investigation of dynamical systems using symbolic

January 17, 2007

13:28

01693

Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 16, No. 12 (2006) 3451–3496 c World Scientific Publishing Company


INVESTIGATION OF DYNAMICAL SYSTEMS USING SYMBOLIC IMAGES: EFFICIENT IMPLEMENTATION AND APPLICATIONS V. AVRUTIN∗ , P. LEVI† and M. SCHANZ‡ Institute of Parallel and Distributed Systems (IPVS), University of Stuttgart, Universit¨ atstrasse 38, D-70569 Stuttgart, Germany ∗[email protected] † [email protected] ‡[email protected] D. FUNDINGER§ and G. OSIPENKO¶ Laboratory of Mathematical Modeling, St. Petersburg State Polytechnic University, Polytechnic av., 29, 195251 St. Petersburg, Russia §[email protected] ¶[email protected] Received November 29, 2004; Revised February 9, 2006 Symbolic images represent a unified framework to apply several methods for the investigation of dynamical systems both discrete and continuous in time. By transforming the system flow into a graph, they allow it to formulate investigation methods as graph algorithms. Several kinds of stable and unstable return trajectories can be localized on this graph as well as attractors, their basins and connecting orbits. Extensions of the framework allow, e.g. the calculation of the Morse spectrum and verification of hyperbolicity. In this work, efficient algorithms and adequate data structures will be presented for the construction of symbolic images and some basic operations on them, like the localization of the chain recurrent set and periodic orbits. The performance of these algorithms will be analyzed and we show their application in practice. The focus is not only put on several standard systems, like Lorenz and Ikeda, but also on some less well-known ones. Additionally, some tuning techniques are presented for an efficient usage of the method. Keywords: Symbolic images; algorithms; applications; verification; unstable invariant sets.

1. Introduction Investigation of the global structure of vector fields is an important task for the analysis of nonlinear dynamical systems. If one simulates an orbit of a dynamical system for a given initial condition one can only observe one attractor of this system but will not be able to locate any other objects

existing in the state space. Although several coexisting attractors might be detected by variation of the initial conditions, one can never be sure to have found all existing attractors. Furthermore, it is not possible to find unstable objects like, for instance, unstable limit cycles. Other investigation methods are required which do not trace single orbits in the

3451

January 17, 2007

3452

13:28

01693

V. Avrutin et al.

state space but investigate the global structure of the vector field instead. Such methods are able to detect invariant sets in the state space, no matter whether these sets are stable or not. The presented method approaches this task. It provides a unified framework for the acquisition of information about the system flow without any restrictions concerning the stability of specific invariant sets. The mathematical theory was presented in a series of works by G. S. Osipenko [1983, 1993, 1994, 2004]. It can be considered close to Cellto-Cell mappings [Hsu, 1980, 1987] and is related with symbolic dynamics [Alekseev, 1976; Bowen, 1982; Lind & Marcus, 1995; Walters 1991]. The main idea is the construction of a directed graph which represents the structure of the state space for the investigated dynamical system. This graph is called the symbolic image of the focused system and can be seen as an approximation of the system flow. From the computational point of view, the usage of such a graph bears the big advantage that, once it is constructed, all investigations are matters of graph analysis, e.g. each strongly connected part represents an invariant set of the flow. This allows us to locate return trajectories of all types without any restrictions concerning their stability. When dealing with limit cycles (both stable or unstable), their periodicity can also be determined. More sophisticated computational analysis of the symbolic image graph allows, among others, the localization of invariant sets [Kobjakov et al., 2003], attractors and their basins [Osipenko & Campbell, 1999; Osipenko, 1999; Fundinger, 2006] as well as the computation of the Morse Spectrum [Osipenko, 1997, 2000; Osipenko et al., 2004] or verification of hyperbolicity [Osipenko, 2004]. Several other approaches exist which use concepts similar to the construction of the symbolic image graph. In [Eidenschink, 1995] and [Mischaikow, 2002] a symbolic image-like graph, called there a multivalued mapping, is constructed in order to compute isolated blocks in the context of the Conley Index Theory, see also [Conley, 1978]. The set-oriented methods of Dellnitz, Hohmann and Junge [Dellnitz & Hohmann, 1996, 1997; Dellnitz & Junge, 1998, 2002] use a scheme similar to our graph and apply a subdivision technique which is also used slightly modified in our implementation. Hruska [2002, 2005] made a box chain construction to get a directed graph with the aim to compute an expanding, or hyperbolic, metric for dynamical systems.

An analogous tool for discretization of dynamical systems was applied by F. S. Hunt [1998] and Diamond et al. [1995]. Furthermore, there are many other constructive and computer-oriented methods, see for instance [Coomes et al., 1994, 1997; Golub & Loan, 1999; Guckenheimer & Holmes, 1983; Malinetskii & Potapov, 2000; Sim´ o, 1989]. The theoretical background of the symbolic images and the constructive methods which are applied on them were described in detail by Osipenko. But although in [Osipenko, 1993] first numerical calculations were presented, the algorithmic basics, which are needed for the implementation of the method, were not reported until now. Therefore, in this paper we describe those aspects of the method which are important for an efficient implementation. In practice, it turned out that they are essential cornerstones for the usability of the discussed method. We present algorithms and data structures for the implementation of symbolic images and provide a theoretical analysis of their performance. Some of the concepts we mention were already introduced in implementations of related investigation methods, especially by the GAIO software package [Dellnitz et al., 2001]. We extend these concepts and combine them with new ones in order to give a complete description of all implementation aspects regarding symbolic image construction. Note that we restricted ourselves to basic operations on the symbolic image, like the localization of chain recurrent sets or periodic orbits. The topic of this paper is not the application of advanced investigation methods on the graph, but rather more the provision of a computational framework which allows their easy and efficient integration. Numerical computations are performed for several dynamical systems in order to verify our theoretical results. We discuss some reasonable parameter settings and the steps to be taken for the acquisition of the data. While doing so, the reader will be introduced to possible fields of application for symbolic image construction. Based on these empirical studies we will propose several techniques for the tuning of our method which help to reduce numerical errors, make the calculations more reliable and allow the user a better control of the desired degree of accuracy. We believe that the results of this work are not only useful in the context of symbolic image construction but also in the larger framework of related techniques which use symbolic image-like graphs, multivalued mappings or set-oriented methods. All

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

of the techniques mentioned are implemented and tested within the AnT project [Home page of the AnT 4.669 project, 2005; Avrutin et al., 2003] The paper is organized as follows. Firstly, in Sec. 2, the main ideas of the symbolic image construction are briefly summarized. A detailed description of these ideas is out of scope of this paper (we refer to the original works [Osipenko, 1983, 1994, 1993], see also [Bathia & Szego, 1970; Broer & Simo, 1998; Conley, 1978; Malinetskii & Potapov, 2000]. Only a short overview is given in order to understand the presented algorithms. After that, in Sec. 3, the main topic of this paper, i.e. the algorithms used for an efficient calculation of symbolic images will be discussed. Several investigation methods which can be applied on symbolic images are discussed in Sec. 4. The performance of these algorithms is analyzed in Sec. 5 in order to show which steps of the calculation are critical for an efficient implementation and usage of the discussed method. Some considerations about the accuracy of a computation are given in Sec. 6. Sections 7–9 provide examples of a practical application of our methods. These examples are also used to point out some problems and difficulties of a successful application. The succeeding Sec. 10 is then addressed to propose tunings of the basic method in order to solve these problems and improve the performance.

2. Theoretical Background Let us consider a dynamical system which is generated by a vector field and defined on the C ∞ -smooth manifold M : f : M → M.

(1)

As a rule, the manifold M is a compact in Rn , and f is a homeomorphism. For simplicity of the description we assume that the dynamical system is discrete in time. But, however, this is not essential for the theory. In Sec. 8 we will show how the concepts can be applied for dynamical systems continuous in time. Let
    n C = M (1), . . . , M (n)  M (i) = M (2) i=1

be a finite covering of the domain M by closed sets. The sets M (i) ⊂ M are named boxes of the covering C. For each box M (i) we consider its image f (M (i)) with respect to the flow f as f (M (i)) = {y|y = f (x), x ∈ M (i)}

(3)

3453

Then we define for the box M (i) the covering C(i) of its image f (M (i)). This covering consists of boxes M (j) ∈ C, whose intersections with f (M (i)) are not empty: C(i) = {M (j)|M (j) ∩ f (M (i)) = ∅} .

(4)

Let us construct the directed graph G with n vertices which matches to each box M (i) the vertex ci . The vertices ci and cj are connected by the directed edge ci → cj iff the cell M (j) is an image box of M (i), i.e. M (j) ∈ C(i). In the following we denote M (i) as a box, and a vertex ci on G as a cell. The graph G constructed as described above is called the symbolic image of f with respect to the covering C. Definition 1.

We can consider the symbolic image as a finite approximation of the flow f , which depends on the covering C. We can change the symbolic image of f by varying C. If the cells ci and cj are connected by the edge ci → cj then there exists a point x in the box M (i) so that its image f (x) lies in the box M (j). Denote by V (G) the set of cells on G. Then the graph G can be considered as a multivalued mapping G : V (G) → V (G) between the vertices. The described construction procedure is illustrated in Fig. 1. In this example an area M ⊂ R2 is covered with boxes C = {M (1), . . . , M (12)}. Let the image f (M (1)) of the box M (1) be the area in the center of the picture. Then the covering C(1) is given by the set {M (3), M (4), M (6), M (7), M (8), M (10), M (11)}. These boxes are colored gray in Fig. 1(a). So the symbolic image of f with respect to the covering C possesses the edges shown in Fig. 1(b). In order to construct the symbolic image, the covering must be found for all boxes M (1) . . . M (12). We also introduce some parameters of a symbolic image. Let δ(M (i)) be the diameter of a box belonging to a covering C, δ(M (i)) = max(ρ(x, y)|x, y ∈ M (i)).

(5)

The largest diameter of the cells from C is denoted by δ(C) = max(δ(M (i))|M (i) ∈ C).

(6)

Definition 2. An infinite in both directions (biinfinite) sequence {cik } of cells in the graph G is called an admissible path (or simply a path) if for each k the graph G contains the edge cik → cik+1 . A path {cik } is said to be p-periodic if cik = cik+p for each k ∈ Z and cik = cik+p
for all p < p.

January 17, 2007

3454

13:28

01693

V. Avrutin et al.

Roughly speaking, an admissible path represents the trace of an ε-orbit and vice versa. Definition 4 [Conley, 1978].

A point x is called chain recurrent if for every ε > 0 a periodic ε-orbit passes through it. The set of chain recurrent points is called a chain recurrent set and denoted by Q. A chain recurrent set is invariant, closed and contains periodic, homoclinic, nonwandering and other singular trajectories [Osipenko, 1993]. Although a chain recurrent point is not periodic, it may become periodic under a small C 0 -perturbation of f .

(a)

Definition 5. A cell of a symbolic image is called

recurrent if there is a periodic path passing through it. Two recurrent cells ci and cj are called equivalent if there is a periodic path containing both, ci and cj .

(b) Fig. 1. Construction of the symbolic image. (a) The box M (1) is marked with a thick square in the upper left part of the picture. The image f (M (1)) is represented by the area in the center of picture. The covering C(1) is colored in gray. (b) Edges of the symbolic image given by c(1). For a more detailed description, see text.

A finite path ω = {ci1 , . . . , cim } with the length |ω| = m is defined in the same way. Definition 3. Fix ε > 0. An infinite in both direction sequence {xk , k ∈ Z} is called an ε-orbit or a pseudo-orbit of f if for any k the distance between the image f (xk ) of the point xk and the next point xk+1 is less than ε:

|f (xk ) − xk+1 | < ε

(7)

A pseudo-orbit {xk } is said to be p-periodic if xk = xk+p for each k ∈ Z and xk = xk+p
for all p < p (see [Anosov, 1967]). In practice, a real orbit is seldom known exactly and, in fact, we usually find nothing more than εorbits for sufficiently small positive ε. A p-periodic ε-orbit (or periodic path) will be denoted by its periodic path {x1 , . . . , xp } ({ci1 , . . . , cip }). There is a natural correspondence between the admissible paths on the symbolic image G and the ε-orbits.

Denote the subset of recurrent cells in G as RV (G). The set RV (G) decomposes into classes H1 , H2 , . . . of equivalent recurrent vertices. In graph theory, a class Hk is called a strongly connected component of the graph G. The boxes M (i) belonging to the cells RV (G) are a neighborhood of the chain recurrent set. The detection of this neighborhood is the basic task for every symbolic image calculation. In order to get a better approximation of the vector field f , a subdivision procedure will be applied (see [Osipenko, 1993] and [Dellnitz & Hohmann, 1996; Dellnitz & Junge, 1998]). This procedure can be described as follows. Let C s be a covering of subdivision level s. Then a subset of boxes belonging to C s gets selected for the next subdivision. The decision which boxes get selected depends on the kind of investigation which is performed, e.g. if the chain recurrent set should be approximated then all the recurrent cells of Gs get selected. In general, only the first covering C 0 covers the whole domain M according to Eq. (2), and the coverings C s , s > 0 are defined as C s = {M (1), . . . , M (n)|M (i) ⊂ M },

(8)

and do only cover parts of M . Suppose now a new covering C s+1 is a subdivision of C s . The boxes of C s+1 are denoted as m(i, k). Each box M (i) which is selected for subdivision will be split up into new boxes m(i, k),

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

k = 1, 2, . . . , so that each m(i, k) ⊂ M (i) and  m(i, k) = M (i).

3455

initialization of the area to be investigated

(9)

k

Denote by Gs+1 the symbolic image for the new covering C s+1 . In this case the cells of the new symbolic image are designated as ci,k and there is a natural mapping h : ci,k → ci from Gs+1 onto Gs . It holds that h(Gs+1 (ci,k )) ⊂ Gs (h(ci,k )),

subdivision construction of the symbolic image investigation of the symbolic image

(10)

so that every path on Gs+1 can be transformed on some path on Gs . This means that all paths of Gs+1 and, respectively, also their corresponding ε-orbits, are included in Gs . However, ε can be fixed to a smaller value for Gs+1 and so the approximation of the investigated area becomes more precise for each application of the subdivision procedure.

3. Implementation of the Basic Framework Any kind of investigation on the symbolic image is embedded into a basic framework of computations. This framework can be considered as an iterative process which will be repeated for increasing levels of the state space discretization. At each level, the calculation involves three main steps which have to be performed several times: 1. Subdivision of selected parts of the state space into smaller parts (see Secs. 3.1 and 3.3). 2. Construction of the symbolic image for the current discretization of the state space (see Sec. 3.2), 3. Application of an investigation method on the symbolic image graph (see Sec. 4). As a result, parts of the state space get selected for further subdivision and a more precise investigation. As it is shown in Fig. 2, these steps will be repeated until a termination criterion is fulfilled. This condition depends on the desired accuracy as well as on the existing computation power (see Sec. 3.3). In this section we describe the implementation of this basic framework and discuss those aspects which have to be taken into consideration for an efficient implementation. The basic framework as proposed here was implemented and tested within the AnT project [Home of the AnT 4.669 project, 2005; Avrutin et al., 2003].

termination criterion

no

fulfilled? yes Fig. 2.

3.1.

Flow chart of the described method.

Box and cell objects

In order to build a symbolic image for a domain M ⊂ Rd of the state space, a finite covering C has to be defined according to Eq. (2). In the theory of symbolic images there are no restrictions concerning the geometry of the domain M and of the boxes M (i) ∈ C, except that they have to be closed and compact sets. In general, the investigated domain M could be any confined part of the state space and has to be provided by the user. For the simplicity of the implementation we assume here that it is an n-dimensional rectangular region: M = [M1min , M1max ] × · · · × [Mnmin , Mnmax ].

(11)

Then we can subdivide the area M into uniform grid boxes. In order to do this, the user has to define for each state space coordinate k, (k = 1, . . . , n), the numbers imax of subdivisions for the domain k M . Then M can be subdivided into n  imax (12) m= k k=1

n-dimensional rectangular boxes. The length of the edge k is given for each box by  1  (13) dk (M (i)) = max Mkmax − Mkmin . ik Note that after the application of the subdivision procedure there is no need for the user to explicitly define the domain of the state space which has to be investigated in the (s + 1)th step. This domain will

January 17, 2007

3456

13:28

01693

V. Avrutin et al.

be determined automatically, based on the results of the sth step (see Sec. 3.3). Every box must hold the information about its position in the n-dimensional state space. In context of an efficient implementation, the positions of boxes are represented as n-dimensional multiindices I ∈ Nn , I = (i1 , . . . , in ) ∈ Nn with ik ∈ [1, . . . , imax ∀ k = 1, . . . , n, k ],

(14)

so that for every box M (I) ∈ C exists a unique multi-index I defining its position in M . This representation was chosen because it allows a fast and easy mapping from x ∈ M (I) to I and vice versa. Furthermore, the mapping range within the domain of usual integer values is much larger if multi-indices are used instead of onedimensional indices. If Nint is defined as the maximal integer value of a computer system and one would only use one-dimensional indices to identify the boxes M (i) then i = 1, . . . , Nint . A domain M could only be subdivided in m = Nint boxes because no more values are available for the description of all possible positions. Note that there must be an index for every position i, no matter if M (i) exists or not. This is a crucial restriction for higher-dimensional systems. If multi-indices are used instead, an ndimensional domain M can be subdivided in m = (Nint )n boxes which means we have an exponential growth for the number of possible box positions. Some mapping functions, which will be introduced in the following, require the definition of a strict weak order relation ≺ for the box indices I of a covering C. It can be given by using the mapping

n k−1   max il (ik − 1) (15) φ(I) = k=1

l=1

with φ(I) ∈ N and defining the relation ≺ for two indices I and I  by I ≺ I

iff φ(I) < φ(I  ).

