collection Px ⊆ Tr(F, K, γ) such that Xn = ∪x∈Px xn ,

n ∈ Z.

˜-semi-hyperbolic with respect to a modified split By Lemma B.4, F is weakly s ˜ and the same splitting, so that for every x ∈ Px , Theorem B.13 guarantees s the existence of some y(x) = {yn (x)} ∈ Tr(Φ, K) such that ky(x) − xk∞ ≤ α(γ + distf,φ (F, Φ)). Let us denote Yn := {yn (x) : x ∈ Px }. According to Lemma B.15, for every accumulation point y of Yn0 there exists some y = {yn } ∈ Tr(Φ, K) such that yn0 = y and yn is an accumulation point of Yn for all n ∈ Z. Thus, the closures Yn∗ of Yn form a set-trajectory Y = {Yn∗ } ∈ STr(Φ, K) with dist(Xn , Yn∗ )H ≤ α(γ + distf,φ (F, Φ)),

n ∈ Z. t u

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List of Symbols

AT

—

transpose of the matrix A

A,D

—

linear operators associated with the sequence of matrices {An }, p. 57

AT

—

transpose of the matrix A

A,D

—

linear operators associated with the sequence of matrices {An }, p. 57

B(r)

—

the closed ball of the radius r centered at 0

B(r, x)

—

the closed ball of the radius r centered at x

C(X, Rd )

—

the Banach space of continuous functions f : X → Rd , p. 26

Cε (S)

—

ε-capacity of the compact set S, p. 17

CR(f, X)

—

the set of chain recurrent points of the mapping f , p. 16

CR(f, ε, X)

—

the set of ε-chain recurrent points of the mapping f , p. 110

Cyc(f, K, γ)

—

the set of γ-pseudo-cycles of the mapping f , p. 104

Cyc(f, K, 0), Cyc(f, K)

—

the set of cycles of the mapping f , p. 104

Df

—

differential of the mapping f

D(K)

—

the space of diffeomorphisms f : K → K

dim E

—

dimension of the linear space E

∂X

—

boundary of the set X

E

—

linear space or Banach space

Ec

—

complexification of the linear space E, p. 54

Exs , Exu

—

subspaces forming a splitting Rd = Exs ⊕ Exu , p. 21

f, ϕ, f˜, ϕ, ˜ ...

—

mappings

f |X

—

restriction of the mapping f to the set X

f ◦g

—

composition of mappings: (f ◦ g)(x) := f (g(x))

217

218

List of Symbols

H (K)

—

the space of homeomorphisms f : K → K

h(f, K)

—

topological entropy of the mapping f , p.17

hε (f, K)

—

topological ε-entropy of the mapping f , p.17

Int X

—

interior of the set X

Lip(X, Rd )

—

the Banach space of Lipschitz functions f : X → Rd , p. 26

`∞ (I, Rd )

—

the space of bounded sequences of vectors xn ∈ Rd with indices n taking values in an interval I, p. 35

M (s)

—

split matrix, p. 35

Oε (X)

—

ε-neighbourhood of the set X

Per(f, X)

—

the set of periodic points of the mapping f .

Pxs , Pxu

—

projectors corresponding to a splitting Rd = Exs ⊕ Exu , p. 21

—

real coordinate space of dimension d

—

splitting of Rd , p. 21

s, s = (λs , λu , µs , µu )

—

split, p. 22

x, y, z, x ˜, y˜, z˜ . . .

—

vectors from the space Rd

TM

—

tangent bundle

Tx M

—

tangent space

T fx

—

tangent mapping

Td

—

torus of dimension d, p. 30

Tr(f, K, γ)

—

the set of γ-pseudo-trajectories of the mapping f , p. 94

Tr(f, K, 0), Tr(f, K)

—

the set of trajectories of the mapping f , p. 94

Tr±k (f, K)

—

the set of trajectories of the mapping f defined over the interval [−k, k], p. 76

Wεs (x), Wεu (x), W s (x), W u (x)

—

stable and unstable sets of the mapping f , p. 12

X, Y, Z . . .

—

subsets of Rd

Z

—

the set of integers

X

—

closure of the set X

x = (x1 , x2 , . . . , xd )

—

coordinate representation of the vector x ∈ Rd

{xn }, x = {xn }, x = {x0 , x1 , . . .}, x = x0 , x 1 , . . .

