on some properties of invariant sets of two

Tutorials and Reviews International Journal of Bifurcation and Chaos, VoI. 7, No.6 (1997) 1167-1194 © World Scientific Publishing Company

ON SOME PROPERTIES OF INVARIANT SETS OF TWO-DIMENSIONAL NONINVERTIBLE MAPS CHRISTOS E. FROUZAKIS

1. C. Engines and Combustion Laboratory,

JET, ETH-Zurich, CH-8092, Zurich, Switzerland

LAURA GARDINI

Facoltd di Economia, Universita di Brescia e Universitd di Urbino, Jtaly

IOANNIS G. KEVREKIDIS Department oj Chemical Engineering, Princeton University, Princeton, NJ 08544 GILLES MILLERIOUX and CHRISTIAN MIRA JNSA-DGE. Complexe Scientifique de Rangueil, 31077 Toulouse, France Received June 11, 1996; Revised October 15, 1996 We study the nature and dependence on parameters of certain invariant sets of noninvertible maps of the piane. The invariant sets we consider are unstable manifolds of saddle-type fixed and periodic points, as well as attracting invariant circles. Since for such maps a point may have more than one first-rank preimages, the geometry, transitions, and generai properties of these sets are more complicated than the corresponding sets for diffeomorphisms. The criticai curve(s) (locus of points having at least two coincident preimages) as well as its antecedent(s), the curve(s) where the map is singular (or "curve of merging preimages") play a fundamental role in such studies. We focus on phenomena arising from the interaction of one-dimensional invariant sets with these criticai curves, and present some illustrative examples.

1. Introduction

The purpose of this paper is to present and explain phenomena associated with the geometry and de­ pendence on parameters of certain one-dimensional invariant sets of noninvertible maps of the pIane (two-dimensional endomorphisms). The sets we ex­ ami ne are attracting invariant closed curves, as well as (global extensions of local) unstable manifolds of saddle-type fixed and periodic points. The class of two-dimensional noninvertible maps, T : R 2 --; R 2 , considered here, possess one or more Critical Curve(s) LC (from Ligne Critique) , which sepa­ rate the pIane locally into distinct adjacent regio ns. Phase points X in these regions possess different numbers ofrank-one preimages (or antecedents) Y : T(Y) = X, (Y = T- 1 (X)). The curves LC, which

may be made up of several segments, constitute the geometrie locus of points X having at least two co­ incident preimages, T- 1 (X). These coincident or double preimages are in turn located on a curve (which again may be made up of several segments) LC- 1 called the curve oj merging preimages. The curve LC, with T- 1 (LC) :2 LC-l and T(LC_ 1 ) = LC , is the two-dimensional generalization of the notion of criticaI point (in the sense of Julia-Fatou) of a one-dimensional endomorphism. When the map T is continuously differentiable , LC_ 1 is included in the set defined by IJ(X)I = O, where IJI denotes the Jacobian determinant of T. Because of this relation with the Jacobian of the map, previous literature has used the terms Jo and J 1 for the LC_ 1 and LC curves, respectively. 1167