(16)

However, one has to be careful when dealing with the implementation of the relation ≺ because for n ≥ 2 the range of the mapping φ defined by Eq. (15) can easily overflow the domain of usual integer data types. Therefore, the relation ≺ should be implemented without a direct usage of Eq. (15). It is better to use an iterative comparison of the components ik (k = 1, . . . , n) of an index I, start-

ing with the largest “digit” in : I ≺ I  iff ∃ k = 1, . . . , n with ik < ik and ij = ij ∀ j > k.

(17)

After the construction of the covering C is completed, an approximation of the symbolic image based on C can be constructed. It represents a graph G, whereby the cells of G correspond to the boxes M (I) of C. Each of these cells can have a list of target cells, which defines the edges of the graph G. A distinctive feature of the cells is, that each of them is uniquely connected with a box M (I). This correspondence represents a link between the domain M and the symbolic image G.

3.2. Construction of the symbolic image The construction of a symbolic image based on numerical calculations is always only an estimation of the “real” symbolic image G. Besides the usual numerical errors which occur by the computation of a vector field f (x) for x ∈ M , another reason for this is the fact that the construction of the image T (I) = f (M (I)) = {y|y = f (x), x ∈ M (I)} ⊂ Rn (18) for a box M (I) would involve the calculation of f (x) for every x ∈ M (I). This is, of course, beyond the limits of every finite numerical computation. One approach for the approximation of T (I) is given by the usage of interval arithmetic, see for instance [Alefeld & Herzberger, 1983]. This technique was used by Hruska [2002] in the context of box chain construction. The basic idea is to perform numerical computations on intervals instead of numbers. The results of such computations are then again intervals which include all solutions. In higher dimensions, the operations can be carried out component-wise on interval vectors. A box M (I) can then be seen as a higher-dimensional interval, and f can be computed using interval arithmetics. As a result, we get an outer covering of T (I). Although this is an interesting approach, we did not consider it to be appropriate for our implementation. The reason for this is, that the error bounds easily tend to increase largely, as was also reported in [Hruska, 2002]. This would lead to an increase of edges in G which is not desirable. Concerning this topic, see also the discussion about the method’s performance, Sec. 5. Additionally, the system function f is limited to the operations which are defined for interval arithmetic.

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

So for our implementation we use a different method. The image T (I) will be approximated by a finite set of points. This technique was also used earlier by Dellnitz et al. [Dellnitz & Hohmann, 1997], and has proved to be a good approach in practice. From each box M (I) a representative set of k points is selected, S(I) = {xj |xj ∈ M (I),

j = 1, . . . , k}

(19)

the so-called scan points of the box M (I). Then the approximation T˜(I) of the region T (I) in the state space is calculated by T˜(I) = f (S(I)) = {yj |yj = f (xj ), xj ∈ S(I)} (20) As one can see, the continuous region T (I) will be approximated by the discrete set T˜(I) ∈ T (I) consisting of k points. The number k of scan points for the boxes as well as the positions of these points within the boxes are parameters of the described method which must be set by the user. There is no general strategy how the scan points should be placed within the box M (I). In Dellnitz et al. [Dellnitz & Hohmann, 1997] it was proposed that the points should lie on the boundary or in the edges of the box. Additionally, there should be one point in the center of the box. Note now that this strategy should not be used for symbolic images. One has to consider that the boundaries of the boxes overlap, and, hence, if the scan points of the boxes lie on the boundary, they also overlap. This leads to the occurrence of clusters of boxes during subdivision. A better strategy is that the scan points are either uniformly distributed within M (I) or that they are put into the neighborhood of the boundaries. Besides the calculation of scan points, a mapping p : M → Nn ,

∀ x ∈ M (I) ⇒ p(x) = I

(21)

of a point x ∈ M (I) onto a box index I is required for further computations. Additionally, we need its inverse mapping p−1 : Nn → M,

∀I

p−1 (I) = x ⇒ x ∈ M (I) (22)

which defines for every M (I) the spatial coordinates of a point within this box. Note that p−1 (I) is only defined if M (I) exists. Due to the fact that all uniform grid boxes have the same size, the mapping p : M → Nn can be

3457

simply defined as p(x) = I = (i1 , . . . , in )  (23) xk − Mkmin + 1, k = 1, . . . , n with ik = imax k The inverse mapping can be described in a similar way. Note, however, that the inverse mapping is not unique and, in practice, requires the definition of an arbitrary reference point within the box M (I). If using, for instance, the minimal point of each box, one can get p−1 (I) = x = (x1 , . . . , xn )T with xk = Mkmin + (ik − 1) · dk (M (I)), k = 1, . . . , n

(24)

After the functions p and p−1 have been defined, ˜ the approximation C(I), ˜ (25) C(I) = {M (I  )|M (I  ) ∩ T˜(I) = ∅}, of the covering C(I), C(I) = {M (I  )|M (I  ) ∩ T (I) = ∅},

(26)

can be computed. Obviously, we have the relation ˜ C(I) ⊆ C(I). Proceeding this task, the following steps have to be performed for each box M (I) in the covering C: 1. The set of scan points S(I) is calculated using a set of globally defined relative coordinates S = {ξ j |ξ j = (ξj,1 , . . . , ξj,n )T, ξj,i ∈ [0, 1], i = 1, . . . , n, j = 1, . . . , k} with respect to the reference point x0 =

(27) p−1 (I):

S(I) = {xj |xj,i = x0,i + ξj,i · di (M (I)), ξ j ∈ S, j = 1, . . . , k}

(28)

This is necessary for scaling the relative coordinates ξ j , which are defined within the hypercube [0, 1]n , onto the area M (I). 2. For every point xj ∈ S(I) the target point yj ∈ T˜(I) will be calculated. Dealing with dynamical systems discrete in time, i.e. x(n + 1) = f (x(n)), the target point yj can be found by a simple one step iteration of the point xj : yj = f (xj )

(29)

Note that also the nth iterated function f [n] can be used in Eq. (29) instead of f . This is necessary if we want to apply symbolic images to dynamical systems continuous in time, see Sec. 8. Despite that, the option is also useful if

January 17, 2007

3458

13:28

01693

V. Avrutin et al.

we want to work with higher iterated functions in order to decrease the critical slowing down phenomenon [Goldenfeld, 1992], see Sec. 10.3 for a more detailed discussion. 3. For each target point yj ∈ T˜(I) the corresponding box object M (I  ) with yj ∈ M (I  ) must be found. The index I  of this box is given by I  = p(yj )

(30)

It is important to check, whether the conditions 1 ≤ ik ≤ imax k

∀ k = 1, . . . , n

(31) I .

hold for the components of the multi-index If not, the index I  exceeds the dimension range and there is no box defined for this index. 4. Within the implementation context, a memory access function g : Nn → M,

∃ M (I  ) ∈ C ⇒ g(I  ) = M (I  ) (32)

is required in order to access a box object M (I  ) for an index I  . In a numerical computation one can consider at least two approaches to define g: ]×· · ·×[1, imax (a) There is an array A = [1, imax n ] 1 covering all possible index positions I in M and containing pointers to every box M (I) if such an M (I) exists. (b) There is a hash map H which contains a key I iff there exists a corresponding M (I). The first approach, the use of an array, would be very fast and allows the detection of M (I) in constant time. Unfortunately, it also requires a large amount of memory space. For every possible index position I in M memory is needed, even though there might be no box objects M (I) at many index positions. This aspect becomes more crucial for every subdivision step and would not allow a precise calculation of nontrivial symbolic images. Therefore, the second approach should be preferred. A hash map H is used to map I onto M (I) if such a box object exists. No memory space is wasted for indices without a corresponding box object. It should be mentioned that a fast and proper implementation of H requires the strict weak ordering ≺ of I mentioned before [see Eq. (17)]. 5. If M (I  ) has been located, a reference to it will ˜ be added to the list of C(I). When the location

of the target boxes M (I  ) for all scan points ˜ xj ∈ S(I) is completed, the list of C(I) is an estimation of the covering C(I) for the box object M (I). Figure 3 illustrates the construction of a covering ˜ C(1) for a box M (1). As one can see, the scan points xi , marked with ◦, are mapped onto the points yi , marked with • (i = 1, . . . , 9). Based on these points, ˜ the following approximation C(1) of the covering C(1) was calculated: ˜ C(1) = {M (3), M (6), M (7), M (8), M (10), M (11)}. (33) In Fig. 3 the covering C(1) is colored gray and the ˜ approximation C(1) is shown as a hatched area. As one can see, the box M (4), which belongs to ˜ C(I), gets lost in the approximation C(1), so that ˜ C(1) = C(1). This is is due to the insufficient number of scan points, a typical problem which might occur in numerical simulations with a discretization of the scan points S. Note that there exists a method [Junge, 2000] for a rigorous computation of the sets C(I) by scan points. This method was introduced for the application on set-oriented methods but could be used in our context as well. If the Lipschitz constants can be estimated for f on M then this method is able to ˜ ˜ compute a set C(I), so that C(I) ⊂ C(I). Though this technique has not yet been implemented by us, we introduce in Sec. 10.2 an error tolerance which can be considered as a basic framework for the

Fig. 3. Implementation of the symbolic image construction. The scan points xi ∈ S(1) ⊂ M (1) are marked with ◦. The images of these points yi ∈ T˜(1) ⊂ C(1) are marked with •. For detailed description, see text.

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

integration of this technique. However, one should consider that the application of this method, like the interval arithmetic mentioned earlier, provides an outer covering. Heuristic experience has shown that such an approach for the approximation of C(I) is often too pessimistic, and in many cases not desirable for practical calculations. See the results of Secs. 7 and 8 for a more detailed discussion. After the described steps were performed for all boxes of the state space discretization, the approximation G of the symbolic image has been constructed. The vertices of the graph are given by the cells corresponding to the boxes M (I), and the ˜ edges by the coverings C(I). If the symbolic image has been built, graph algorithms can be applied for its investigation. A wide range of different kind of investigations is possible, see Sec. 4 for further details. All these investigation methods have in common that some cells of the graph get selected. The boxes which correspond to these selected cells are the parts of the state space which are subject to further subdivision and a more precise investigation.

3.3. Subdivision process In the previous section the construction of a symbolic image G was described. It was already mentioned that the precision of the state space discretization for such a symbolic image must be increased by an iterative process. To describe this process, we introduce an index s which indicates the level of state space discretization, s = 1, 2, . . . , or, in other words, the subdivision depth of a symbolic image. In the following the notation for the symbolic image G is extended to Gs . The notation C s , I s and so on is introduced in the same way. After Gs is constructed, it is necessary to decide if the process has to be continued in order to construct the next (and hopefully more precise) image Gs+1 , or if the construction process should be stopped (see Fig. 2). There are two typical reasons to terminate the process: • The user-defined maximal number of subdivision steps has been reached. This should be the normal case and means that the symbolic image Gs was calculated with the desired accuracy. • Although the maximal number of subdivision steps has not yet been reached, the number of cells in the symbolic image Gs is so huge that the next subdivision of cells would overflow the memory space of the used computer. This can be

3459

interpreted as some kind of failure. An appropriate output will be produced by the software and the next subdivision will not be performed. If the subdivision process will be continued in order to get the graph Gs+1 then a new covering C s+1 has to be calculated. This covering depends on a selection of cells SV (Gs ) ⊆ V (Gs ) chosen by the application of an investigation method to the graph Gs . The covering C s+1 usually covers only parts of the area covered by C s . Let cI s be the cell in Gs which matches to the box M (I s ). Then the new area of investigation is given by the joint of all boxes which belong to a selection of cells SV (Gs ):  M (I s ). (34) M s+1 = cI s ∈SV (Gs )

Each of these boxes will be divided into m subboxes M (I s+1 ) [see Eq. (12)] which build together the new covering C s+1 . For every I s = (i1 , . . . , in ) the m new indices I s+1 are defined as follows: I s+1 = (j1 + (i1 − 1) · imax , . . . , jn + (in − 1) · imax 1 n ) ∀ k = 1, . . .,n with jk = 1, . . . , imax k (35) After the subdivision, the graph Gs+1 is constructed for the covering C s+1 as described in Sec. 3.2, and the whole calculation process is repeated.

3.4. Comparison with a similar implementation In order to summarize the description of the implementation details, we compare our approach with those of others. To our knowledge, there exists only one implementation which is comparable with symbolic image construction, and for which details were published. Namely, this is the implementation by Dellnitz et al. [Dellnitz & Hohmann, 1997; Dellnitz & Junge, 2002] for set-oriented methods. Other authors, like Mischaikow [2002], also refer to this implementation as the basic framework for the application of their methods. Another implementation was introduced by Hruska [2002, 2005] but only a few details are given about those aspects of the implementation which we described here. For the comparison of our implementation with the one of Dellnitz et al., we focus on aspects of storage, subdivision and mapping. As mentioned before, the selection of scan points is similar in both implementations, while the main difference is that the approach of Dellnitz et al. uses a binary search tree

January 17, 2007

3460

13:28

01693

V. Avrutin et al.

for the storage of the collection of sets (or boxes in our terminology). This tree spans over the elements of all subdivision steps. An indexation of the sets is not explicitly given. Instead, it is implicitly determined by a set’s position in the binary tree. In order to reach a set belonging to a covering C k , one has to start at the initial covering C 0 and then traverse the binary tree downwards through the coverings C 1 , . . . , C k . As in our approach, the coordinates of a box will not be stored but calculated if needed. In our opinion, this approach has several disadvantages in comparison with an explicit indexation and the storage in a hash-map as it is proposed by us: Flexibility: A binary search tree dictates a special geometric structure for the boxes. The basic covering is a rectangle, and in every subdivision this rectangle can only be subdivided into two pieces. In our approach, one can use uniform grid boxes at every stage of subdivision and divide them into as many parts as desired. So it is possible to integrate several subdivision steps on the binary search tree into one. Furthermore, the geometry of a box is not bounded to its storage scheme. This makes it possible to easily extend the implementation on other geometric forms, or even “intelligent” boxes which decide their geometry themselves. The explicit indexation allows in general a larger degree of freedom. Memory requirements: The binary tree needs to store the boxes of all subdivision steps, while in our approach, all boxes except those of the current subdivision step are deleted. Performance: In order to find the corresponding box object to a value f (x), one must traverse the binary tree. During this traversal, the coordinates of every visited box must be calculated. This operation gets more expensive, as more subdivision steps must be traversed. In our approach, the coordinates of a box are only calculated once — to compute the index of a box belonging to a value f (x) [Eq. (21)]. The search on the hash-map in order to find the box for this index does not involve the computation of further coordinates [Eq. (32)]. Note that the hash-map can also be represented by a binary tree. However, this tree is only spanned over the elements of one subdivision step in contrast to the binary tree of Dellnitz et al. which is spanned over the elements of all subdivision steps.

4. Investigating the Symbolic Image In the preceding section we described the basic framework for the symbolic image calculation. This included all aspects that must be performed for every kind of investigation. In this section, we discuss the basic investigation techniques, that have to be applied to the symbolic image graph in order to analyze the properties of the underlying dynamical system. Note, that we use variations of some standard graph algorithms. A discussion of the original algorithms is out of scope of this paper, therefore we refer to [Aho et al., 1987; Tarjan, 1991; Sedgewick, 1993; Rivest, 2000].

4.1. Localization of recurrent cells and the chain recurrent set The most important kind of investigation technique on the graph is the localization of the recurrent cells, see Definition 5. Applying this technique, we can determine a neighborhood of the chain recurrent set Q. Let us denote such a neighborhood as  M (i) with ci ∈ RV (Gs ), Ss = i

where s is the subdivision depth. Recall that a M (i) ⊂ M is the box corresponding to the cell ci of the symbolic image Gs , and RV (Gs ) is the subset of the recurrent cells in Gs . Obviously, this neighborhood is an outer covering of the chain recurrent set. Furthermore, recall that δ(C s ) is the largest diameter of the covering C s , see Eq. (6). In our implementation, δ(C s ) depends on the length of the box edges, see Eq. (13). Theorem 1 [Osipenko, 1994]. The sequence of sets S 0 , S 1 , S 2 , . . . has the following properties:

1. the neighborhoods S k are embedded into each other, i.e. S0 ⊃ S1 ⊃ S2 ⊃ · · · ⊃ Q 2. if the largest diameter δ(C s ) → 0 as s tends to infinity then

lim S s = S s = Q. s→∞

s

This theorem states, that by increasing the subdivision depth, the chain recurrent set can be approximated with an arbitrary precision. The cells which have to be selected for further subdivision are recurrent ones. Hence, we propose now an algorithm for the localization of the recurrent cells in a symbolic image. Note, that almost all other investigation

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

methods based on symbolic images also require the detection of the recurrent cells. Therefore, this technique is considered as a computation which is of general importance. An efficient approach to detect the recurrent cells is the variation of Tarjan’s algorithm for the calculation of strongly connected components in directed graphs [Tarjan, 1972]. This algorithm locates the strongly connected components of a directed graph G by a depth-first search. Two vertices a and b of G are said to be strongly connected (a ∼ b) if there exists a path from a to b and from b to a. Furthermore, the relation a ∼ a (reflexivity) always holds by definition. It can be proved that ∼ is an equivalence relation and that therefore G will be partitioned by the relation ∼ into equivalence classes, the strongly connected components. Although recurrent cells of G and strongly connected vertices are not the same, they are closely related to each other. If ζa = {b|a ∼ b}

(36)

is a strongly connected component for which there is a path from a to each b and vice versa with a = b then, for a as well as b, exists a periodic path (see Definition 2). It follows that, if |ζa | > 1, then for all cells c ∈ ζa exists a periodic path and therefore all these cells are recurrent. The special case to look at is |ζa | = 1. Due to reflexivity, if there is only one cell in the set it could mean that this cell is either nonrecurrent or, if there is an edge a → a, its least period size is 1. So, for the localization of recurrent cells, Tarjan’s algorithm needs a minor extension. What has to be done in addition is to perform a test for each set ζa if |ζa | = 1 holds. In this case, it has to be checked for the single cell of this set whether it is one-periodic (or recurrent), which means one of its target cells is itself, or not. If the cell is not oneperiodic, it is non-periodic (or non-recurrent). All cells belonging to a set ζa with |ζa | > 1 are periodic, i.e. recurrent. All recurrent cells that belong to the same set ζa can be considered as one of the equivalence classes H1 , H2 ,. . . , i.e. as a set of equivalent recurrent cells, see Definition 5. We like to mention here that the approach to localize the chain recurrent set by computation of strongly connected components is standard, and was also used by other authors. In [Eidenschink, 1995; Mischaikow, 2002] a slightly different relation between strongly connected components and chain recurrent sets was drawn. However, the resulting

3461

algorithm, like Tarjan’s, is also based on a breadthfirst search and in O(n). In [Dellnitz & Junge, 2002] the exact algorithm was not outlined but has also the same performance properties.