—

sequence of elements x0 , x1 , . . ., trajectory, pseudo-trajectory, cycle, pseudo-cycle

xT

—

transpose of the vector x

α(s, h)

—

bi-shadowing parameter, p. 95

Rd Rd

=

Exs

⊕

Exu

List of Symbols

219

β(s, h, δ)

—

bi-shadowing parameter, p. 95

γ(s)

—

split parameter, p. 36

δ(E1 , E2 )

—

separation between linear subspaces E1 , E2 , p. 9

ν(s)

—

split parameter, p. 22

Ω(f, X)

—

the set of non-wandering points of the mapping f , p. 16

ρN (x, y)

—

metric or semi-metric on the space of sequences, p. 17

Σ

—

the set of bi-infinite binary sequences x = {xn }, p. 15

Σ(X)

—

the set of bi-infinite sequences x = {xn } ∈ X, p. 109

σ:Σ→Σ

—

the shift mapping, p. 77

σ(A)

—

spectral radius of the matrix or linear operator A

σ(s)

—

spectral radius of the split matrix M (s), p. 36

#(·)

—

cardinality of a set, p. 12

|·|

—

absolute value of a number

|t − s| mod k

—

distance between real t and s modulo k, p. 30

k·k

—

a norm in Rd , a norm of a matrix or a linear operator

k · kx

—

Riemannian norm on a manifold

k · k∗

—

weighted ‘cubic’ norm in R2 , p. 40

k · k∞

—

supremum norm in `∞ (I, Rd ), p. 35

k · k∗∞

—

‘cubic’ supremum norm in `∞ (I, Rd ), p. 41

k · kα ∞

—

weighted ‘cubic’ supremum norm in `∞ (I, Rd ), p. 40

kf − ϕk∞

—

supremum distance between mappings f and ϕ, p. 87

k · kC

—

norm in the Banach space of continuous functions C(X, Rd ), p. 26

k · kLip

—

norm in the Banach space of Lipschitz functions Lip(X, Rd ), p. 26

(X, ρ)

—

metric space X with the norm ρ

220

List of Symbols

Index

Anosov automorphism, 10, 30 endomorphism, 30 system, 9 automorphism Anosov, 10, 30 hyperbolic, 30 Axiom A, 16 bi-shadowing, 94 cyclic, 104, 130 system, 94, 129 canards, 175 cardinality, 12 chain connected components, 125 points, 125 chaos Devaney, 20 Li–Yorke, 19 shift mapping, 18 symbolic dynamics, 18 threshold, 119 chaotic mapping, 20 complexification, 54 components chain connected, 125 condition A1–A3, 9 C0–C2, 156 CH1–CH3, 19 DCH1–DCH3, 20 H1∗ , H2∗ , 10 H1, H2, 11 HCF1, HCF2, 11, 12 N1, N2, 158

P1–P3, 174, 175 R1–R3, 163 S1–S4, 118 SH0(Lip)–SH2(Lip), 25, 130 SH0–SH2, 22, 139 SH2(Diff), 24 SH2(Diff)∗ , 23 SH2(Mod), 79 cone field, 11 hyperbolic, 11 cycle, 8, 104 pseudo, 104 decomposition, 8, 21 Devaney chaos, 20 direct shadowing problem, 83 discrepancy, 156 disk, 19 duck-trajectories, 175 endomorphism Anosov, 30 hyperbolic, 30 entropy, 17 topological, 17 equation hyperbolic, 148 equivariance, 8 equivariant splitting, 49 expanding fixed point, 20 expansive system, 75 expansivity, 13 factorization, 109 ferromagnetic hysteresis, 183 fixed point expanding, 20 generic property, 15

221

222 Hausdorff distance, 197 semi-distance, 144, 197 homeomorphism expansive, 13 hyperbolic, 12 homoclinic point, 18 transversal, 18 horseshoe mapping Smale, 128 twisted, 142 hyperbolic automorphism, 30 cone field, 11 endomorphism, 30 equation, 148 homeomorphism, 12 linear operator, 55 mapping, 10 matrix sequence, 52 point fixed, 8 periodic, 8 set, 10, 11 hyperbolicity, 11 on a manifold, 10 hysteresis ferromagnetic, 183 nonlinearity, 157, 158 inverse shadowing, 88 problem, 84 Lang–Kobayashi equations, 164 laser, 163, 164 Li–Yorke chaos, 19 mapping (V, W )-hyperbolic, 162 (U , k)-chaotic, 174 (ε, k)-chaotic, 118 (s, h)-semi-hyperbolic, 24 s-semi-hyperbolic, 23, 24, 130 uniformly, 139 chaotic, 19, 20 compatible, 163 strongly, 159 conjugate, 15 weakly, 110 continuously semi-hyperbolic, 26 expansive, 122 exponentially, 78 horseshoe Smale, 128