1168

C. E. Frouzakis et al.

It is convenient to classify noninvertible maps according to the number of solutions of the inverse map for all possible points X E R 2 , and the rela­ tive arrangement of regions with different numbers of preimages. A region of the plane is labeled "a Zk region" when its points X possess k distinct preim­ ages. Zo regions are, as we will see , particularly important in determining certain unique features of endomorphisms. They consist of points that have no preimages (and thus have been given the pic­ turesque name "Gardens of Eden" for their lack of "previous history" in the cellular automata litera­ ture [Amoroso, 1970; Skyum, 1975]). Obviously, neither an attractor, nor an unstable manifold of a saddle can penetrate such a Zo region. We will present a number of dynamic phenom­ ena characteristic of noninvertible maps of the plane and illustrate them through a number of specific ex­ amples. We will first examine the so-called Zo - Z2 class. In this case , LC separates R2 into two open regions Zo and Z2; a point X E Z2 has two distinct preimages of rank one, and a point X E Zo has no preimages. This, in some sense, constitutes the simplest class of planar endomorphisms. We will then consider the so-called Zl < Z3 classi here the plane contains a wedge-shaped region of points with three preimages, separated from the complement , each point of which possesses a unique preimage, by a critical curve that contains a cusp point. The curve LC- 1 is the two-dimensional exten­ sion of the local extrema of a differentiable one­ dimensional map. Non-differentiable, piecewise linear maps like the tent map, have been used in the literature of one-dimensional endomorphisms to facilitate the analysis of their dynamics. In such cases, the points of non-differentiability where the slope of the map changes sign (like the apex of the tent map) play the role of the extremum of the smooth logistic map. In two dimensions, if T is not differentiable , the role of LC_ 1 will then, in gen­ eral, be played by the curve of non-differentiability, where the Jacobian determinant of the m!J,p changes sign [Gurnowski & Mira, 1977, 1980a]. To our knowledge, the notion of crìtica.l curves in the study of two-dimensional endomdrphisms was introduced in 1964 in relation to its role in the determination ofbasin boundaries [Gumowski & Mira, 1965 ; Mira, 1964]. Since 1969 several papers studied the role of critical curves in transitions of the type simply con­ nected basin ;...-+ non-connected basin [Gumowski & Mira, 1980, 1980a; Mira & Roubellat, 1969; Mira et al., 1994] and simply connected basin ;...-+ multiply

connected basin [Barugola et al., 1986; Bernussou et al., 1976; Cathala, 1989]. These (qualitatively re­ lated) codimension-one transitions result from the interaction of a basin boundary with a critical curve segmento Critical curves are essential in the de fini­ tion of absorbing areas and chaotic areas and their dependence on map parameters [Barugola, 1980, 1984; Barugola & Cathala, 1985, 1992; Cathala, 1983, 1987, 1990; Gumowski & Mira, 1977, 1978, 1980a; Gardini, 1991a, 1991b, 1992; Kawakami & Kobayashi, 1979; Mira, 1990; Mira & Narayanin­ samy, 1993]. Global bifurcations, including homoclinic and heteroclinic interactions of invariant sets of fixed points (see below for the definition of invariant sets, the extension of stable and unstable manifolds of fixed or periodic points to endomorphisms), are also crucially affected by the existence of critical curves, and often differ fundamentally from the correspond­ ing invertible global bifurcations. The "seeds" of these important differences were Introd uced and discussed in the phenomenology of one-dimen­ sional endomorphisms [Sharkovskij, 1969] (see also [Gumowski & Mira, 1978; Mira, 1987], mentioning higher-dimensional such phenomena in an embry­ onic form). Gardini has recently studied, in a series of papers [Gardini, 1991a, 1991b, 1992, 1993, 1994; Gardini et al., 1992a, 1992b, 1994], global bifur­ cations and invariant manifold interactions for the noninvertible case, in contrast to the corresponding invertible phenomena. During the last few years the study of two- (and higher) dimensional noninvertible maps is becoming a subject of increasingly wider interest and research, and some of the results mentioned were sporadi­ cally rediscovered by other authors. A forthcoming monograph [Mira et al., 1996a] will systematically address these (and related) phenomena. In this paper we begin a systematic attempt to characterize the interaction of critical curves (and their iterates and preiterates) with one-dimensional invariant sets of planar endomorphisms. We will demonstrate that this interaction is essential in the understanding of phenomena 1ike the formation of self intersections of the global unstable set of a saddle-type fixed or periodic point. It will also ex­ plain certain properties of stable invariant closed curves; these include more "global" phase space features, like the possibility of obtaining periodic points (of the same periodic orbit) located on both sides of an invariant closed curve. More than fif­ teen years ago, such phase portraits , "pathological"