4.2. Localization of periodic orbits A related investigation is the localization of pperiodic points for a given value p. Let us denote the set of p-periodic points by P (p) = {x|f p (x) = x, x ∈ M ⊂ Rd }.

(37)

and the neighborhood of P (p) as  M (i) with ci ∈ V (Gs )is p-periodic, SP s = i

where s is the subdivision depth. Obviously, this neighborhood is an outer covering of P (p). Theorem 2 [Osipenko, 1993]. The sequence of sets SP 0 , SP 1 , SP 2 , . . . has the following properties:

1. the neighborhoods SP k are embedded into each other, i.e. SP 0 ⊃ SP 1 ⊃ SP 2 ⊃ · · · ⊃ P (p) 2. if the largest diameter δ(C s ) → 0 as s tends to infinity then

SP s = P (p). lim SP s = s→∞

s

This theorem states that the set of p-periodic points can be approximated with an arbitrary precision. The cells which have to be selected for subdivision are the p-periodic ones. Hence, we propose now an algorithm for the localization of the periodic cells in a symbolic image. For practical applications of Theorem 2, we have to consider that there might be more than one admissible path to which a cell ci belongs to, especially in the case of a coarse phase space discretization. Indeed, the number of admissible paths to which a cell belongs can even be infinite. However, if each cell of a symbolic image represents exactly one periodic point in the state space then this cell ci belongs only to one p-periodic path {. . . , cp0 , . . .} with cp0 = ci . Although such a precise covering is typically not achieved by numerical computation, a reasonably fine phase space discretization is usually sufficient to get a unique path for a cell which belongs to a covering of a periodic point. Considering these facts, the p-periodic points can be located by selecting all cells for subdivision which have a

January 17, 2007

3462

13:28

01693

V. Avrutin et al.

shortest periodic path for a period p ≤ p. Obviously, such a selection contains all cells which belong to a p-periodic path. After several subdivisions, we check that there exists a unique periodic path for each cell of the symbolic image. In case this cannot be achieved, a higher precision of the symbolic image is required, i.e. more subdivisions must be applied. Consequently, an algorithm is needed which is able to find the shortest periodic path ci0 → ci1 → · · · → cin−1 → ci0 for every recurrent cell c0 on the symbolic image. Furthermore, the length of such a path must be detected. Note that Tarjan’s algorithm is not capable to solve this task. Instead, we introduce a different algorithm which is based on the idea of Dijkstra’s algorithm for calculation of shortest paths in directed graphs [Dijkstra, 1959]. It belongs to the class of so-called greedy algorithms and performs a breadth-first search. The Dijkstra algorithm does not only find the shortest paths from each cell ci0 to all other cells of the graph, but also locates at the kth step first the path ci0 → · · · → cu so that the following equation is fulfilled: d(ci0 , cu ) = min{d(ci0 , cv )|cv ∈ V (G) / Dk−1 }, ∧ (ci0 → · · · → cv ) ∈

(38)

whereby d(ci0 , cu ) is defined as the length n of the shortest path between ci0 and cu , and Dk−1 is the set of all shortest paths which have already been detected in the previous steps. Then the shortest periodic path of a cell ci0 can be found by checking for the first detected shortest path ci0 → · · · → cu whether the edge cu → ci0 exists. If so, the algorithm can be stopped because the path ci0 → · · · → cu → ci0 is the shortest periodic path for the cell ci0 and the period of ci0 is the length of this path. If not, then the next shortest path has to be detected and checked for the same condition until a periodic path has been found or until all shortest paths have been visited. There are still several improvements to speed up Dijkstra’s algorithm within our context. First of all, the original Dijkstra algorithm is developed for weighted graphs while the edges of G are unweighted. This means that the edge weight γ(ci → cj ) is 1 for all edges of G. Therefore, the outer edge of visited but not yet examined cells can be implemented as a queue. Every cell which is visited first time and becomes a part of the outer edge will be pushed into the queue, while the next cell which will be examined can be popped out of

the queue. This works fine because our edges are unweighted and so the distance between ci0 and the first element in the queue is always the minimum distance between ci0 and every other element in the queue. Next it should be considered that all periodic cells of G have to be inspected. So in the worst case the modified Dijkstra algorithm must be started once for every cell ci ∈ V (G). The following improvements allow it to spare out some of the cells: • If a cell has no incoming edges it cannot be periodic and there is no need to start the algorithm for this cell. • We can first run the Tarjan algorithm to detect the recurrent cells and the sets H1 , H2 , . . . of equivalent recurrent vertices. The Dijkstra algorithm must then only be started for the recurrent cells. Furthermore, it is sufficient to check for each of these cells only the paths to the equivalent recurrent cells which belong to the same set Hk . Cells which do not belong to the same set cannot belong to the same shortest periodic path. Despite all improvements, the modified Dijkstra’s algorithm cannot compete with the performance of the aforementioned Tarjan’s algorithm. So it should only be chosen by the user if the additional information about the periodic paths and/or the least period sizes are really required for the calculation. The idea of using a breadth-first search algorithm like Dijkstra’s for investigations on a symbolic image-like graph is not new, see for instance [Hruska, 2002; Mischaikow, 2002]. However, to our knowledge it was not yet modified and improved in order to apply it for the detection of periodic orbits. Another option to compute shortest periodic paths on a graph is given by the Floyd–Warshall algorithm [Floyd, 1962; Rivest, 2000]. This algorithm uses the principles of dynamic programming and solves the all pairs shortest path problem. As with Dijkstra’s, this algorithm could easily be modified to compute the shortest periodic paths, see also the method proposed in [Junge, 2003]. For several reasons, we consider this approach not to be advantageous. First of all, this algorithm works on the adjacency matrix of the graph, while we use an adjacency list for the storage of edges, i.e. target cells, because the symbolic image graph is in general huge but sparse. A storage of edges in

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

an adjacency matrix would immensely increase the memory requirements. Furthermore, the time complexity of the Floyd–Warshall algorithm is Θ(n3 ). Even if modified for our problem, this complexity will not change. On the other hand, we show in Sec. 5 that our approach has a significantly better time complexity of O(n2 ) if the number of edges in the graph is fixed, i.e. the construction depends on a fixed number of scan points per box.

4.3. Further investigation methods As already mentioned, symbolic images represent a basic framework for the implementation of a broad spectrum of investigation methods. In this section we give an overview of the methods which have already been implemented . Localization of invariant sets: Let us define a cell cI as outgoing if f (M (I)) ∩ M = ∅ for its corresponding box M (I) and as leaving if all paths lead to an outgoing cell. Recall, that M is the investigated domain, see Eq. (11). Furthermore, a cell is called nonleaving if it is not leaving. Then a neighborhood of an invariant set inside M can be located by the detection of the nonleaving cells of the symbolic image. This investigation is closely related to the computation of the recurrent cells as described in Sec. 4.1. Further details and a graph algorithm for the detection of nonleaving cells are presented in [Kobjakov et al., 2003]. Construction of attractors, repellers and basins of attraction: Let us define L ⊂ V (G) as a set of cells which generates a subgraph G(L). The set of cells En(L) = {cj ∈ L : there exists an edge / L} (39) ci → cj , ci ∈ is called the entrance of L and Ex(L) = {ci ∈ L : there exists an edge / L} ci → cj , cj ∈

(40)

is called the exit of L. A set L ⊂ V (G) is invariant if for each cell ci ∈ L an admissible path through ci is in L. An invariant set L is an attractor of a symbolic image if Ex(L) = ∅. An invariant set L∗ is a repeller of a symbolic image if En(L) = ∅. Definition 6 [Osipenko, 1999].

3463

Let L be an attractor of a symbolic image. A basin or domain of attraction is the set of vertices D(L) = {cj : each path through cj finishes in L}, (41) i.e. for each path {. . . , cj , . . . , ck , . . .} there exists a number k∗ so that the vertices ck with k > k∗ belong to L. Note that the boxes corresponding to L, L∗ and D(L) form the neighborhoods of real attractors, repellers and basins [Osipenko, 1999]. In [Fundinger, 2006] algorithms are presented to detect attractors and repellers on symbolic images by graph analysis. Again, the computation of the recurrent cells is required. If attractors are detected, their domain can be computed as well, namely by a breadth-first search algorithm. Slight variations allow also the computation of connecting orbits as well as stable and unstable manifolds. Construction of filtrations: In [Fundinger, 2006] it was also shown how filtrations [Nitecki & Shub, 1975] can be constructed. A finite sequence φ = {∅ = B0 , B1 , . . . , Bm+1 = V (G)} of cells is a filtration on the symbolic image G, if ∅ = B0 ⊂ B1 ⊂ · · · ⊂ Bm = V (G) if there is no exit from Bk (k = 1, . . . , m). Let Lk be a maximal invariant set in Bk . Then each Lk ⊂ Bk is an attractor and ∅ = L0 ⊂ Ls ⊂ · · · ⊂ Lm = V (G). The area covered by the boxes corresponding to the cells of such a sequence φ is a neighborhood of a filtration, and can be constructed for a symbolic image graph. Due to the fact that its maximal invariant set is an attractor, the computation of an filtration is of special interest for the localization of attractors and the analysis of their structure. Estimation of the Morse Spectrum: A rather sophisticated investigation is the estimation of the Morse Spectrum. Note, that the Morse Spectrum can be interpreted as the union of the Lyapunov exponents of all periodic pseudo-orbits. Therefore, it allows the verification of hyperbolicity and structural stability [Osipenko, 2004]. For an estimation of the Morse Spectrum, an extension of the symbolic image is needed. More precisely, we consider a mapping F : E → E on a linear bundle E over the manifold M . The mapping F covers a homeomorphism f : M → M and is linear on fibers. In practice, usually the differential Df on the tangent bundle T M = E is used as F . The symbolic image

January 17, 2007

3464

13:28

01693

V. Avrutin et al.

is then built for the dynamical system induced by F . Hence, for a homeomorphism f defined on Rd , the symbolic image is constructed in a (2d − 1)dimensional space. In [Osipenko et al., 2004] a graph algorithm is introduced which allows the estimation of the Morse Spectrum for such an extended symbolic image by analyzing the growth rate of the vectors in the tangent bundle T M along periodic paths. Adaption to related approaches: As mentioned above, there exist approaches similar to the investigation methods based on the symbolic image graph. These investigation methods can also be integrated into the basic framework of our implementation. The algorithms presented in [Mischaikow, 2002] in the context of the Conley Index Theory are graph algorithms and, hence, can be applied on the symbolic image graph. The same holds for the computation of an expanding metric according to Hruska [2002, 2005]. The metric is calculated by investigation of a symbolic image-like graph with special algorithms based on a breadth-first search. Furthermore, some set-oriented investigation methods of Dellnitz et al. [Dellnitz & Junge, 2002] could also be integrated into the presented framework. Though most of these algorithms are not graph-oriented, they can be re-formulated as such.

5. Performance Analysis The performance of symbolic image construction as described above will be analyzed by studying the worst-case scenario. In this section we show that the construction of a symbolic image, as well as most operations on it, can be done in the time O(ns log(ns )) (Proposition 4). In the following, ns denotes the number of cells in the symbolic image Gs . For the construction of Gs , a function getBoxMapping(M (I)) can be called for every box ˜ M (I) ∈ C s which calculates an estimation C(I) for  C(I) by first locating all indices I with M (I  ) ∈ ˜ C(I) and afterward accessing these boxes by calling the function g(I  ) [see Eq. (32)]. We discuss the performance of these steps in the following. Remark 1. The localization of the box index I  for

˜ the box M (I  ) with M (I  ) ∈ C(I) is in O(1).

In order to find an index Ii with i = 1, . . . , k and k is the number of scan points, first for each point xi ∈ S(I) the value yi ∈ T˜(I) has to be processed. The calculation of yi requires only the calculation

of the system function f for mf times and is therefore ∈ O(mf · 1). The constant mf depends on the type of the dynamical system (discrete or continuous in time) and the applied tunings. For discrete systems it is usually 1. Afterward, the mapping p [see Eq. (21)] must be applied in order to calculate I  = p(yi ), whereby the calculation takes time tp ∈ O(d), where d is the dimension of the state space. Because both, mf as well as d, are constants, it follows that the calculation of a I  is in O(mf + d) ⊆ O(1). Remark 2. For a given index I  the access of a box

M (I  ) is in O(log(ns )).

The function g(I  ), see Eq. (32), can be implemented by using a hash map. Its find(I  ) operation is in O(d · log(ns )) ⊆ O(log(ns )) whereby d is the state space dimension and also the maximal number of operations which have to be performed for determination of the order of two multi-index objects using the relation ≺ [see Eq. (17)]. Proposition 3. The construction of the symbolic

image Gs with respect to the covering C s is in O(ns log (ns )). Proof. In order to get the scan points xi ∈ S(I) (i = 1, . . . , k, k is the number of scan points), the getBoxMapping(M (I)) function requires first of all the calculation of p−1 (I) = x0 ∈ M (I). This takes time tp−1 ≈ tp ∈ O(1). If this was done, the scan points, which depend on x0 , can be computed in O(k · d) [see Eq. (28)]. Then for every yi ∈ T (I) the box index Ii with yi ∈ M (Ii ) has to be found and the corresponding box M (Ii ) must be located. According to Remarks 1 and 2 this can be done in O(k ·(1+log(ns ))). So getBoxMapping(M (I)) needs a total time of

O(1 + (k · d) + k · (1 + log(ns ))) ∈ O(log(ns )) (42) ˜ Next, all cells cI
∈ V (Gs ) with M (I  ) ∈ C(I) must be added as targets to the edge list of cI . This list can also be implemented as a hash map so that the obligatory check whether cI
already belongs to the )), whereby emax is the maxilist is in O(log(emax I I mal number of target cells which is also limited to a constant by the number k of scan points per cell = k). So the complexity with regard to the (emax I calculation of all target cells cI
for the cell cI is in O(k · log(k)) ⊆ O(1). In order to get Gs , the target cells for all cI ∈ Gs must be found. So the final complexity

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

concerning the construction of the symbolic image Gs for the covering C s is O(ns · log(ns )).