Index twisted, 142 hyperbolic, 10 semi-conjugate, 14 upper-semicontinuous, 16 Markov chain, 159 matrix (s, h)-semi-hyperbolic, 37 nonlinearity hysteresis, 158 normal, 158 Preisach, 158 von Mises, 158 operator hyperbolic, 55 semi-hyperbolic, 54 spectrally hyperbolic, 55 Period Three Implies Chaos, 19 point ε-chain recurrent, 110 chain recurrent, 16, 110, 124 critical, 176 non-degenerate, 176 fixed expanding, 20 hyperbolic, 8 homoclinic, 18 transversal, 18 hyperbolic fixed, 8 periodic, 8 non-wandering, 16 points chain connected, 125 Preisach nonlinearity, 158 problem Direct Shadowing Problem, 83 Inverse Shadowing Problem, 84 property generic, 15 shadowing, 14 pseudocycle, 104 trajectory, 14, 84 finite, 84 infinite, 84 scrambled set, 19 semi-distance Hausdorff, 144 semi-hyperbolic linear operator, 54

Index mapping, 23, 24, 26, 130 matrix, 37 semiconductor laser, 165 separation, 9, 46 sequence of maps split hyperbolic, 156 of matrices (s, h)-semi-hyperbolic, 39, 51 hyperbolic, 52 set hyperbolic, 10, 11 invariant, 9 scrambled, 19 stable, 12 unstable, 12 shadow, 14, 84 shadowing, 14 direct, 85 problem, 83 inverse, 88 problem, 84 theorem, 156 shift, 159 mapping, 77 chaos, 18 operator, 109 slow surface, 175 Smale horseshoe mapping, 128 split, 22 hyperbolicity, 156 matrix, 35 positive, 22, 35 splitting, 8, 21 continuous, 25 equivariant, 49 uniform, 21, 139 stability Ω, 16 topological, 15 stop, 152, 158 subspaces

223 stable, 8 unstable, 8 symbolic dynamics chaos, 18 system Anosov, 9 bi-shadowing, 94, 129 cyclically, 104, 130 expansive, 75 theorem Equivariant Splitting Theorem, 49 Marotto Theorem, 20 Period Three Implies Chaos Theorem, 19 Perturbation Theorem, 38 Shadowing Theorem, 14, 157 Smale’s Homoclinic Theorem, 18 Topological Stability Theorem, 15 topological ε-entropy, 17 entropy, 17 Markov chain, 159 trajectory, 75, 117, 159 α-robust, 88 finite, 75 homoclinic, 119 infinite, 75 periodic, 104 pseudo, 14, 84 finite, 84 infinite, 84 periodic, 104 transducer stop, 152 turning subsurface, 175 twisted horseshoe mapping, 142 uniform splitting, 21, 139 von Mises nonlinearity, 158 weak factorization, 109

Authors Phil Diamond Mathematics Department The University of Queensland QLD 4072 Australia Email: [email protected] Peter Kloeden Institut f¨ ur Mathematik Goethe Universitaet D-60054 Frankfurt am Main Germany Tel: +49-(0)69 798 28622 Fax: +49-(0)69 798 28846 Email: [email protected] http://www.math.uni-frankfurt.de/~numerik/kloeden/ Victor Kozyakin Institute for Information Transmission Problems Russian Academy of Sciences Bolshoj Karetny lane 19 Moscow 127994 GSP-4, Russia Tel: +7-495-699-83-54 Fax: +7-495-650-05-79 Email: [email protected] http://www.iitp.ru/en/users/46.htm Alexei Pokrovskii Department of Applied Mathematics Room G49, Western Gate Building National University of Ireland University College Cork Ireland Tel: +353 21 420 5826 Fax: +353 21 420 5364 Email: [email protected] http://euclid.ucc.ie/pages/staff/pokrovskii/apokrovskii.htm