Properties of Planar Endomorphisms 1169

in the sense that they are not encountered in pla­ nar diffeomorphisms, were observed in a Zo - Z2 quadratic map (cf. [Gumowski & Mira, 1980, 1980a]), and in two Zl - Z3 - Zl maps (cf. [Gumowski & Mira, 1980a; Maribe, 1982]) . An interesting noninvertible scenario of the breakup of stable invariant circles was observed later by Lorenz [1989], who described the transition of a smooth in­ variant closed curve to a chaotic attractor by cre­ ation of local loops. In the same spirit Frouzakis [1992] discusses the formation of self-intersecting loops of the unstable set of a saddle fixed point in a model of an adaptively controlled system (in the form of a two-dimensional noninvertible map) . In addition to the mentioned adaptive control application (also cf. [Adomaitis & Kevrekidis, 1991 ; Frouzakis et al., 1996a]), many systems in engineer­ ing, particularly in control theory and electronics, lead to models in the formof noninvertible maps . This is particularly the case in systems using either sampled data, switching elements, or pulse modula­ tion [Gumowski & Mira, 1980a; Mira & Roubellat, 1969; Mira, 1990], and in signai processing [Dinar, 1994; GicqueI, 1995; Rico-Martinez et al., 1995]. Moreover, modeling in economics and biology of­ ten gives rise to noninvertible maps [Beddington et al., 1975; Delli Gatti et al., 1995; Gardini, 1993; Guckenheimer et al., 1977]. The interest in better understanding the dynamics of higher-dimensionai endomorphisms is therefore not only motivated by theory, but aiso by a wide spectrum of practicai applications. The paper is organized as follows: Section 2 presents some properties of criticaI curves. Sections 3-6 deai with basic types of interactions between the unstable set of a saddle fixed point with criticaI curves; Sections 7 and 8 describe certain noninvertible features involving attracting invariant circles.

2. Some Definitions: CriticaI Sets, Preimages and Foiding of the Phase PIane In this paper we describe phenomena involving rank-one preimages of phase points for a planar en­ domorphism T and their relation to its criticaI curve LC. These phenomena can be extended to the mth iterate T m of the endomorphism, involving rank m phase point preimages; they will be then related to the critical set, EC(T m ), a generalization of the criticaI curve, whose definition we now recaii.

The critical set EC(T m ) of T m , m > 1, (from Ensemble Critique) is the Iocus of points X = (x, y) having at Ieast two coincident preimages T-m(x). The following proposition differentiates endomor­ phisms depending on the (non)existence of a "garden of Eden" or Zo regioni it is due to Gardini [1994], who improves on that of Barugola [1984] .

p 1. lf T is a map without a Zo region, the crit­ ical set EC(T m ) of T m , m > 1, is the union of the critical curve LC and its successive images LCi of rank i = 1, 2, ... , m-I : m-l

EC(T m ) =

U LCi ,

LCo == LC .

i=O

A critical curve LCi (called critical curve of rank i) belonging to EC(T m ), separates the (x, y) -plane locally into two regiOriS, one with points having p preimage,s of rank m, the other with points having q preimages of rank m, p > O, q > O. In the general case q = p + 2. • When a region Zo exists , the critical set EC(~) ofTm , m > 1, is given by: EC(T m ) = LCm -

1

UT m (LC_2)'"

UTm(LC_ m ). In the above,

Indeed, due to the presence of Zo, for integers 1 < k $ m , it is guaranteed that Tm(LC_k) C LCm-k. So, for a Zo - Z2 map, LC_2 = T-I(LC_d = T-I(LC_ I n Z2), but T 2(LC_2) is only part of LC. Since rank-one criticaI curve segments define on the phase pIane regions Zk each point X of which has k first rank preimages, it is useful to consider these regions as the superposition of k sheets , each of them associated with a given first-rank preimage. Two such sheets may join in pairs via a conceptuai "fold" (the fold "projects" on one of the segments of the criticaI curve). Three sheets may join at a singular criticaI point, a cusp point at the junction of two fold segments, which has three coincident first rank preimages. Each of these conceptuai sheets is associated with a well defined expression of the inverse map T-I(X), i.e. with one of the roots Y