(43)


Remark 3. The localization of the recurrent cells

RV (Gs ) ⊆ V (Gs ) using Tarjan’s algorithm for calculation of strongly connected components is in O(ns ). It is shown in [Sedgewick, 1993] that the strongly connected components can be found in linear time with Tarjan’s algorithm. The extensions needed to locate recurrent cells are first the distinction, if a set ζa [see Eq. (36)] contains only a single recurrent cell, second the creation of equivalence classes (sets H1 , H2 , . . . ) and third the assignment of the recurrent cells to these sets. These operations can easily be embedded into Tarjan’s algorithm and do only require constant time. Therefore, the localization of recurrent cells is still in O(ns ). Remark 4. The localization of RV (Gs ) ⊆ V (Gs )

and of the shortest periodic path for each cell c ∈ RV (Gs ) is in O(n2s ). The analysis of Dijkstra’s algorithm (see [Aho et al., 1987; Tarjan, 1991; Rivest, 2000]) leads to the result that all shortest paths for a cell ci can be found in time       O(1) + |R| tDijkstra (ns ) ∈ O  cj ∈Gs

ek ∈T (cj )

⊆ O(es + ns · |R|)

(44)

whereby T (cj ) is the list of target cells for cj , R is the outer edge with the number of elements |R| ≤ ns and es is the number of edges in Gs . As mentioned above, if the graph is unweighted, the outer edge R can be implemented as a queue. This increases the performance time significantly to     O(1) + O(1) t Dijkstra (ns ) ∈ O  unweighted

cj ∈Gs ek ∈T (cj )

⊆ O(es + ns · 1)

(45)

Furthermore, the number of edges per cell is limited by the number of scan points k. So the complete number es of edges in the graph Gs cannot be larger than k · ns . Therefore we obtain t

Dijkstra (ns ) unweighted

3465

The modified version of this algorithm, which locates periodic paths, does also not require more time. It is even faster because it terminates immediately whenever a shortest periodic path has been found: tShortest periodic path (ns ) ≤ t

Dijkstra (ns ) unweighted

⇒ tShortest periodic path (ns ) ∈ O(ns )

(47)

Note that the modified Dijkstra’s algorithm for calculation of shortest periodic paths needs some more checks than the original Dijkstra’s algorithm for unweighted graphs. However, the performance time of these operations can be neglected for theoretical analysis. In order to calculate the shortest path not only for one cell ci , but for all cells in Gs , it is now necessary to start the modified Dijkstra’s algorithm for each cell once. Hence tAll periodic paths (ns ) ≤ ns · tShortest periodic path (ns ). (48) Therefore the resulting time is ∈ O(n2s ): tAll periodic paths (ns ) ∈ O(n2s )

(49)

In Remark 4 the worst-case scenario is considered. One should note that some more improvements of the algorithm were presented. These improvements, although not important for theoretical analysis, can lead to a significant increase of performance time for the average case. However, this depends strongly on the properties of the dynamical system in focus. Remark 5. The subdivision of a covering C s into a

covering C s+1 is in O(ns ). For the subdivision of C s , all boxes M (I s ) which correspond to selected cells csI ∈ SV (Gs ) must be subdivided. For each box M (I s ) the subdivision requires m calls [see Eq. (12)] of a function σ(I s , j) = Ijs+1 ∀ j = (j1 , . . . , jn ) ∀ jk = 1 . . . imax k with

Ijs+1 = (j1 + (i1 − 1) · imax ,..., 1

(50)

jn + (in − 1) · imax n )

∈ O(es + ns ) = O(k · ns + ns )

which creates a new index Ijs+1 (compare with Eq. (35)). The number of these boxes, |SV (Gs )|, is not larger than ns . Taking into consideration that m is a constant value and tσ ∈ O(1), it follows that

⊆ O(ns )

tsubdivide(ns ) ≤ ns · m · tσ ∈ O(ns · k) ⊆ O(ns ) (51)

(46)

January 17, 2007

3466

13:28

01693

V. Avrutin et al.

Proposition 4. The complete symbolic image con-

struction process for a subdivision phase s can be done in time O(ns · log(ns )) if only recurrent cells are located, and in time O(n2s ) if the least period size for each cell should also be calculated. Proof. The complete construction process for the (s + 1)th discretization of the state space involves at first the subdivision of all boxes M (I s ) with cI s ∈ RV (Gs ) into C s+1 , then the construction of Gs+1 and finally the location of all recurrent cells RV (Gs+1 ) ⊆ V (Gs+1 ). According to Remarks 3–5, the following performance time can be achieved:

tconstruction(ns+1 ) ∈ O(ns+1 + ns+1 · log(ns+1 ) + ns+1 ) (52) ⊆ O(ns+1 · log(ns+1 )) If the least period size of every cell has to be calculated then the cell location cannot be done with Tarjan’s algorithm. Instead, the modified Dijkstra algorithm for location of shortest periodic paths must be used. According to Remark 4, the performance time changes as follows: t least (ns+1 ) ∈ O(ns+1 + ns+1 · log(ns+1 ) + n2s+1 ) period

⊆ O(n2s+1 )

(53)


In order to sum up the results obtained so far, one can say that, except for the calculation of the least period sizes, the time required by the discussed method is within O(ns · log(ns )), and therefore almost ideal from the theoretical point of view, especially for large ns . Note that these results are closely related to the proposed method for the approximation of the covering C(I), Eq. (26). The number of edges for a cell is limited by the scan points per box, and in our case this is a constant. If another approach is chosen for the approximation of the covering like, e.g. interval arithmetics or the calculation of the Lipschitz constant [Junge, 2000], the number of edges per cell is no longer bounded by a constant. For our results of performance analysis this would mean that most of the terms must be multiplied by ns . It turns out that the most critical factor for the performance is not the computation time of the algorithms but the size of the input value ns . During the subdivision process, ns could grow exponentially for every subdivision step s. Additionally, it is important to point out that, although generally the size and growth rate of ns should not depend on

the dimension of the phase space but on the dimension of those objects which are the subject of localization, in practice, this is unfortunately not always the case. It rather more heavily depends on the specific properties of the focused dynamical system. A potentially high growth rate of ns leads to the conclusion that in general the required memory is the most crucial factor for the computations. Hence, it should be the main concern to keep the number of cells ns low and avoid a high growth rate during subdivision. In many cases this can be achieved by appropriate parameter settings or tunings of the method, see Sec. 10. To give a rough overview about the computation times necessary for specific investigations, we present in the following parts some reference times. The used reference machine for all these calculations was an Asus L3000D laptop with an AMD Athlon XP-M 1400+ processor and 512MB SDRAM.

6. Accuracy of the Computations Let us next consider the accuracy of the numerical calculation. One should recall that we do not calculate specific points in the state space but boxes M (I) with some extent. These boxes are always only an outer covering of a solution. In our implementation, a box M (I) is defined as a uniform grid box. The size of such a grid box defines the accuracy ε of the calculation. Let us denote by dk the edge length of a grid box M (I) on the axis k of the state space, and by S s the union of boxes corresponding to the selected cells SV (Gs ) in the symbolic image Gs constructed after the sth subdivision. Then these boxes in S s are neighborhoods (or an outer covering) of a solution L. The basic principle of symbolic analysis is that the sequence of embedded neighborhoods S 0 ⊃ S 1 ⊃ · · · ⊃ S m gets for every subdivision step s = 0, . . . , m closer to L in such a way that, if the largest edge length tends to zero as s becomes infinite, then,

S s = L. (54) lim S s = s→∞

s

See also Theorems 1 and 2 as examples of this principle. Unfortunately, in any numerical calculation there is a minimal edge length which restricts the accuracy. Reason for this is that the n-dimensional state space covered by M [see Eq. (11)] gets divided into regions which are identified by multi-indices I ∈ Nn , Eq. (14). As mentioned above, a value ik of the kth component of the multi-index I is represented by an integer value, and for every digital

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

computation machine there exists a constant Nint giving the largest number which can be represented as an integer value. Consequently, we have the lim≤ Nint . Therefore, every edge length dk itation imax k is limited to = dmin k

Mkmax − Mkmin Nint

(56)

This is the limit for the accuracy of every calculation on our reference machine.

7. Dynamical Systems Discrete in Time In order to demonstrate the capabilities of the symbolic image based analysis, we present in the following some typical investigation examples. The main aim hereby is not to reproduce some wellknown results but to demonstrate, what kind of

(a)

results can be obtained by the symbolic image analysis and how the parameters of this method have to be adjusted for specific investigation tasks.

7.1. Logistic map (55)

which means that the minimal error εk is restricted to εk ≥ dmin k . Note, that this limit is not specific for the presented method, but only for its implementation presented in our work. It is possible to extend this limit to any size by taking a different representation of a number ik , though this implies a higher memory consumption. Using our reference machine mentioned above, symbolic images of a size up to ≈ 2 000 000 cells can be constructed. Furthermore, the largest number representable by the used hardware architecture is Nint = 232 , so that we obtain ≈ 10−9 · (Mkmax − Mkmin ). εk ≥ dmin k

3467

Let us start with one of the most standard and wellknown examples in the field of nonlinear dynamics, namely with the logistic map defined by: x(n + 1) = fl (x(n)), fl : [0; 1] → [0; 1], (57) fl (x) = αx(1 − x). For this system, a period-doubling bifurcation scenario can be observed, formed by a sequence of flip bifurcations, as described for instance in [Grossmann & Thomae, 1977; Feigenbaum, 1978, 1979]. As one can see in Fig. 4, the results obtained using the symbolic image analysis contain more information about the investigated bifurcation scenarios, than the results of forward iterations. For the period doubling scenario one observes not only stable fixed points and limit cycles, but also fixed points and limit cycles, which become unstable at the points of flip bifurcations.

7.2. Ikeda map As an example for the application of the symbolic image based analysis in the case of a 2D map, let us consider the Ikeda map [Ikeda, 1979] defined by: x(n + 1) = fI (x(n)),

fI : R 2 → R 2 ,

x = (x, y)T

(b)

Fig. 4. Logistic map: results obtained by forward iterations (red) and by symbolic image analysis (blue). Rectangle marked green in (a) is shown enlarged in (b).

January 17, 2007

3468

13:28

01693

V. Avrutin et al.

 fI (x) = with

r + a(x cos g(x, y) − y sin g(x, y)) b(x sin g(x, y) + y cos g(x, y)) g(x, y) = c1 −



c2 (1 + x2 + y 2 ) (58)

This systems occurs in the modeling of optical recording media. The Ikeda map is interesting for symbolic image analysis because at the considered parameter values it has stable and unstable periodic orbits as well as a chaotic attractor. Note, that the Ikeda map has an explicit inverse [England et al., 2004]. Therefore, it is possible to verify the calculation of the unstable fixed points or unstable periodic points by forward iteration of this inverse map if both manifolds of these sets are unstable. However, for most invariant sets of the Ikeda map this is not the case and hence the forward iterations of the inverse map is not applicable. In contrast to this, the symbolic image analysis is applicable in any case, especially for saddle-like invariant sets. A comprehensive study of the system by symbolic image methods can be found in [Osipenko, 2004]. However, in that work only the results are presented. No details about the numerical calculation are mentioned. For this reason, some of the computations are discussed here again. The numerical simulations have been carried out for the parameter values a = b = 0.9, c1 = 0.4, c2 = 6.0 and r = 0.9. According to other numerical results obtained up to date (see, for instance [Osipenko, 2004; Miller & Yorke, 2000] and references therein) there exists at these parameter values a chaotic attractor A, two unstable fixed points P1,2 (saddle points) and the stable fixed point P3 :   0.4819 , P1 ≈ 0.2545   1.1987 (59) , P2 ≈ −2.3769   3.0027 P3 ≈ 3.8945 Additionally, there exists the unstable 2-periodic orbit T 12 , as well as two unstable 3-periodic orbits 3 : T 1,2     0.5964 0.4497 2 T1 ≈ , 0.6394 −0.6453       0.8091 0.9960 −0.0280 3 , , T1 ≈ 0.7834 −1.0090 −0.8758

T 23

 ≈

     1.3512 0.6568 −0.2418 , , −0.0707 −1.1932 −0.4462 (60)

3 lie close to the chaotic Note, that P1 , T 12 and T 1,2 attractor A. We start the analysis of the system by localization of the chain recurrent set. As already mentioned, the chain recurrent set contains all kind of return trajectories and, hence, should give an overview about the areas of interest for further investigation. We set the area of investigation in the state space to M = [−5.0; 5.0] × [−5.0; 5.0]. This area M is initially divided into a covering C 0 of 20 × 20 boxes. Then the symbolic image G0 is constructed for C 0 . We apply Tarjan’s algorithm, see Sec. 4.1, to detect the recurrent cells. The construction and cell detection process was repeated for four subdivisions. In each step the boxes are subdivided into 4 × 4 smaller boxes. The results of such a calculation can be seen in Fig. 5. After four subdivision steps, the distinct features of the Ikeda map can be found in the different recurrent sets of the symbolic image. Three areas in the state space are detected. One of them represents the stable point P3 , the other one the unstable saddle point P2 , and the last one contains all cells representing the chaotic attractor A and all

Fig. 5. Ikeda map: numerical computation of an outer covering of the chain recurrent set for the Ikeda map, Eq. (7.2). This outer covering of the chain recurrent set contains the chaotic attractor A as well as the fixed points P2,3 .

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

unstable periodic orbits close to this attractor. It is worth mentioning that chaotic attractors can be found in symbolic images because their skeleton is typically build up from unstable periodic cycles [Cvitanovi´c, 1991, 1992] and, therefore, of recurrent cells. The calculation of the symbolic image takes less than 2 minutes. In Table 1 the number of cells in the symbolic image and the number of located recurrent cells for every subdivision level are shown. We observe that the number of recurrent cells grows during the subdivision process by factor ≈ 10. This high growth rate is due to the fact that it depends on the dimension of those objects which are the subject of localization, see also [Junge, 1999; Mischaikow, 2002]. In this case, one of the subjects of localization is the chaotic attractor A which has a dimension close to 2. Note also that the periodic 3 as well as the fixed point P1 orbits T 12 and T 1,2 lie inside the computed outer covering of the chain recurrent set. Due to the fact, that they are close to the attractor A, they cannot be distinguished from it by this kind of computation. Our next goal is the localization of the peri3 . This requires, of course, odic orbits T 12 and T 1,2 the usage of the time consuming variant of the cell location algorithm based on the Dijkstra algorithm, see Sec. 4.2. We consider the same area M = [−5.0; 5.0] × [−5.0; 5.0], as for the last computation but select only the cells with period size p < p = 3 for further subdivisions. After an initial subdivision of M into 20 × 20 boxes, the boxes are subdivided into 8 × 8 smaller boxes in every subdivision. The construction and cell detection process was repeated for 9 subdivisions. The results of the calculation in the area [−1.5; 2.5] × [−1.5; 2.0] can be seen in Fig. 6(a), namely the points belonging to 3 . Note, that also the points P P1 , T 12 and T 1,2 2,3 were detected but no other orbits with a period ≤ 3. Here

Table 1. Computation of the recurrent cells in the Ikeda map, Eq. (7.2). The phase space discretization of the area M = [−5.0; 5.0] × [−5.0; 5.0] is shown. Furthermore, the number of cells belonging to a symbolic image, |V (Gs )|, and the number of localized recurrent cells |RV (Gs )| are presented depending on the subdivision depth s. s

Phase Space Discretization

|V (Gs )|

|RV (Gs )|

0 1 2 3 4

20 × 20 80 × 80 320 × 320 1280 × 1280 5120 × 5120

400 1264 6640 48 288 446 048

79 415 3018 27 878 284 727

3469

it turns out, that the usage of a sufficient number of scan points is important. In this example, every box contains 8×8 scan points distributed uniformly in the box, and five additional points close to the corners and the center of the box. Such a high number is needed to acquire all the periodic orbits. If less scan points are chosen, another parameter must be set for error tolerance (see Sec. 10.2), or some of the periodic orbits will not be detected. Although this calculation uses the more time consuming period detection algorithm, the computation takes less than 30 seconds on the reference machine. This is due to the fact that only very few cells have a period size smaller than or equal to 3. In our calculations, not more than 97 cells per subdivision step fit to this criterion. So the size of the symbolic images remains small. However, one should notice that the performance time can increase exponentially if the parameter p is set to a higher value and more such cycles with p ≤ p exist. A serious problem we have to deal with in this computation is, that for most points not only one box corresponding to a periodic cell is found, but several boxes in the neighborhood. In this case we get up to five boxes as an outer covering for each periodic point instead of one box. In the following, we will refer to this phenomenon as clustering. Empirically, it has turned out, that clustering can be considered as one of the most common and crucial numerical problems occurring in the context of symbolic image analysis. Due to clustering, the growth rate of the number of cells in a symbolic image increases, and the accuracy of the computation decreases. Moreover, the analysis of the computed data becomes more difficult. For this reason, further techniques are necessary in order to avoid or at least reduce clustering. Theoretically, the following accuracy, compare Sec. 6, could be achieved for the calculated points: ε≤

1 1 · 4 ≈ 2 · 10−9 . · 20 89

However, in practice, the error is larger because of clustering. Taking this into account, we state, that the error is approximately ε ≤ 1 · 10−7 . Because A is a chaotic attractor, one can expect to find in its vicinity some unstable limit cycles with periods higher than 3. So first we increase p to 6 and then to 14. Some results of these calculations are presented in Fig. 6(b), which shows two of the detected unstable 5- and 6-periodic orbits. In Fig. 6(c) an overview of all detected 6- and

January 17, 2007

3470

13:28

01693

V. Avrutin et al.

(a)

(b)

(c) Fig. 6. Ikeda map: (a) P1 – blue square, T 21 – green triangles, T 31 – empty circles, T 32 – blue circles. (b) Some detected unstable limit cycles with periods 5 (empty circles) and 6 (blue circles). (c) All detected unstable 6-periodic (empty circles) and 13-periodic (blue circles) points. The chaotic attractor A in the background is shown for a better orientation but was not calculated by the same computation.

13-periodic points is given. Remarkably, the symbolic images for p = 6 contain not more than 325 cells, for which the corresponding boxes have to be subdivided. Hence, the calculation do not take much more computation time than in the first case (≈ 30 seconds). However, the location of cells with a period size ≤ 14 consists of up to 27 000 marked cells. Boxes corresponding to each of them are subdivided into 8×8 smaller boxes, so that the symbolic images had up to 1 700 000 cells. Therefore, the calculation takes around eight hours in this case. Next, we present some results of more sophisticated investigation methods for the Ikeda map. The intention here is to show the possibilities of

symbolic image construction. In Fig. 7(a) the basin of attraction for the Ikeda attractor A is shown. The figure contains the attractor A and its basin as well as the saddle point P2 . It becomes clearly visible, that the stable manifold of the saddle point P2 separates the basins of the chaotic attractor A and of the stable fixed point P3 . For a more detailed analysis of these results we refer to [Fundinger, 2006]. In Fig. 7(b) the stable and unstable manifolds for P2 are computed. We like to mention that there exist already other methods for the computation of the stable and unstable manifolds, see e.g. [Dellnitz & Hohmann, 1997; Krauskopf & Osinga, 1998a, 1998b; Krauskopf et al., 2005;

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

(a)

3471

(b)

Fig. 7. Ikeda map: (a) chaotic attractor A (green), its basin of attraction (red) and the saddle point P2 . (b) The stable manifold (green) and the unstable manifold (red) of the saddle point P2 .

England et al., 2005; You et al., 1991]. However, the application of symbolic image construction overcomes some of the drawbacks of the other methods. Firstly, the computation of higher-dimensional manifolds is, in principle, possible. Secondly, in case of discrete dynamical systems, neither the computation nor the existence of the system function’s inverse is required in order to compute the global stable manifold, or, respectively, the global stable set. Due to the fact that additional research is still required in this field of symbolic image construction, a more detailed discussion of this topic would be out of focus for this work. We consider the presented results as a motivation for further investigations.

Until now we investigated the Ikeda map for fixed parameter values, as described above. Using the symbolic image based methods under variation of some parameters, interesting results can be obtained as well. For instance, one can observe the bifurcations causing the unstable periodic orbits to emerge, which determine the structure of the chaotic attractor discussed above. Performing this task, we consider the area M = [−0.4; 1.5]×[−1.7; 1] in the state space and calculate the periodic orbits up to period six. Using an initial subdivision into 20×20 boxes and performing four subdivision steps, whereby each box is divided into 2 × 8 × 8 smaller boxes, we obtain the results shown in Fig. 8. The parameters a and r are varied in the interval [0; 0.9]

(a)

(b)

Fig. 8. Ikeda map: periodical points up to period 6 under variation of parameters a respectively r. The chaotic attractor A is shown for a better orientation.

January 17, 2007

3472

13:28

01693

V. Avrutin et al.

the other parameters are kept fixed to the same values as above. In both experiments we observe a period doubling bifurcation scenario and a large number of saddle-node bifurcations.

7.3. Coupled logistic map We take a look at another two-dimensional map, the coupled logistic map defined by: x(n + 1) = fC (x(n)), fC : R2 → R2 , x = (x, y)T   (1 − r)g(x, a) + rg(y, b) fC (x) = rg(x, a) + (1 − r)g(y, b) with g(x, m) = mx(1 − x).

(61)

The system, as presented here, can be considered as a two-dimensional case study of coupled map lattices [Kaneko, 1993] for the logistic map [Gyllenberg et al., 1993]. In our context, the study of this system reveals some interesting dynamics. As for the Ikeda map, we computed the chain recurrent set and periodic orbits. We did not only find an attractor but also fractal structures which, in contrast to the attractor, cannot be revealed by forward iterations. For all our investigations, we fixed the parameter settings to a = b = 3.8 and r = 0.07. Analytically, it is easy to show that, due to a = b, we have symmetric behavior with respect to the diagonal y = x. This means that orbits become symmetric if one interchanges the x- and y-coordinates, and that all points on the diagonal at y = x form an invariant set D. By numerical analysis based on forward iterations and calculation of Lyapunov exponents, one can find out, that the system is governed by a single attractor A which consists of two symmetric parts in the phase space, see Fig. 9. Our first investigation of the system by symbolic image analysis was the computation of the chain recurrent set. We initially divided the area M = [0.0; 1.0] × [0.0; 1.0] into 5 × 5 boxes. In each subdivision step, a box gets divided into 3 × 3 new ones. After five subdivisions the chain recurrent set consists of 430 000 boxes with a side length ≈ 1 · 10−3 . It is important to mention that a high number of scan points is required. If one takes to few scan points, large parts of the chain recurrent set will not be detected during the first subdivisions. Hence, for our investigation we covered each box with a regular grid of 100 scan points. Applying these settings, the computation takes around 8 minutes, and its results can be seen on Fig. 10(a).

Fig. 9. Coupled logistic map: numerical approximation of the attractor A.

The chain recurrent set does not only contain the chaotic attractor but also of fractal structures which are symmetric with respect to the diagonal. Note that these fractal structures are unstable entities. Orbits started in a neighborhood of the chain recurrent set are attracted by the attractor A. We observed that even orbits started in the area covered by the computed fractal structures are attracted by A. However, this can be explained by the fact that our numerical computation produced an outer covering of the real chain recurrent set and, hence, covers also the chain recurrent set’s neighborhood. To the authors’ knowledge there is no other method of numerical investigation, except maybe the related set-oriented approach by Dellnitz et al. [2001], which is capable to reveal these structures. In Fig. 10(b) we colored the boxes of each equivalent recurrent set differently. It is clearly seen, that there are four distinct equivalent recurrent sets, one of them represents A, and another one a 2-periodic unstable orbit in the holes of A. In order to verify our results, we also computed periodic orbits. We used the cell location algorithm based on the Dijkstra algorithm and computed all periodic points with a periodicity ≤ 8. We applied 17 subdivisions so that the error ε ≤ 1 · 10−8 . The computation took around 25 minutes, and we got 614 periodic points, see Figs. 11(a) and 11(b). It can be observed that periodic orbits are scattered over the whole area designated by the chain recurrent

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

(a)

3473

(b)

Fig. 10. Coupled logistic map: (a) numerical approximation of the chain recurrent set (red) and the attractor A (green). (b) Sets of equivalent recurrent sets are shown in different colors.

(a)

(b)

Fig. 11. Coupled logistic map: (a) detected periodic points. Periodicity: 1 (brown), 2 (cyan), 4 (magenta), 5 (green), 6 (black), 7 (blue) and 8 (red). (b) Periodic points and the chain recurrent set.

set. We confirmed our results by applying a Quasi– Newton method to locate periodic orbits as proposed in [Nusse & Yorke, 1997]. This method solves the equation f p (x) − x = 0,

whereby p is the periodicity of the orbit. As start values for the Newton iteration we took our computed periodic points. For each of them we found a periodic orbit of the same period in the immediate neighborhood, giving evidence for the correctness of

January 17, 2007

3474

13:28

01693

V. Avrutin et al.

our calculations. Furthermore, we also checked that each periodic orbit is unstable. Combining the results of our numerical computations so far, we find strong evidence for the hypothesis that the computed fractal structure of the chain recurrent set is an outer covering of a set of unstable periodic orbits of any size. This reminds us of the hypothesis of Cvitanovi´c [1991] regarding periodic orbits as the skeleton of chaotic attractors. However, the fractal structure we observe here is not an attractor.

7.4. Discrete food chain model Next we analyzed a discrete system of mathematical biology. The three-dimensional dynamical model describes a discrete food chain model, studied by Lindstr¨om [2002]. It is based on the continuous food chain model introduced by Rosenzweig [1973] as an extension of the Rosenzweig–MacArthur model [Rosenzweig & MacArthur, 1963]. The system is defined by x(n + 1) = fdfc (x(n)), fdfc : R3 → R3 , x = (x, y, z)T   µ0 xe−u  1 + x max (e−u , g(z)g(u))    fdfc (x) =  ,  µ1 xye−z g(y)g(µ2 yz)  (62) µ2 yz  −s   1 − e , if s = 0, s with g(s) =   1, if s = 0. In this work we will only focus on the following parameter setting: µ0 = 3.4001, µ1 = 1 and µ2 = 4. As a first step, it is our intention to find the fixed points. The analytic results of Lindstr¨ om showed, that Eq. (62) possesses at most four fixed points and that three of them are given by: P0 = (0, 0, 0)T P1 = (µ0 − 1, 0, 0)T   T  µ1 µ0 (63)   µ log   0 µ µ 1 + µ 1 0 1  P2 =   (µ0 − 1)µ1 − 1 , log 1 + µ1 , 0 This allows us to confirm some of our numerical results. We set the parameter pmax = 1 to detect all one-periodic cells. For the domain space

M = [−1.0; 4.0] × [−1.0; 4.0] × [0; 1.6] and an initial subdivision into 50 × 50 × 8 boxes, we get, after several subdivisions, four areas in the state space around the following points: P˜0 = (0, 0, 0)T , P˜1 = (2.4001, 0, 0)T , P˜2 = (1.289, 0.531, 0)T ,

(64)

P˜3 = (2.116, 0.25, 0.423)T . Taking into consideration that the domain space was eventually divided into 204 800 × 204 800 × 32 768 regions, and that in each area exist some neighboring boxes where the corresponding cells are detected as 1-periodic, the values of P˜0,1,2 correspond with a numerical error of ε ≤ 1 · 10−3 to the analytical values of P0,1,2 . Additionally, a fourth point, P˜3 , was detected, which cannot be determined analytically for this parameter setting. The overall calculation took less than 15 seconds which is due to the fact that not more than 32 cells are considered 1-periodic in each subdivision step. As for the Ikeda map, we do not only want to find specific p-periodic cells, but also the complete symbolic image which includes all stable and unstable invariant sets of the domain space M . That there exists at least one attractor can be demonstrated by a forward iteration (see Fig. 12). We tried to approve these results by symbolic image calculation. Unfortunately, this is for a three-dimensional system not as easy as for a two-dimensional one. Due to the additional dimension, usually one has to deal with much more cells. In this case, it was not possible to get an appropriate approximation of the attractor by means of usual symbolic image construction. Some additional techniques to tune the method must be applied to get satisfiable results. These techniques will be introduced later, in Secs. 10.3 and 10.6. By applying them, the equilibrium points, and maybe some other information, may disappear in the symbolic image after several subdivision steps. But, however, the attractor can then be approximated very precisely. Furthermore, an unstable quasi-periodic cycle is detected in between, see Fig. 12. Both invariant sets are represented as different recurrent cell sets. Such a calculation takes around one hour and the symbolic image grows up to ≈ 1 100 000 cells. The long calculation time is hereby mainly caused by the application of the tuning-techniques. In order to get a better impression how the construction process works, Fig. 13 shows the results of

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

(a)

3475

(b)

Fig. 12. Discrete food chain model: (a) + (b): two different views of the results of symbolic image analysis. The attractor (red), an unstable quasi-periodic cycle (green) and the stable fixed points (blue) are shown. The fourth fixed point at (0, 0) cannot be seen.

(a) second subdivision step

(b) third subdivision step

(c) fourth subdivision step Fig. 13. Discrete food chain model: numerically calculated fixed points and three subdivision steps of the symbolic image construction (green). For better orientation, an approximation of the attractor by forward iteration (red) is also shown.

January 17, 2007

3476

13:28

01693

V. Avrutin et al.

several subdivision steps. The rough position of the attractor can be located after the second subdivision of the domain space M into 200 × 200 × 32 regions [Fig. 13(a)], then, in the third subdivision (Fig. 13(b), 400 × 400 × 64 regions), the principal shape of the attractor becomes visible. But only after the fourth subdivision (Fig. 13(c), 1200 × 1200 × 192 regions) the symbolic image splits into two different recurrent cell sets, which correspond to the stable and unstable invariant sets.

8. Dynamical Systems Continuous in Time Only dynamical systems discrete in time have been discussed so far. The symbolic image for such a system x(n + 1) = f (x(n)) can be constructed by performing one iteration, which means simply applying the system function f (x) on the points x ∈ M (I) lying in a box M (I) of a certain covering, see Eq. (29). If we are dealing now with systems continuous in time, some kind of mapping is required which transforms an orbit continuous in time into one discrete in time. Several approaches exist toward this task. In [Mischaikow, 2002] the use of local Poincar´e sections is proposed. The dynamics of multiple (n − 1)dimensional hypersurfaces are studied, which are transversal to parts of the system’s flow. An advantage of this approach is that the dimension of the grid is reduced by one. In our application we use a more general approach — a stroboscopic mapping with a sufficiently small discretization time t. Then the values y, which represent the images of the points x, can be calculated by solving the equation ˙ x(t) = f (x(t))

(65)

for the time t and initial conditions x(0) = x. In the context of a computer implementation it is suitable to use t = ∆t · n, whereby ∆t is the integration step size of the applied integration method, and n ∈ N+ is the number of integration steps. Note that similar approaches were proposed in [Junge, 1999] and [Pilarczyk, 1999]. Both values ∆t and n are user-defined parameters. Suitable settings depend on the properties of the investigated dynamical system as well as on the characteristics of the particular symbolic image construction. Hereby, the choice of an appropriate integration step size ∆t should mainly depend on the properties of the investigated dynamical system and the used integration method. The aim is

to keep the numerical error reasonably small. On the other hand, a suitable setting for the number of integration steps n = (1/∆t)t should be chosen with respect to the properties of the symbolic image, especially the used box size, as well as the velocity of the dynamical system. It turns out, that an appropriate setting of ∆t and n is a nontrivial task, which strongly influences the symbolic image construction process. This is mainly due to the fact, that a continuous trajectory does not jump from a point xn to xn+1 in the domain space M but more rather moves either from the area covered by a box M (I) to the area covered by a neighboring box M (I  ), or it stays within the same area. Hence, we distinguish two critical cases: 1. If t is chosen too small, some trajectories might not leave their boxes M (I) in our simulation although they would do so at a later time t > t. Then a cell cI corresponding to such a box M (I) appears to be 1-periodic although it is not periodic. Note that this behavior can also happen for systems discrete in time. However, systems continuous in time are usually much stronger affected. 2. If t is chosen too large, the simulation of the trajectory started in a box M (I) might not stop after the trajectory has entered the next neighboring box M (J) but only after it has entered a later box M (K) on the path of the trajectory. Then edges between some cells cI and cJ are not detected. In this case, the symbolic image graph is not an appropriate representation of the system’s flow, especially if the velocity of the flow strongly fluctuates. As a result, for instance, some cycles might not, or only partly, be located. Before we discuss some possible solutions of this problem, we present the investigation results for two dynamical systems continuous in time, Van der Pol and Lorenz, to show what kind of information can be gathered by symbolic image construction and how the problems mentioned above can be controlled.

8.1. Van der Pol system In order to demonstrate, that symbolic image construction is applicable for dynamical systems continuous in time, let us firstly consider the Van der Pol system, defined by ˙ x(t) = fVdP (x(t)), fVdP : R2 → R2 , x = (x, y)T

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

 fVdP (x) =

y γy(1 − x2 ) − x

3477

 (66)

Investigation results for this two-dimensional dynamical system continuous in time can be found for instance in [Argyris et al., 1994]. Especially for γ = 1.5 the system possesses an unstable fixed point P0 = (0, 0)T and a stable limit cycle around this point. We try to reproduce these results. Therefore, the symbolic image is constructed for the state space area M = [−3.5; 3.5] × [−3.5; 3.5]. An initial subdivision of M into 50 × 50 boxes and further divisions of boxes into 4 × 4 new ones are used. The trajectories are approximated by setting the integration step size ∆t = 0.001 and the number of integration steps n = 100. It is important to consider that the approximation of the trajectory x(t) is not to be very precise because the calculated trajectory is short. So the integration error is by far not as crucial as if a long run would be performed. One should recall that the symbolic image method constructs limit cycles or other distinct features by the combination of many small trajectories instead of tracking a few trajectories for a longer time. A too small setting of the time ts = n · ∆t would in many cases only increase the size of the symbolic image and the calculation time, but not the quality of the results. A more detailed analysis of this topic will be given in Sec. 10.4. The construction of four subdivisions requires a computation time of around 15 minutes and produces images of up to 800 000 cells. The results for several subdivision steps can be seen in Fig. 14. The stable limit cycle as well as the unstable fixed point P0 were found with an error of ε ≤ 10−2 after M was divided into 12 800 × 12 800 boxes. Clustering can be observed for the performed calculation. P0 is not represented by a single cell but rather more by a bundle of 426 distinct recurrent cell sets, each containing only one periodic cell, which corresponds to a box in the neighborhood of the fixed point. Furthermore, the stable limit cycle is represented by a single recurrent cell set, containing 202 886 cells. Considering Fig. 14, one can recognize that the principal shape of the limit cycle was found at a very early stage of the calculation while the separation into different areas (the limit cycle and the fixed point), or different recurrent cell sets, takes a longer time. This phenomenon was already mentioned in the last section, Sec. 7.4, and seems to be a typical characteristic of symbolic image construction.

(a) first subdivision step

(b) second subdivision step

(c) fourth subdivision step Fig. 14. Van der Pol system: three subdivision steps of the numerically calculated symbolic image.

January 17, 2007

3478

13:28

01693

V. Avrutin et al.

8.2. The Lorenz system Next we consider the well-known dynamical system continuous in time introduced by Lorenz [1963] and defined by fL : R3 → R3 , x = (x, y, z)T   σ(y − x) (67)   fL (x) =  x(r − z) − y  . xy − bz

˙ x(t) = fL (x(t)),

This system is considered in order to validate our investigation method in a three-dimensional state space and to demonstrate its capability to detect unstable orbits of dynamical systems continuous in time. We use the standard values of the parameters σ = 10, b = 8/3 and investigate the Lorenz system at two values of the parameter r, namely r1 = 14.6 and r2 = 20. As shown in [Sparrow, 1973], for these settings exists an unstable fixed point S = (0, 0, 0)T and two stable ones C1 and C2 , each of them accompanied by an unstable limit cycle. The value r1 is chosen close to the so-called homoclinic explosion which occurs at r ≈ 13.926, where the unstable manifolds of S return to the origin. In order to reproduce some of these results with symbolic image construction we define for r1 and r2 the domain spaces M1 = [−35.0; 35.0] × [−35.0; 35.0] × [0.0; 30.0] and M2 = [−20.0; 20.0] × [−20.0; 20.0] × [0.0; 30.0]. The division of these spaces is initially set to 4 × 4 × 2 and 2 × 2 × 2. In the following subdivision stages each box is divided into 2 × 2 × 2 smaller boxes. It is our main intention to detect the unstable limit cycles, and not the fixed points, so we chose settings which are not suitable to find S but which allow the location of the limit cycles. Therefore, ∆t is set to 0.001, and the number of iteration steps to n1 = 100, n2 = 200. Figures 15 and 16 show the results of the calculations for the parameters r1 and r2 . The computations took 30 minutes for r1 and 2 hours for r2 , the symbolic images contained up to 1 400 000 cells. The high computation time is mainly due to a relative high setting of the iteration time ts . To produce the final cycles in Figs. 15(c) and 16(c), some advanced tunings are applied, see Sec. 10.6 for a further discussion. Remarkably, one can see that the limit cycles for r1 still contact each other, which is due to some numerical inaccuracy, while for r2 the cycles shrank closer around C1 and C2 . Again, we see that the principal shape of the cycles becomes visible very early in the subdivision process, while the distinction into fixed points and cycles happens

(a) fifth subdivision step

(b) sixth subdivision step

(c) tenth subdivision step Fig. 15. Lorenz system: three subdivision steps of the numerically calculated symbolic image at r1 = 14.6.

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

3479

later. If n would be set smaller, this distinction would not be visible in the numerical computation. Too many cells would be considered recurrent and the size of the symbolic image would be too big for calculation after a few subdivision steps.