1170

C. E. Frouzakis et al.

ofthe equation X = T(Y), where X is given. Points on a criticaI segment correspond to double roots of this equation, and a cusp point of LC may corre­ spond to a triple root (or in generaI a multipli city m root, so that more complicated situations are also possible). The Implicit Function Theorem (and, more generally, singularity theory, [Golubitsky & Schaeffer, 1985]) can be used to classify the arrange­ ment of the criticaI curves on the pIane, the occur­ rence of singular points on them (where they are not smooth) and their dependence on map parame­ terso These superposed sheets are "unfolded" back to the entire pIane through the various branches of T-l. Thus , one can think of the entire pIane as tiled by regions, each associated with an expression (branch) for T-l (for more details see [Mira et al., 1996a]). Knowledge of the "coverage" of the pIane by regions resulting from different branches of the inverse map can be indispensable for the charac­ terization of the dynamics of the map backward in time. In particular, it plays an important role in the determination of the boundaries of absorbing areas, chaotic areas, and boundaries of basins of attraction of coexisting attractors. Before we start, we also discuss briefiy the effect of noninvertibility on the nature of the "insets and outsets" of saddle-type fixed and periodic points; for invertible maps these constitute the saddle sta­ ble and unstable manifolds, and are invariant both forward and backward in time. Consider a saddle­ type fixed point p for the map T, which is assumed to be C k , and a neighborhood U' of p. Following Robinson [1995] the iocal stabie manifoid for p in the neighborhood U' is defined to be the foUowing set:

WI~c(P, U' , T)

= {q

EU' : Tj(q) E U' for j > O and

d(Tj (q), p)

-t

O as j

-t

oo}

where d(q , p) defines the distance between points p and q in an appropriate norm. The locai unstabie manifold for p in U' is defined to be the following set:

Wl~c (p, u', T) ;:::: {q E U

i :

there exists some

choice of preimages of q,

{q_j}~oCU' , suchthat d(q_j, p)

-t

O as j

-t

oo}.

According to the Stable Manifold Theorem, these local stable and unstable manifolds are Ck

---- -

- --

_.

embedded manifolds which can be represented as the graph of a map from a disk in one of the (stable, unstable) subspace to the other (unstable, stable) subspace [Robinson, 1995]. Once the local unstable manifold is constructed, the globai unstabie mani­ fold is obtained by

W U(p) =

UTjWl~c(P, U' , T); j?O

this global unstable manifold can, as we will see, self-intersect, and thus wiU no longer be a smooth manifold; technically it is an "immersed submani­ fold". In accordance with the definitions in [Mira et al. , 1996a]; we will call it the giobal unstabie set (R. P. McGehee suggests the term "unstable man­ ifold" for reiations [McGehee, 1992]). This globai unstable set, which is invariant forward in time, is not simply invariant backwards. Indeed , there may exist a Iarger set of points which forward in time map into the globai unstable set. Through a combination of first forward and then backward ap­ plieation of the map, points in this set wiU asymp­ totically approach the saddle p. This more generaI set remains, as yet, unnamed. There is no comparabie "global stabie mani­ fold". The set of points which, forward in time, asymptotically approach the saddle wiU, in the same spirit, be referred to as the globai stabie set of the saddie. Dependingon the map and the parameters, it is possible for these sets to consist of either a finite or an infinite number of segments or branches (an "arborescent" structure [Mira et al. , 1996a]) since every ti me the map is iterated backwards the num­ ber of preimages of a Iocal manifoid segment may grow geometrically. The numericai approximation of such structures is a higqIy nontrivial computa­ tional pro biem.

3. Interaction of Curve Segments with the CriticaI Curves It is convenient, befote considering the intersection of parametrized curves, invariant under a map T, with the criticaI curves of the map, to describe what happens when a "smaU" curve segment interacts with (crosses) the criticaI set. In particular, we de­ scribe the interactions of such segments with LC and LC_ l , since all other interactions of one­ dimensionaI curves with the criticaI set can be thought of as stemming from these.

P'T'Il'lll''T'I.il' .•

DJ Planar

l"in/10rtW'T'1Jn2!.ms

1171

a' o

Ao

a~ )~ ~1 Le (a) 1.

3.1.