8.3. Chua’s circuit Let us now apply the symbolic image analysis to the well-known model of Chua’s circuit, defined by fC : R3 → R3 , x = (x, y, z)T   kα(y − x − f (x))   fC (x) =  k(x − y + z) , k(−βy − γz)

˙ x(t) = fC (x(t)),

(a) sixth subdivision step

(b) seventh subdivision step

(c) tenth subdivision step Fig. 16. Lorenz system: three subdivision steps of the numerically calculated symbolic image at r2 = 20.

1 with f (x) = m1 x + (m0 − m1 )(|x + 1| − |x − 1|). 2 (68) This system has been studied widely and many interesting phenomena can be observed, see e.g. [Chua, 1994]. Note that the system is piecewiselinear and contains several symmetries. In this section, we focus on the localization of unstable invariant curves which occur due to period-doubling bifurcations. It is well known that in period-doubling scenarios of dynamical systems, stable limit cycles become unstable at perioddoubling bifurcation points, where a new stable limit cycle with twice the period emerges [Feigenbaum, 1978]. In Sec 7.1 we already applied the symbolic image method to a system discrete in time, in order to compute both, stable and unstable limit cycles of the period-doubling scenario. Now it is our intention to compute the unstable limit cycles after a period-doubling bifurcation for Chua’s circuit, i.e. a dynamical system continuous in time. The system is first investigated at the parameter setting k = 1.0, α = 15.6, β = 35.0, c = 0.0, m0 = −1.142857143 and m1 = −0.714285714. At these parameter values, two coexisting attractors with period two can be found which are symmetric to each other with respect to the origin. By decreasing the parameter β, a period-doubling cascade can be observed. In order to perform the computation of the symbolic image, we set the area of investigation in the state space to M = [−2.5; 2.5] × [−0.4; 0.4] × [−4.0; 4.0]. The area M is initially divided into a covering of 50 × 8 × 80 boxes. Eight scan points per box are used for the construction of the symbolic image. The integration step size is set to ∆t = 0.01

January 17, 2007

3480

13:28

01693

V. Avrutin et al.

and the number of iteration steps to n = 98. Our aim is to compute an outer covering of the chain recurrent set. The subdivision process is repeated six times. In each subdivision step, the boxes are subdivided into 2 × 2 × 2 smaller boxes. Using these settings, the symbolic image contains up to 2 364 900 cells. The final results of the computation are illustrated in Fig. 17. Obviously, the computed approximation of the chain recurrent set contains the two period-2 invariant curves which are the

attractors of the system. Additionally, also the two unstable period-1 limit cycles belong to the chain recurrent set and, hence, can be detected. Note that the tuning method for reconstruction of fragmented limited cycles as described in Sec. 10.6 was applied to obtain these results. We repeated the computation for the modified parameter β = 33.83. At this parameter value, a period-doubling bifurcation has occurred, and the two attractors of the system are period-4 invariant curves. Due to the fact that the attractors are symmetric to each other with respect to the origin, we restricted our investigation to one of them. Hence, the symbolic image is computed for the area M = [−1.5; 2.5] × [−0.4; 0.4] × [−4.0; 2.0]. The initial covering consists of 40 × 8 × 60 boxes. In order to achieve a good approximation of the limit cycles, it is necessary to set ∆t = 0.0025 and the number of iteration steps to n = 2800. Obviously, such high settings result in a high amount of computation time of around 24 hours. After seven subdivisions, the symbolic image contains 688 800 cells. The computed outer covering is shown in Fig. 18(a). Again, the tuning method for reconstruction of fragmented limited cycles was applied. As can be seen, the approximated chain recurrent set contains one of the stable period-4 invariant curves, and also the unstable period-2 invariant curve. However, the period-1 invariant curve, which also exists, is not detected. The reason for this is the large setting of the parameter n which leads to the loss of this

(a)

(b)

Fig. 17. Chua’s circuit: computation of an outer covering of the chain recurrent set by the symbolic image method (red) and the two attractors with period two, calculated by a usual forward iteration (green) for β = 35.0.

Fig. 18. Chua’s circuit: (a) computation of an outer covering of the chain recurrent set by symbolic image construction (red) and one of the two period-four attractors calculated by usual forward iteration (green) for β = 33.83. Due to tuning methods, the period-one invariant cycle was not detected. (b) Computation of an outer covering of the chain recurrent set for a Poincar´e section (blue) and parts of the symbolic image and the attractor of Fig. (a) projected on the x × z-plane.

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

3481

part of the chain recurrent set. A more detailed discussion of this phenomenon follows in Secs. 10.4 and 10.6. However, we were also able to compute the period-1 cycle using a smaller setting of n and the drawback that the period-2 cycle cannot be properly detected. The computations presented above show that the symbolic image method can successfully be applied to detect unstable limit cycles without any preliminary information of the system and the limit cycles. However, the computations can be time and memory consuming, especially in the case of

dynamical systems continuous in time. To overcome these problems, a Poincar´e section can be computed, as described in Sec. 10.5. We applied the technique at the parameter setting β = 33.83. The area of investigation, i.e. the Poincar´e section, is set to M p = [1.7; 2.5] × [−0.001; 0.001] × [−4.0; −2.0] and represents a thin layer around the plane y = 0. The covering of this area is set to 8 × 1 × 20 boxes which are subdivided in 2 × 1 × 2 boxes in every subdivision. In order to ensure that trajectories return to M p , an integration step size of ∆t = 0.001 is applied and the

(a)

(b)

(c) Fig. 19. Chua’s circuit with piecewise-linear nonlinearity: (a) computation of an outer covering of the chain recurrent set which contains three attractors. (b) Numerical approximation of the three coexisting attractors (red, green, blue) by forward iteration. (c) Computation of an outer covering of the chain recurrent set for a Poincar´e section (magenta), and an approximation of three coexisting attractors projected on the x × z-plane.

January 17, 2007

3482

13:28

01693

V. Avrutin et al.

8.4. Chua’s circuit: piecewise-linear versus smooth nonlinearity

number of integration steps is set to n = 1 800. The computation of 12 subdivision steps takes around 6 minutes and, hence, only a fraction of the former computation time. This is due to the fact that the symbolic images consist of not more than 3096 cells. A projection of the results on the x × z plane is shown in Fig. 18(b). For comparison, also the results of the former computation are illustrated. Obviously, all limit cycles which cross M p can be localized, also the unstable period-1 curve which was missing in the computation of the whole area. Furthermore, the approximation of the limit cycles is also more precise on M p .

In the previous section we discussed Chua’s circuit with a piecewise-linear function as it was originally proposed by Chua [1992]. Recall that Chua’s circuit is a nonlinear electronic circuit, and the only nonlinearity of this system is a diode. The characteristic of this diode is usually assumed to be piecewise-linear because this approach has some advantages with respect to rigorous mathematical analysis. However, the characteristics of nonlinear components in real circuits are always smooth. In [Tsuneda, 2005] this topic was addressed and a comparison of attractors

(a)

(b)

(c) Fig. 20. Chua’s circuit with smooth nonlinearity: (a) computation of an outer covering of the chain recurrent set which contains two attractors. (b) Numerical approximation of the two coexisting attractors (red, green) by forward iteration. (c) Computation of an outer covering of the chain recurrent set for a Poincar´e section (blue), and an approximation of two coexisting attractors projected on the x × z-plane.

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

for the models of Chua’s circuits with piecewiselinear and smooth nonlinearity was presented. The model with smooth nonlinearity as proposed in [Tsuneda, 2005] is defined by a modification of Eq. (68). Hereby, the piecewise-linear function f (x) is replaced by a cubic polynomial fˆ(x) = ax3 + bx which is similar to f (x). Indeed, in [Tsuneda, 2005] a formula was proposed to convert parameter settings of the piecewise-linear model to similar settings for the smooth model. It is now our aim to show how the symbolic image method can contribute to this research topic. We investigate the attractors of both systems at parameter settings where the characteristics of the systems are assumed to be similar. Namely, at α = 15.6, β = 28.58, γ = 0 and k = 1.0. The parameters specific for f (x) are set to m0 = −1.142857 and m1 = −0.7142857, and the ones specific for fˆ(x) are set to a = 0.0659179490 and b = −1.1671315463. In [Tsuneda, 2005], one attractor for the piecewiselinear and one for the smooth case were computed. An investigation with the symbolic image method reveals that there exist not only one but multiple coexisting attractors in both cases. The computation of the chain recurrent set for the piecewise-linear case is illustrated in Fig. 19(a). Three different limit cycles close to each other can be identified. In order to verify these results, the symbolic image was computed for a Poincar´e section along the second coordinate axis. Afterwards, we performed forward iterations using initial values from the outer covering of the computed chain recurrent set in order to get a precise approximation of the three different attractors. The outer covering of the chain recurrent set at the Poincar´e section and the three attractors approximated by forward iterations are shown in Fig. 19. We also calculated the Lyapunov exponents for all attractors in order to verify that each of them is chaotic. Next we approximated the chain recurrent set for the model with the smooth nonlinearity using the same Poincar´e section. As can be seen in Fig. 20, we again detected multiple attractors. However, in contrast to the piecewise-linear case, only two coexisting attractors can be found. Both of them are also chaotic.

investigated system possesses a continuous system function. Because piecewise-smooth systems, which do not fulfill this restriction, have become more and more important in the last years, the question arises, whether the symbolic image based investigation methods can also produce some valuable results in this case. For several examples it turned out that the produced results are correct as well, especially if the borders between specific partitions in the state space coincide with the border of the boxes used in each subdivision step. However, one has to verify the results obtained at the borders of the partitions in the state space as shown in the next example.

9.1. Bernoulli map A good opportunity to test the capabilities of the symbolic image analysis for piecewise-smooth systems is given by the Bernoulli map [Argyris et al., 1994; Hao & Zheng, 1998] defined by: x(n + 1) = fb (x(n)), fb : [0; 1] → [0; 1), fb (x) = rx mod 1.

(69)

For the parameter value r = 2 an infinite number of limit cycles of the Bernoulli map can be determined analytically. It can also be shown, that all these limit cycles are unstable. Especially, the number of unstable limit cycles with any period p can ˆ be easily calculated. Let B(p) be the complete set of fixed points of the pth iterated function of fb : [p] ˆ B(p) = {x|fb (x) = x}.

(70)

ˆ Furthermore, let B(p) ⊂ B(p) be the set of pperiodical points of fb : ˆ ˆ  ), ∀ p < p}. (71) B(p) = {x|x ∈ B(p), x ∈ B(p ˆ Note that although some of the points x ∈ B(p) have the period p, also points with lower periods ˆ includes p < p belong to B(p). More precisely, B(p) p -periodic points for all p which are divisors of p. ˆ (p) the number of points in Let us now denote by N ˆ B(p), and by N (p) the number of points in B(p). Using the binary representation (see for instance [Argyris et al., 1994]) for initial values in the interval [0, 1], the following can be shown: ˆ (p) = 2p − 1, N  ˆ (p) − N (p ) N (p) = N

9. Dynamical Systems with Discontinuous System Function As mentioned in Sec. 2, the theory of symbolic images is developed for the case, that the

3483

(p
)

with

1≤

p

p < p,  ∈ N, N (1) = 1 p

(72) (73)

January 17, 2007

3484

13:28

01693

V. Avrutin et al.

Consequently, the number of unstable limit cycles with period p is given by (1/p)N (p). We remark, that for the sequence N (p) with p = 1, 2, . . . there is no closed formula known. The sequence is denoted as A038199 in the encyclopedia of integer sequences [Sloane, 2004]. For the period ˆ (10) = 210 − p = 10, for instance, we obtain N 1 = 1023 fixed points for the tenth iterated function. This number includes a fixed point, a limit cycle with period two and six limit cycles with period five. The remaining 990 points are 10periodic and form 99 limit cycles (see Table 2). As already mentioned, all these limit cycles are unstable. An inherent problem which eventually occurs by the investigation of piecewise-smooth dynamical systems like the Bernoulli map can be observed for the point x = 1. As one can see, this point is a direct preimage of the unstable fixed point x = 0. However, it turned out, that the point x = 1 behaves similar to a fixed point with respect to the construction of symbolic images. In order to show this, one considers an ε-neighborhood Uε (1) of this point. As one can show, for any arbitrary small ε a point x(ε) ∈ Uε (1) exists in this neighborhood, so that its image lies in the neighborhood as well: fb (x(ε)) ∈ Uε (1). Therefore, the point x = 1 is not distinguishable from a fixed point during the construction of symbolic images. Note, that this is not a phenomenon caused by the finite numerical precision, but exists for any arbitrary high

accuracy. As one can see from Table 2, there are two numerically detected points, which are interpreted as fixed points. The first one, x = 0, is really a fixed point, and the second one is the point x = 1. We apply now symbolic image construction and determine the unstable limit cycles of the Bernoulli map numerically by calculating the neighborhoods of periodic orbits. Table 2 presents the results obtained for periods up to p = 15. Hereby the subdivision depth is 9. The area of the domain space, given by the interval M = [0; 1], was initially subdivided into 20 boxes. For every following subdivision, each box corresponding to a recurrent cell gets divided into eight new ones. Hence, the precision of the calculation is given by ε < (1/20) · (1/89 ) ≈ 4 · 10−10 . Five scan points are defined for every box. As one can see, the results obtained by numerical computation are in exact accordance with the expected values given by Eqs. (72) and (73), except that two fixed points are found due to the reasons explained above. The three limit cycles with period four and the six limit cycles with period five are shown in Fig. 21. An interesting task related to the Bernoulli map is the investigation of its behavior under variation of the parameter r. It follows directly from definition (69), that the points xi = i/r (with i = 1, . . . , r − 1) represent the points of discontinuity of the system function. Using symbolic image analysis we demonstrate, that for r > 1 the behavior of

Table 2. The invariant sets of the Bernoulli map detected by numerical calculation. P denotes a fixed point and L denotes a limit cycle, whereby its period is given by the upper index. For a detailed description see text. p

ˆ (p) N

N (p)

Detected Recurrent Cells in G9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 4 8 16 32 64 128 256 512 1024 2048 4096 8196 16 384 32 768

2 2 6 12 30 54 126 240 504 990 2046 4020 8190 16 254 32 730

L2 , 2 × L3 , 3 × L4 , L2 , 6 × L5 , 9 × L6 , 2 × L3 , L2 , 18 × L7 , 8 30 × L , 3 × L4 , L2 , 56 × L9 , 2 × L3 , 99 × L10 , 6 × L5 , L2 , 186 × L11 , 12 6 4 335 × L , 9 × L , 3 × L , 2 × L3 , L2 , 630 × L13 , 14 1161 × L , 18 × L7 , L2 , 2182 × L15 , 6 × L5 , 2 × L3 ,

2×P 2×P 2×P 2×P 2×P 2×P 2×P 2×P 2×P 2×P 2×P 2×P 2×P 2×P 2×P

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

(a) Fig. 21.

3485

(b)

Bernoulli map: unstable limit cycles with periods (a) four and (b) five at r = 2.

(a)

(b)

Fig. 22. Bernoulli map: unstable limit cycles with periods three (a) and four (b) under variation of r. Red curves mark the discontinuity points of the system function. As one can see, all limit cycles emerge via border collision bifurcations at these discontinuity points. In (b) the points of the limit cycles shown in Fig. 21(a) are marked.

the Bernoulli map is determined by a sequence of border collision bifurcations. As an example, Fig. 22 shows the behavior of unstable limit cycles with periods three and four in dependence on the parameter r. As one can see, all unstable limit cycles of the Bernoulli map emerge at the points, where the system undergoes the border collision bifurcations. Note, that three four-periodic limit cycles presented in Fig. 21(a) emerge at different bifurcation points, namely the first one at r ≈ 1.381, the second one at r ≈ 1.755, and the third one at r ≈ 1.929.

9.2. Piecewise-quadratic map The second example we consider is the piecewisequadratic map, defined by x(n + 1) = fpq (x(n)), fpq : [0; 1] → [0; 1]  1   if x ≤  βx(1 − x) 2 fpq (x) =    1 − αx(1 − x) if x > 1 2

(74)

It is shown in [Gambaudo et al., 1986; Procaccia et al., 1987] that this map represents a special kind

January 17, 2007

3486

13:28

01693

V. Avrutin et al.

of Poincar´e return map for vector fields like the one of the well-known Lorenz system [Lorenz, 1963]. In [Avrutin & Schanz, 2004, 2005b] the bifurcation scenario of system (74) for the case α = β is investigated in detail. This scenario is denoted as border-collision period doubling, and represents a sequence of pairs of bifurcations. The first bifurcation in each pair is a border-collision bifurcation (see for instance [Zhusubaliyev & Mosekilde, 2003] for an up-to-date overview about this topic), and the second one is a pitchfork bifurcation. Similar to the case of the usual period doubling, the periods of the attractors in the case of a border-collision

period doubling scenario form a series pn = p0 · 2n . However, the periods are doubled not at the flip bifurcations, as in the case of the usual period doubling, but at border-collision bifurcations. Remarkably, at the nth border-collision bifurcation two coexisting attractors with periods p0 · 2n , which are symmetric to each other with respect to the point of discontinuity x = 1/2, merge and form a symmetric attractor with period p0 · 2n+1 . The border-collision period doubling scenario investigated by using symbolic images is shown in Fig. 23. As one can see, both unstable limit cycles and pairs of coexisting attractors are detected simultaneously.

(a)

(b)

Fig. 23. Piecewise-quadratic map: results obtained by forward iterations (red) and by symbolic image analysis (blue). The rectangle marked green in (a) is shown enlarged in (b).

(a)

(b)

Fig. 24. Piecewise-quadratic map: (a) analytical results related to limit cycles with period two. (b) Numerical results obtained by forward iterations (red) and symbolic image analysis (green).