Line segment

(b) '''''H' O of Do, Eo are non-transverse contact points of WU(P) with a segment of LCm , m ~ n-I, LCo == LC. This results in a folding of WU(P) along LCm , leading to "oscillations" in the shape of the global unstable set, because WU(P) cannot cross through LC, i.e. it cannot have points in "Zoo Furthermore, consider the area (hd C Z2 n R2, bounded by the global unstable set segment (DI, El) and the LC segment with the same end­ points. Its first-rank preimage is (ho) = (h6) u (h6), where (h6) C RI is bounded by the WU(P) seg­ ment (Do , Eo) and a LC- I segment, and (h6) C R2, bounded by the (Do, Eo) segment on the addi­ tional preimage of WU(P) [the part shown with a dashed line in Fig. 7(c)] and the same LC- I seg­ mento Then, for purposes of backward invariance of the global unstable set , the segment (Do, Dd can be thought of belonging to one, S Hl, of the two sheets folding along LC, while the segment (DI, Ed C (DI, D 2 ) belongs to the other sheet, SH 2 . It should also be noted that a point in (ho), located on the right (left) side of (Do, Eo) is mapped into a point in (hd located between (DI, Ed and LC; stated differently, certain points that are "internaI" to the curve can become external and vice versa. This mechanism of phase space "transport" across invariant curves is, in some sense, reminiscent of the "lobe dynamics" of in­ vertible maps of the pIane [Rom-Kedar & Wiggins, 1990]. As the parameter is further varied, the "lower" point Eo of intersection of the global unstable set with LC-I and its "rightmost" point of tangency DI with LC approach each other and the intersec­ tion lo of the LC and LC- I curves. At À ::::::: 1.245, they coincide with lo, and after that the picture of Fig. 8( d) develops. Notice that now the iterate D 2

- - - ..•._ - - ­

1178

C. E. Frouzakis et al. (c)

(o)

2

y

x

x

Fig . 9. Self-intersections in the form of loops and appearance of a cusp point on the unstable set for the map G: (a) and (b) for .\ = 1.58 the loops on the global unstable set are clearly visible, (c) for .\ = 1.593 a cusp forros.

of Dl lies at a new tangency between the global unstable set and the "other" branch LC? of the it­ erate LCI of LC. The global unstable set acquires an increasingly "wavy" shape .

5.4. Loop formation

5.5. Global unstable set intersects LC_ 1 with excess slope The global unstable set initially becomes quadrati­ cally tangent to LC_ l for À ::::: 1.187. For nearby pa­ rameter values the angles at which it subsequently intersects with LC_ l are far away from the (criticai) angle that the map null vector forms with LC_ l . As the parameter value increases , however , the "bulge" of the global unstable set in the area Rl becomes more pronounced , and the angles of its two intersec­ tions with LC_ l grow. At À ::::: 1.593 [Fig. 9(c)], the angle at which the global unstable set locally inter­ sects LC_ l becomes equal to the one corresponding to the null vector at Do, and WU(P) develops a cusp similar to that of Fig. l(b) ; the cusp point is the image DI = G(Do). For À < 1.593, the self­ intersections of the unstable set in the form of loops are present [Fig. 9(a)].

As we continue to increase À, the location of the unstable set and its additional preimage on the phase piane also changes. After becoming tangent, two intersection points of the set and its additional preimage appear away jrom LC_ l [Fig. 7( d) and the corresponding schematic 8(e)] ; this should be con­ trasted to the previously considered intersections of the unstable set with its excess preimage which oc­ curred on the curve LC_ l . This then means, that there exists a point Fl on the unstable set having two preimages, F~l E Rl and F: 1 E R2, on the unstable set (these two points are the intersections of the unstable set with its preimage away from LC_ l ). In terms of backward invariance, the two 6. Interaction of the Global segments intersecting at Fo should be thought of as Unstable Set of a Saddle with the lying on alternate "sheets": alternate branches of CriticaI Curves for Zl < Z3 Maps the inverse map should be used on each one of them to retain their first rank preimages on the global 6.1. Global unstable set intersects unstable set. As discussed in Sec. 4.1, a loop must LC, but not LC_ 1 appear around Fo; the point Fo is a seij intersec­ tion oj projection. Its forward images [Fl , F2 in This situation is important in the case of Zl - Z3 Fig. 8( e)] are simply seij intersections: in terms of Zl, or Zl < Z3 noninvertible maps, since by con­ backward invariance, a single branch of the inverse struction in the Zo - Z2 case a global unstable set map is needed. The loops persist for À values up cannot cross into the zero-preimage region. As a to À = 1.593 (Fig. 9); after a global bifurcation at parameter varies, a smooth curve (e.g. part of the À ::::: 1.5673 the unstable set crosses over to the other global unstable set for a smooth map) can start to interact with the curve LC. The typical case, side of the saddle-type fixed point P.