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

For the case α = β the behavior of system (74) becomes more complex. It is shown in [Avrutin & Schanz, 2005a, 2005c], that this behavior can be described adequately only in the 2D parameter space α × β. In this parameter space there exist areas leading to attractors with specific periods, which posses a characteristic shape like the one presented in Fig. 24 for the period two. These areas are bounded by the border-collision and saddle-node bifurcation curves and are dominated by cusp catastrophes, as shown in Fig. 24(a). For this reason, in each of these areas there are parts leading to one stable (red), one stable and one unstable (yellow), two stable and one unstable (blue) as well as one unstable (green) limit cycles. Figure 24(b) demonstrates, that all these invariant sets are detected using the symbolic image analysis.

10. Tuning of the Method In order to achieve the most proper results using the symbolic image analysis, the user has to adjust the parameters of the method to the specific properties of the investigated dynamical system. Empirical studies demonstrate, that the quality of the results is often strongly depending on this adjustment, and hence on the experience of the user. Unfortunately, there are only a few rules for this task known so far, and therefore we would like to support the reader and report typical situations occurring in our experiments. Additionally, based on the performed experiments, we found some tuning techniques, which can lead to a more efficient and/or more precise calculation of the symbolic images. In this section we will briefly discuss some of these techniques and, if necessary, also mention some of the implementation aspects and useful heuristics for an efficient usage of the method. The tunings which are presented are motivated by the results of our experience. It has turned out, that one of the most crucial limitations of investigations based on symbolic image analysis is due to a high growth rate of the number of cells during the subdivision process. As already stated in Sec. 5, this growth rate can be exponential but should depend on the dimension size of the localized objects. However, due to the complexity of the dynamics, this is often not the case. Instead, we observed in many scenarios that much more cells are selected for subdivision than those covering the solution. The phenomenon of clustering, which we already mentioned earlier, see Sec. 7.2, is an example of such behavior.

3487

From the theoretical point of view, the selection of too many cells does not matter. By successive application of the subdivision process, the solution will eventually be detected. However, from the practical point of view, one has to deal with limited resources and hence the number of applicable subdivisions is limited. Firstly, by the memory of the computation machine, which only allows the storage of a symbolic image graph of a limited size. Secondly, by the fact that the division of the phase space is limited by a constant Nint , see Sec. 6. For the above mentioned reasons, it is now our strong concern to avoid the selection of cells for subdivision which do not contain a solution. This aim can only be achieved by a change of paradigm. The target is not anymore the rigorous construction of the symbolic image graph for a phase space discretization. We are not interested in providing all existing edges between the cells as requested in the theoretical approach, but rather more only those edges which are necessary for the detection of the solution. The aim of the proposed tunings is to approach this goal. By doing so, we are also aware of the fact that some important information might disappear. However, our empirical studies have shown that computational investigations are mostly limited by performance resources instead of an insufficient approximation of the symbolic image. A reason for this is that the method is typically quite robust. Note, that also the decision to approximate the image of a cell by a finite number of scan points instead of a continuous covering is driven to a large extent by these considerations.

10.1. Basic adjustments There are several basic parameters one has to adjust for the construction of symbolic images. We have already mentioned that there are no general rules for proper adjustments. However, we give here an overview about the results of our heuristic experiences. Area of investigation: The area of investigation, M , should not be chosen too big. If one already knows by other investigations, like forward iteration and analytic results, which areas are of interest, the symbolic image computation should first be focused on them. Then, if some computations were successfully performed, the area M could be adapted. Of course, in order to detect an invariant set, M must cover the area containing this set.

January 17, 2007

3488

13:28

01693

V. Avrutin et al.

Number of initial boxes: As a general rule, the more boxes one chooses, the better the approximation. It is preferable to use as the initial covering not only one box, but a relatively large number of them. Note that the main idea of subdivision is to remove areas of no interest in the phase space. Hence, it does not matter if the initial covering already consists of a large number of boxes. For instance, we usually started with an initial covering of not less than 20 × · · · × 20 boxes. Number of scan points: As for boxes, also for scan points a general rule is, that the more we use the better the approximation. However, a lot of scan points lead to a decrease of the performance. Also, too many scan points might result in a too high growth rate of the number of boxes in a symbolic image during subdivision. So we are usually interested in finding the lower bound of a reasonable setting. A good start proved to be 3d scan points for state space dimension d, one in the center of the box, and the others close to the boundaries. If no or only a few recurrent cells will be detected, then the number of scan points should be increased. If one knows already the position of some invariant sets, one can check if the chosen amount of scan points is sufficient for their detection. If no invariant sets are known, it is a good idea to start with at least 3d scan points. Subdivision of a box: Our experience is, that it is of minor importance how a box will be subdivided. This parameter is mainly of interest in order to decrease the number of subdivision steps of a computation. Integration step size and number of steps: For dynamical systems continuous in time the integration step size ∆t and the number of steps n must be set by the user. As already mentioned, only short trajectories are computed, hence, the precision of the computed trajectory is of minor importance. A setting ∆t = 0.001 in combination with a high number of steps, n = 100, proved in many cases to be more successful than a precise computation. An indicator for a bad setting is the number of boxes which are selected for subdivision in the first subdivision steps. If all boxes get subdivided, then it is likely that no trajectory leaves the box it is started in. Hence, it is advisable to either increase ∆t and n, or the number of boxes in the initial covering. Scans over parameter values: For scans over areas in the dynamical system’s parameter space,

an efficient computation at each position is crucial. So one should tend to restrict the number of subdivisions and the maximal number of cells in a symbolic image appropriately. It might also be a good choice to compute only fixed points and orbits with low periods. As one can see, the adjustment of the parameter settings for symbolic image computation is a complex task, requiring many considerations. Nevertheless, it was our intention to give the user of our implementation a considerably large amount of freedom in choosing appropriate settings. Reason for this is that symbolic image computation is still an open field of research, and it has shown that also unusual changes of the basic parameter settings can lead to surprisingly good results. However, a future goal of research could also be to find appropriate standard settings by analysis of system-specific properties.

10.2. Error tolerance for location of recurrent cells As already stated, the construction of symbolic ˜ images requires the approximation C(I) of the covering C(I), Eqs. (25) and (26). Note, that for our ˜ approach C(I) ⊆ C(I), and so some boxes may not be detected. This behavior can be reduced by usage of a large number of scan points for each box. A disadvantage of this approach is, that the computation time may become inappropriately large in this case. As another solution of this problem, one can extend ˜ the covering C(I) by boxes, which correspond to boxes in its neighborhood. We define a small constant and introduce the extended covering ˜ ∃ x ∈ M (J), C ext (I) = {M (I  )|∃ M (J) ∈ C(I),  (75) ∃ y ∈ M (I ), |x − y| ≤ }. Note that a suitable setting of the constant depends on the size di of the used boxes, and hence on the subdivision level. Therefore, in our implementation the user can define the parameter e so that i = e · di

(76)

for the ith state space component and use the condition |xi − yi | ≤ i instead of |x − y| ≤ . In practice, this parameter was used to detect p-periodic trajectories if they cannot be found otherwise, whereby a setting of e = 0.1 proved to be sufficient in our simulations. One should keep in mind that the use of this parameter usually

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

Fig. 25. Influence of the error tolerance parameter. For details, see text.

increases the size of the symbolic image and, therefore, it should only be applied if necessary. Furthermore, it is often a good alternative to increase the number of scan points S(I) for a more precise calculation instead of applying error tolerance. Some of the scan points should then be placed close to the corners of the boxes. The influence of the error tolerance parameter is illustrated in Fig. 25. The shape in the middle part of the picture represents the image of the box M (I). The gray area is the theoretical covering of this image C(I). As one can see, some boxes ˜ of this covering may be lost in the covering C(I) obtained by numerical calculation (shown red). In order to avoid this, the numerical obtained covering is extended by the area marked by the blue line. Then the resulting covering C ext (I) (shown green) contains the complete theoretical one. Note that this approach was motivated by the technique described in [junge, 2000]. Actually, if is calculated by Lipschitz constants as proposed by ˜ Junge, it can be guaranteed that C(I) ⊂ C(I).

10.3. Use of higher iterated functions When dealing with dynamical systems discrete in time x(n + 1) = f (x(n)), the points y ∈ T˜(I), which represent the images of x, will be calculated as direct successors of the scan points: y = f (x). However, in some cases it is more suitable to use an iterated function of f and calculate the image points by y = f [n] (x),

n>1

(77)

3489

In other words, the symbolic image is constructed in this case not for the function f , but for its nth iterated function f [n] . The practical usage of such a calculation is illustrated by the following example. Let us consider the logistic map (57) in the vicinity of the first flip bifurcation point α = 3. At this point the fixed point x∗ = 1 − (1/α) becomes unstable and a two-periodic limit cycle consisting of the points √ 2 − 2a − 3) emerges. = (1/2) + (1/2α)(1 ± a x∗∗ 1,2 We assume, that these points should be found using symbolic image analysis. For this reason the interval M = [0.58; 0.74] is divided initially into 50 boxes, further the boxes are subdivided into four new ones. In each box five scan points are selected. The results of the calculation of symbolic images for the parameter range α = [2.98, 3.03] are presented in Fig. 26(a). As one can see, the points x∗ and x∗∗ 1,2 are found, but in the vicinity of the first flip bifurcation point α = 3 a large number of boxes are detected as recurrent, although it is obvious from the analytical point of view, that these boxes correspond to the transient dynamics. This behavior is known as critical slowing down behavior [Goldenfeld, 1992] and leads to the blurring of the bifurcation diagram close to the point α = 3. In order to avoid the numerical errors mentioned above, we have to consider the influence of the critical slowing down behavior on the calculation of symbolic images. It is well known, that in the vicinity of a bifurcation point the number of iterations to reach the asymptotic dynamics with a given accuracy grows drastically. In Fig. 26(b) this number is shown in a logarithmic representation for the accuracy ε < 10−8 . Taking into account the dynamic behavior described above, the blurring of the bifurcation diagram presented in Fig. 26(a) can be explained easily. The orbits started in boxes, which lie close to fixed points (stable or unstable) do not leave these boxes within a single iteration step. Therefore, the corresponding cells of the symbolic image are marked as one-periodic. In an analogous way, the orbits started in the boxes close to the two-periodic limit cycle returns into these boxes after two iteration steps, and the corresponding cells are detected as two-periodic. In principle, the problem that an orbit do not leave a box where it was started can take place at any parameter value. However, in the vicinity of the bifurcation point due to the critical slowing down phenomenon the area in the state space, where this problem occurs, becomes large.

January 17, 2007

3490

13:28

01693

V. Avrutin et al.

(a)

(b)

(c) Fig. 26. Reduction of the critical slowing down phenomenon by calculation of the bifurcation diagram for the logistic map in the vicinity of the bifurcation point α = 3. (a) Analytical and numerical results using the function fl . (b) Number of iterations [51] which the system needs in order to reach the asymptotic dynamics. (c) Numerical results using the 51th iterated function fl .

As one can see, there are two possibilities to improve the quality of the calculation. The first one could be to take a covering consisting of smaller boxes than the ones used here. However, in this case the number of boxes grows and the obtained quality improvement is not essential. As a second possibility, we can take into account some additional information about the dynamics. Therefore let us first consider the fact that a fixed point of the function f is obviously a fixed point of the iterated function f [n] for any n as well. Secondly, for any even number n a two-periodic limit cycle of the function f corresponds to a fixed point of the iterated

function f [n]. For odd numbers n, a two-periodic limit cycle of the function f represents a limit cycle with the same period of the iterated function f [n]. Therefore, the bifurcation diagrams of the function f and of the higher iterated f [n] for any odd n are identical. On the other hand, the trajectories, which do not leave a box within a single iteration step, may do it after some sufficient large number of steps. Therefore, we can use the iterated function f [n] with a sufficient large odd n instead of the function f , in order the avoid the critical slowing down behavior which leads to the blurring of the bifurcation diagram. The results for this case are shown

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

in Fig. 26(c), where the 51th iterated function of the logistic map is used. As one can see, the critical slowing down behavior could not be avoided at all, but becomes almost not observable.

10.4. Discretization time for systems continuous in time For dynamical systems continuous in time, the integration of the function must be considered in a different context than the iteration of discrete systems. The number of integration steps n, which determine (for a fixed integration step size ∆t) the discretization time ts is an essential part of the parameter setting. By variation of ts one changes the length of the trajectory between a point x ∈ M (I) in a box M (I) and its image y ∈ T˜(I). In an ideal setting, the time ts should always be chosen in such a way that the image y for a point x ∈ M (I) lies in the next neighboring box M (I  ) to which the trajectory started at x moves to. It is also possible, that such an y does not exist, that means the trajectory x(t) will never leave its initial box. This situation occurs, if the box contains an invariant set, whereby it can be assumed for the box size shrinking to zero, that this invariant set is a fixed point. Unfortunately, empirical experience showed that the described approach may fail for many practical applications. Even for simple dynamical systems like the Van der Pol system (Sec. 8.1) the symbolic image grows too much in each subdivision step because too many cells will be detected as recurrent. A possible solution in this case may be to enlarge the discretization time ts using a corresponding large number n of iteration steps. Due to the following of the trajectories for a longer run, more cells can be detected as nonrecurrent. This is in most cases a necessity for reasonable numerical simulation. Note that the detection of stable fixed points and limit cycles is not affected by a high setting of ts . However, unstable points and cycles may not be detected because the trajectories started close to them diverge. So it is after all still essential that the symbolic image is constructed by the combination of many short trajectories instead of a few long ones. Otherwise the distinctive features of this investigation method cannot be used. This means that ts must be set to a value so that the symbolic image does not grow too big, but that also the information about unstable cycles and fixed points will persist. It is not guaranteed, that such a setting exists for every system in focus, and if it

3491

exists, there is not yet a general rule how to derive it. Only user experience and heuristic testing can lead to the most proper setting of ts . For the dynamical systems we tested, settings in the range ts ∈ [0.1, 0.2] turned out to be a good choice. The trajectories are then long enough to detect and exclude many nonrecurrent cells at an early stage. Nevertheless, the symbolic image can be kept small in the subdivision process. On the other hand, unstable cycles are still recognized, as can be seen for the Lorenz system in Sec. 8.2.

10.5. Poincar´ e sections for systems continuous in time We introduced stroboscopic mappings as the general approach to apply the symbolic image method to systems continuous in time. A disadvantage of this approach is that it can require high time and memory resources. In order to improve performance, the use of a Poincar´e section can be considered. If the system flow is transversal to a plane which is parallel to one of the coordinate axes, a Poincar´e section can be simulated with the methods we already introduced. Note that this approach is related to the use of local Poincar´e sections as proposed in [Mischaikow, 2002]. Let us consider a Poincar´e section defined by a plane P which is transversal to the flow of the investigated system described by f and parallel to an axis k in the phase space. Consequently, P = {(. . . , xk , . . .)|xk = r} for some r ∈ R and the state space coordinate k of a n-dimensional system. This Poincar´e section can be simulated by restricting the area of investigation, see Eq. (11), to min max , Mk−1 ] × [r − ε, r + ε] × · · · , M p = · · · × [Mk−1

for a small ε and setting the number of subdivisions = 1. in boxes along the kth coordinate to imax k Obviously, the area of investigation M p describes an = 1, outer covering of an area on P . Due to imax k the kth axis is not affected by any subdivision of M p and the area of investigation is effectively reduced to dimension (n − 1). Note, however, that the accuracy of the computation depends on the parameter ε. Besides the restrictions of the area of investigation, some further modifications of the symbolic image construction must be considered. Reason is that M p only covers a part of the system’s flow. Trajectories leave the area M p and later return to it. Hence, it is necessary to follow the trajectory started at a scan point x ∈ M (I) ⊆ M p until it hits a point y ∈ M p after leaving the box M (I). Then y

January 17, 2007

3492

13:28

01693

V. Avrutin et al.

In the former sections we have discussed the usage of large discretization times for symbolic image

calculation. In many calculations this option turned out to be an adequate technique to tune the investigation method. But it was also mentioned that unstable limit cycles might not be detected if the time ts is chosen too large. In practice, we observed, that for crucial settings ts the limit cycles do not completely disappear at once, but rather more fall apart. Some parts of them are still recognized while others vanish, as it can be seen in Figs. 27(a)–27(c) for the Lorenz system. Such a behavior is a result of applying a high number of integration steps n to the function and therefore following a relatively long run of trajectories. This leads to the loss of information about the structure of trajectories representing unstable limit cycles. It is not our intention to give here a detailed analysis of this problem, but rather more a solution for the reconstruction of such unstable cycles. However, one should keep in mind that not all of these disappearing cycle structures are necessarily fragments of limit cycles. In some cases it turned out that they represent errors caused by noncyclic orbits. So further tests have to be applied to approve the correctness of obtained results. The reconstruction of the limit cycles can be done by application of an extension to the symbolic image construction algorithm. The basic idea here is to add and/or mark as recurrent all cells belonging to boxes M (I) of the symbolic image which will be passed by the trajectory x(t) on its way from x to its image y. Therefore, first the symbolic image G will be constructed according to the standard

(a) seventh subdivision step

(b) eighth subdivision step

is considered as the image of x on the Poincar´e section. In order to implement the concept, a stroboscopic mapping is used with a integration step size ∆t and a maximum number of integration steps n. For each integration step i = 1, . . . , n it is checked if for y = f [i](x), x ∈ M (I) the following condition is fulfilled: y ∈ Mp

and ∃ j ≤ i : f [j](x) ∈ / M (I).

In case the condition is fulfilled for an i, or f [n] (x) ∈ M (I), then f [i](x) or, respectively, f [n] (x) is the image of x on the Poincar´e section. Note, however, that ∆t and n must be set appropriately so that all trajectories return to M p . The method described allows the computation of the symbolic image for a Poincar´e section. It was successfully applied in practice, as illustrated in Secs. 8.3 and 8.4. Due to the reduction of the area of investigation to a dimension size (n − 1), the performance can be significantly improved. Besides the advantages of this tuning, one must also consider the limitations of it. Firstly, a plane P is required as Poincar´e section and, secondly, the parameters ε, ∆t and n must be set appropriately which is not always possible. Furthermore, the interpretation of the symbolic image and the approval of the correctness of the computed results can be more difficult.

10.6. Reconstruction of fragmented limit cycles

Fig. 27. Lorenz system: reconstruction of unstable limit cycles at parameter r1 = 14.6 with a large discretization time. The fragmented limit cycles will be completed using the technique described in Sec. 10.6.

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

(c) tenth subdivision step

3493

(d) completion of the fragmented limit cycle Fig. 27.

approach. Then the recurrent cells RV (G) will be located. Afterward the following extension must be applied before the next subdivision. For every recurrent cell cI ∈ RV (G) its corresponding box M (I) is detected. Then, for every scan point x ∈ S(I) it has to be checked, whether its target point y ∈ T˜(I) lies in a recurrent cell which is equivalent, i.e. belongs to the same set of strongly connected components. If so, we locate the boxes M (I  ) lying along the trajectory between x and y. If a corresponding cell cI
does not exist for a visited area, it will be added to the symbolic image. Furthermore, the cell cI
will be marked as recurrent. If this extension is applied, the course of a trajectory, which connects recurrent cells, will be reconstructed. Note that the symbolic image can only become more precise by this extension. If a ˜ source cell M (I) and a target cell M (I  ) ∈ C(I) are recurrent, then, consequently, all the cells which are passed by the connecting trajectory are also recurrent. As already mentioned, this operation might add new cells to the symbolic image. So it can still be applied in a stage of subdivision when the limit cycle has already fallen apart to a large extent. This can be seen in Fig. 27, where unstable limit cycles of the Lorenz system will be reconstructed in the tenth subdivision step. Note that the described reconstruction procedure was applied for all final limit cycles presented in Sec. 8.2.

11. Summary In this work, we discussed the construction process for symbolic images as well as the localization of the

(Continued )

sets H1 , H2 , . . . containing equivalent recurrent vertices and the detection of periodic orbits. The basic steps of the implementation are described in detail, whereby the main objectives were efficient algorithms and adequate data structures. Furthermore, we introduced some basic investigation methods for the symbolic image graph. Then the performance of the used algorithms was analyzed. It was shown that almost all steps of the calculation can be performed in the time O(ns · log(ns )) with respect to the number of boxes ns . Only if the length of the periodic paths on the symbolic image must be calculated, the complexity of this task is O(n2s ). The number ns itself grows, in worst case, exponentially during subdivision, or, in other words, with increasing accuracy of the calculation. However, this exponential growing of the number ns can be significantly reduced using some heuristics we described in detail. We also presented examples of the practical application of symbolic image calculation. For dynamical systems discrete and continuous in time we showed some typical fields of application, like the detection of the chain recurrent set or the localization of periodic orbits, without any restrictions with respect to their stability. The comparison with analytic results and observations known so far, allowed us to verify most of our numerical calculations and showed, that the presented investigation method can be successfully applied in practice. Furthermore, we also encountered the limits of the investigation method’s applicability. It turned out, that the crucial factor of numerical

January 17, 2007

3494

13:28

01693

V. Avrutin et al.

computation is the consumption of computer memory. In worst case we have an exponential growth rate for the number of cells per subdivision step. Hence, an early detection and elimination of nonrecurrent cells is essential. Also the precision of our computation mainly depends on the number of cells in the symbolic image. In some cases the performance time becomes a crucial factor. This might happen if one detects periodic orbits with relatively high periods. Other problems we have to deal with in practice are related to the fact, that we often get more than one recurrent cell set gathered around a chain recurrent point, which we called clustering. This is caused by the slow convergence or divergence of orbits in the vicinity of invariant sets. We have demonstrated by an example how this problem can be treated. Further problems may occur when dealing with systems continuous in time. Although we have discussed some ways how these problems can be reduced, still a lot of work must be done in this field. Additionally, we have seen that an efficient calculation of symbolic images requires some experience. Reason for this is that the method possesses a large number of parameters, which must be set depending on specific properties of the dynamical system in focus and on the specific investigation task. Currently, there are no general rules concerning the parameter settings leading to the most precise results. Therefore, another aim of this work was to present some useful heuristics and to introduce advanced techniques for the tuning of the method.

12. Outlook As mentioned before, an important intention of this paper was to introduce a basic framework for other, more sophisticated investigation techniques. It is, for example, not only possible to locate invariant sets but also to find basins of attraction and filtrations. The calculation of the Morse Spectrum is also a topic which is worth mentioning. This spectrum is closely related to the Lyapunov exponents and can be located by the use of symbolic images. So we remark that the construction of a symbolic image opens the door to the application of several new methods for the investigation of dynamical systems. Quite a lot of information can be gathered by its application, and there might be even some more techniques, yet undiscovered, which could be built around symbolic images in future.

References Aho, A., Hopcroft, J. & Ullman, J. [1987] Data Structures and Algorithms (Addison-Wesley). Alefeld, G. & Herzberger, J. [1983] Introduction to Interval Computations (Academic Press). Alekseev, V. [1976] Symbolic Dynamics, 11th Mathematical School (Kiev) (in Russian). Anosov, D. [1967] Geodesic Flow on Closed Riemannian Manifold of Negative Curvature, Trudy Mathematical Steclov Institute (in Russian). Argyris, J., Faust, G. & Haase, M. [1994] An Exploration of Chaos: An Introduction for Natural Scientists and Engineers (Elsevier Science). Avrutin, V., Lammert, R., Schanz, M., Wackenhut, G. & Osipenko, G. [2003] “On the software package AnT 4.669 for the investigation of dynamical systems,” in Fourth Int. Conf. Tools for Mathematical Modelling, Vol. 9, ed. Osipenko, G. (St. Petersburg State Polytechnic University, Russia), pp. 24–35. Avrutin, V. & Schanz, M. [2004] “Border-collision period-doubling scenario,” Phys. Rev. E 70, 026222/ 1–11. Avrutin, V. & Schanz, M. [2005a] “Border-collision induced bifurcation phenomena and multi-parametric bifurcations in a piecewise-quadratic map on interval,” in Proc. Fifth EUROMECH Nonlinear Dynamics Conf. ENOC’2005, accepted for publication. Avrutin, V. & Schanz, M. [2005b] “Period doubling scenario without flip bifurcations in a one-dimensional map,” Int J. Bifurcation and Chaos 15, 1267–1284. Avrutin, V. & Schanz, M. [2005c] “Self-similar structures in 2D parameter space and the concept of big bang bifurcations,” submitted to Physica D. Bathia, N. & Szego, G. [1970] Stability Theory of Dynamical Systems (Springer, NY). Bowen, R. [1982] “Symbolic dynamics,” Ann. Math. Soc. 8. Broer, H. & Simo, C. [1998] “Hill’s equation with quasiperiodic forcing: Resonance tongues, instability pockets and global phenomena,” Bul. Soc. Bras. Mat. 29, 253–293. Chua, L. O. [1992] “The genesis of Chua’s circuit,” ¨ Archiv f¨ ur Elektronik und Ubertragungstechnik 46, 250–257. Chua, L. O. [1994] “Chua’s circuit: An overview ten years later,” J. Circuits Syst. Comput. 4, 117–159. Conley, C. [1978] “Isolated invariant set and the morse index,” CBMS Regional Conf. Series 38. Coomes, B., Kocak, H. & Palmer, K. [1994] “Periodic shadowing, chaotic numerics,” Contemp. Math. 172. Coomes, B., Kocak, H. & Palmer, K. [1997] “Computation of long period orbits,” Aust. Math. Soc. Gaz. 24, 183–190. Cvitanovi´c, P. [1991] “Periodic orbits as the skeleton of classical and quantum chaos,” Physica D 51.

January 17, 2007

13:28

01693

Investigation of Dynamical Systems Using Symbolic Images

Cvitanovic, P. [1992] “Focus issue on periodic orbit theory,” Chaos 2. Dellnitz, M. & Hohmann, A. [1996] “The computation of unstable manifolds using subdivision and continuation,” in Nonlinear Dynamical Systems and Chaos, Vol. 19, eds. Broer, H., van Gils, S., Hoveijn, I. & Takens, F. (Birkh¨auser), pp. 449–459. Dellnitz, M. & Hohmann, A. [1997] “A subdivision algorithm for the computation of unstable manifolds and global attractors,” Numer. Math. 75, 293–317. Dellnitz, M. & Junge, O. [1998] “An adaptive subdivision technique for the approximation of attractors and invariant measures,” Comput. Vis. Sci. 1, 63–68. Dellnitz, M., Froyland, G. & Junge, O. [2001] “The algorithms behind GAIO — Set oriented numerical methods for dynamical systems,” in Ergodic Theories, Analysis, and Efficient Simulation of Dynamical Systems, ed. Fiedler, B. (Springer), pp. 145–175. Dellnitz, M. & Junge, O. [2002] “Set oriented numerical methods for dynamical systems,” in Handbook of Dynamical Systems II: Towards Applications, eds. Fiedler, G. I. B. & Kopell, N. (World Scientific, Singapore), pp. 221–264. Diamond, P., Kloeden, P. & Pokrovskii, A. [1995] “Cycles of spatial discretizations of shadowing dynamical systems,” Math. Nachr. 171, 95–110. Dijkstra, W. [1959] “A note on two problems in connection with graphs,” Numer. Math. 1, 269–271. Eidenschink, M. [1995] Exploring Global Dynamics: A Numerical Algorithm Based on the Conley Index Theory, PhD thesis, Georgia Institute of Technology. England, J., Krauskopf, B. & Osinga, H. [2004] “Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse,” SIAM J. Appl. Dyn. Syst. 3, 161–190. England, J., Krauskopf, B. & Osinga, H. [2005] “Computing one-dimensional stable manifolds and stable sets of planar maps without the inverse,” SIAM J. Appl. Dyn. Syst. 3, 161–190. Feigenbaum, M. J. [1978] “Quantitative universality for a class of nonlinear transformations,” J. Stat. Phys. 19, 25–53. Feigenbaum, M. J. [1979] “The universal metric properties of nonlinear transformations,” J. Stat. Phys. 21, 669–707. Floyd, R. W. [1962] “Algorithm 97 (shortest path),” Commun. ACM 5, 345. Fundinger, D. [2006] Investigating Dynamics by Multilevel Phase Space Discretization, Unpublished doctoral dissertation, University of Stuttgart. Gambaudo, J.-M., Procaccia, I., Thomae, S. & Tresser, C. [1986] “New universal scenarios for the onset of chaos in Lorenz-type flows,” Phys. Rev. Lett. 57, 925–928. Goldenfeld, N. [1992] Lectures on Phase Transitions and the Renormalization Group (Perseus Books).

3495

Golub, G. & van Loan, C. [1999] Matrix Computations (Mir, Moscow). Grossmann, S. & Thomae, S. [1977] “Invariant distributions and stationary correlation functions of onedimentional discrete processes,” Z. Naturforsch. 32. Guckenheimer, J. & Holmes, P. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, NY). Gyllenberg, M., S¨ oderbacka, G. & Ericsson, S. [1993] “Does migration stabilize local population? Analysis of a discrete metapopulation model,” Math. Biosci. 118, 25–49. Hao, B. & Zheng, W. [1998] Applied Symbolic Dynamics and Chaos (World Scientific, Singapore). Home Page of the AnT4.669 Project [2005] (Available at http://www.AnT4669.de). Hruska, S. [2002] On the Numerical Construction of Hyperbolic Structures for Complex Dynamical Systems, Unpublished doctoral dissertation, Cornell University. Hruska, S. [2005] “Constructing an expanding metric for dynamical systems in one complex variable,” Nonlinearity 18, 81–100. Hsu, C. [1980] “A theory of cell-to-cell mapping dynamical systems,” J. Appl. Mech. 47, 931–939. Hsu, C. [1987] Cell-to-Cell Mappings (Springer, NY). Hunt, F. [1998] “Unique ergodicity and the approximation of attractors and their invariant measures using Ulam’s method,” Nonlinearity 11, 307–317. Ikeda, K. [1979] “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. Commun. 30, 257–261. Junge, O. [1999] Mengenorientierte Methoden zur numerischen Analyse Dynamischer Systeme, PhD thesis, University of Paderbonn. Junge, O. [2000] “Rigorous discretization of subdivision techniques,” Proc. Equadiff ’99, Berlin. Junge, O. [2003] “Computing specific isolating neighborhoods,” Progress in Analysis, Vols. I, II (World Scientific, Singapore). Kaneko, K. (ed.) [1993] Theory and Applications of Coupled Map Lattices (Wiley, NY). Kobjakov, S., Matiassevitch, D. & Osipenko, G. [2003] “Location of the invariant set,” in Fourth Int. Con. Tools for Mathematical Modelling, Vol. 9, ed. Osipenko, G. (St. Petersburg State Polytechnic University, Russia), pp. 300–307. Krauskopf, B. & Osinga, H. [1998a] “Globalizing twodimensional unstable manifolds of maps,” Int. J. Bifurcation and Chaos 8, 483–503. Krauskopf, B. & Osinga, H. [1998b] “Growing 1d and quasi-2d unstable manifolds of maps,” J. Comput. Phys. 146, 406–419. Krauskopf, B., Osinga, H., Doedel, E., Henderson, M., Guckenheimer, J., Vladimirsky, A. et al. [2005] “A survey of methods for computing (un)stable manifolds

January 17, 2007

3496

13:28

01693

V. Avrutin et al.

of vector fields,” Int. J. Bifurcation and Chaos 15, 763–791. Lind, D. & Marcus, B. [1995] An Introduction to Symbolic Dynamics and Coding (NY). Lindstr¨om, T. [2002] “On the dynamics of discrete food chains: Low- and high-frequency behavior and optimality of chaos,” J. Math. Biol. 45, 396–418. Lorenz, E. N. [1963] “Deterministic non-periodic flows,” J. Atmos. Sci. 20, 130. Malinetskii, G. & Potapov, A. [2000] Modern Problems in Nonlinear Dynamics (Moscow) (in Russian). Miller, J. & Yorke, J. [2000] “Finding all periodic orbits of maps using Newton methods: Sizes of basins,” Physica D 135, 195–211. Mischaikow, K. [2002] “Topological techniques for efficient rigorous computations in dynamics,” Acta Numerica. Nitecki, Z. & Shub, M. [1975] “Filtrations, decompositions, and explosions,” Amer. J. Math. 97, 1029–1047. Nusse, H. & Yorke, J. [1997] Dynamics: Numerical Explorations (Springer-Verlag). Osipenko, G. [1983] “On a symbolic image of dynamical system,” Interuniv. Collect. Sci. Works (in Russian), pp. 101–105. Osipenko, G. [1993] “The periodic points and symbolic dynamics,” Prog. Nonlin. Diff. Eq. Appl. 12, 261–267. Osipenko, G. [1994] “Localization of the chain recurrent set by symbolic dynamics methods,” Proc. Dynamics Systems and Applications, Vol. 1 (Dynamic Publishers Inc.), pp. 227–282. Osipenko, G. [1997] “Morse spectrum of dynamical systems and symbolic dynamics,” Proc. 15th IMACS World Congress 1, pp. 25–30. Osipenko, G. [1999] “Construction of attractors and filtrations,” Banach Center Publ. 47, 173–192. Osipenko, G. & Campbell, S. [1999] “Applied symbolic dynamics: Attractors and filtrations,” Discr. Contin. Dyn. Syst. 5, 43–60. Osipenko, G. [2000] “Spectrum of a dynamical system and applied symbolic dynamics,” J. Math. Anal. Appl. 252, 587–616. Osipenko, G. [2004] Lectures on Symbolic Analysis of Dynamical Systems (St. Petersburg State Polytechnic University).

Osipenko, G., Romanovsky, J., Ampilova, N. & Petrenko, E. [2004] “Computation of the morse spectrum,” J. Math. Sci. 120, 1155–1166. Pilarczyk, P. [1999] “Computer assisted method for proving existence of periodic orbits,” TMNA 13, 365–377. Procaccia, I., Thomae, S. & Tresser, C. [1987] “Firstreturn maps as a unified renormalization scheme for dynamical systems,” Phys. Rev. A 35, 1884–1900. Rivest, T. C. C. L. R. [2000] Introduction to Algorithms (MIT Press). Rosenzweig, M. & MacArthur, R. [1963] “Graphical representation and stability conditions of predator–prey interactions,” Am. Nat. 97, 209–223. Rosenzweig, M. [1973] “Exploitation in three trophic levels,” Am. Nat. 107, 275–294. Sedgewick, R. [1993] Algorithms in Modula 3 (AddisonWesley, MA). Sim´o, C. [1989] “On the analytical and numerical approximation of invariant manifolds,” in Les M´ethodes Modernes de la M´ecanique C´eleste, eds. Benest, D. & Froeschl´e, C. (Goutelas), pp. 285–330. Sloane, N. J. A. [2004] The On-Line Encyclopedia of Integer Sequences, www.research.att.com/njas/ sequences/ Sparrow, C. [1973] The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (Springer, NY). Tarjan, R. [1972] “Depth-first search and linear graph algorithms,” SIAM J. Comput. 1, 146–160. Tarjan, R. [1991] Data Structures and Network Algorithms (Philadelphia, PA). Tsuneda, A. [2005] “A galley of attractors from smooth Chua’s equation,” Int. J. Bifurcation and Chaos 15, 1–49. Walters, P. (ed.) [1991] Symbolic Dynamics and Its Applications. You, Z., Kostelich, E. & Yorke, J. [1991] “Calculating stable and unstable manifolds,” Int. J. Bifurcation and Chaos 1, 605–623. Zhusubaliyev, Z. & Mosekilde, E. [2003] Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems, World Scientific Series in Nonlinear Science, Series A, Vol. 44 (World Scientific, Singapore).