Properties of Planar Endomorphisms 1179 1.6

. . •Xx .; x

(a)

x

1.2

~

t
---'--........---1

-0.9

-0.7

-0.5

-03

Y

Y

c

Fig. 11 . G10bal unstable set interacting with Le (t = 0.16): (a) self-intersecting global unstable set, (c) in black is its preimage (also part of the global unstable set) and in blue and violet its additional preimages, (b) and (d) schematics corresponding t o (a) and (c ) showing the details of the intersections .

set has to first (in an arclength sense ) approach a neighborhood of C-l before it subsequently ap­ proaches C . Figure 11 (a) shows a computation (and Fig. Il (b) an enhanced schematic) of the interaction of the global unstable set of a saddle-period-30 with LC in the neighborhood of the cusp point C, while Fig. l1(c) [resp. l1(d)] shows a computation (resp. enhanced schematic) of the "preceding" interaction of this global unstable set with LC_ l in the neigh­ borhood of C-l. Notice that all three preimages of the global unstable set AlElBlFlElDI cross each other and LC_ I ; one of them, the "main branch" AbE6B6FJE6D6 belongs to the global unstable set. Figure 12 shows several phases of the inter­ action of the global unstable set of the period-30 saddle solutions of the map H with the criticaI set (first with LC_ l and, subsequently, with LC ),

culminating in the situation in Fig. 11. In Fig. 12(a) both intersections of the global unstable set with LC_ l are on the same side of the cusp preimage C-l ; therefore, the global unstable set will "later on" (in an arclength sense) become twice quadrat­ ically tangent to LC. In Fig. 12(b) the global un­ stable set crosses through C-l , and therefore it later passes through C; the first quadratic tangency is re­ tained. In Fig. 12(c) the intersections of the global unstable set with LC_ l lie on alternate sides of C-l giving rise to subsequent tangencies of the global unstable set with LC on alternate sides of C. In Fig. 12( d) the slope of the leftmost intersection of the global unstable set with LC_ l coincides with that of the null vector, Eo, of the map H there, giving rise to a cusp of the global unstable set with its apex on LC; later this cusp "opens up" in a loop

Properties oJ Planar Endomorphisms ll81

(a)

(c)

(d)

e

Le (e)

(t)

(g)

e

LC

Le

Le

Fig. 12. Global unstable set or the period-30 saddle solutions interacting with LC and LC- 1 : the way to a self-intersecting global unstable set.

with a self-intersection [point E in Fig. ll(b)] of the unstable set with itself. Figure ll(a) shows a com­ puted version ofFig. 12(f). It is important to notice the entanglement of the three distinct preiterates of the AIEIBIFIEIDI part of the global unstable set in the neighborhood of LC- I ; notice the existence of intersections of different preimage branches both away from LC- I (at the different preimages of the self-intersection point E), as well as intersections on LC-I.

-

7. Interaction of an Invariant Closed Curve with a G lobal U nstable Set The study of the interaction of global unstable sets with themselves, their additional preimages and crit­ icaI curves, leads naturally to the study of similar interactions involving attractors - since it is these objects in phase space that the global unstable sets often asymptotically approach. The "trivial" case

- - - - -- - - - - - - - _ . --- --------_._---

­

1182

C. E . Frouzakis et al.

(a)

(b)

Fig. 13 . Invariant circle r , interacting with a global unstable set: (a) global unstable set and its additional preimage become tangent t o r , (b) global unstable set intersecting r.

involves fixed point attractors, whose interactions with the criticaI set are comparatively easy to study. Periodic point attractors can, with some care, be similarly studied as fixed points of some iterate of the map T. An important next step is to consider an ex­ tended (in phase space) attractor. An invariant cir­ ele, resulting from a Ropf bifurcation of a fixed or periodic point, constitutes the subject of the follow­ ing sections. It is important to notice that elose to the onset of its existence, an invariant circle will not interact with the criticaI set for a map of the pIane. This is because at the Ropf bifurcation point two eigenvalues of the linearization of the map are on the unit circle, hence far away from zero; the criticaI set lies far away in phase space from the bifurcation point, and thus also from the (initially small) in- ' variant cirele that will be born in its neighborhood. It is therefore only Iater in parameter space that the "expanding" invariant circle may encounter the criticaI set . Before we consider the interactions with the criticaI curves, however, an invariant elosed curve can interact with the global unstabie set in a differ­ ent way than with manifolds for diffeomorphisms. For the G map we considered above, the stable fixed point O Ioses stabiIity at ). = 1, and a small amplitude invariant circle bifurcates supercriticalIy from it. This invariant cirele grows, and up to ). ~ 1.1873, it is still entireIy contained in the R2 region (which means that it is invariant forward in time and aiso invariant backward in time us­ ing a single branch of the inverse map - it can be thought of as Iying entireIy on one sheet) .

Consider the global unstable set WU(P) of the saddie fixed point P for a Zo - Z2 map, and the invariant elosed curve r, such thàt WU(P) asymp­ totically approaches r . For a diffeomorphism, the unstable manifoid WU(P) can only spiraI towards the closed invariant curve without ever intersecting it . In the case of the two-dimensionai noninvert­ ible quadratic map G, Gumowski and Mira [1980a] have observed a set of parameter values Ieading to intersections of the saddie global unstabie set with r (schematic ofFig. 13). In our exampIe, at). ~ 1.528 the invariant curve initialIy becomes tangent to the global unstabie set at Ko; in a sense this tangency "follows" the tangency between the invariant circle and the additionai first-rank preimage [Fig. 13(a)] of the global unstabie set at one of the preimages of Ko (the point marked K_ 1 ). This, after one forward iteration of the map, resuIts in the first tangency of the global unstabie set with the invariant circle at the point Ko, and, of course, at an infinity of tangencies at alI forward iterates of this point . As the parameter changes, in a situation reminiscent of that of Sec. 5.2, the global unstabie set intersects r at an infinite number of points [Fig. 13(b)] .

8. Interactions of Attracting Invariant Circles wi th CriticaI Curves As long as the attracting invariant circle for a map T does not intersect LC_ 1 (i.e. rnLC_ 1 = 0), which is the case for parameter values close to a Ropf bifurcation, the following properties hold

- - -- -- - -

-

-

-

Properties 01 Planar Endomorphisms 1183

(cf. Gumowski & Mira, 1980 for a particular exam­ pIe, and [Mira et al., 1996a] for the generaI case of Zo - Z2 maps):

P2 (a)

r

and its first rank preimage distinet from itselfr -l (r -l in this ease is Tl-l(r)), do not interseet: r n r -l = 0, T-l (r) = r u

r

r -l.

Le

(b) The area enclosed by r, say a(r), is invari­ ant under the map, T[a(r)]

= a(r).

(c) The (Jorward) invariant eircle is also in­ (Zo"!

........ .. variant baekward in time under the braneh T;l of the inverse map; its interior a(r) is also baekward invariant under the same Fig. 14. Invariant circle intersecting Le-l. braneh of the inverse map. (d) Consider a neighborhood U(r) ofr and its image T(U) C U (sinee r is attraeting). T (d) Consider a neighborhood U(r) of r. In or­ has a unique inverse T 2- 1 : T(U) -7 U. der for the inverses of points in this neigh­ (e) When a periodie orbit of period k, Pk, ex­ borhood to remain in a neighborhood of the ists that does not lie on the invariant eirele, invariant eirele, both b.ranehes of the in­ the Pk points are either alt inside a(r), or verse map must be used. In other words, they are alt outside a(r). for "baekward neighborhoorf' (as an exten­ sion of "baekward invarianee") purposes, 8.1. The attracting invariant circle it is neeessary to use both branehes of the intersects LC_ 1 inverse at different loeations in U(r). (e) If periodie points Pk exist in the vieinity of As in Secs. 5.1 and 5.3, we consider the case when the invariant eircle r (but not on r), it is the attracting invariant circle, r, intersects LC_ l at possible that some of these periodi e points two points Do and Eo (Fig. 14) . Tangency points Pk may lie inside a(r), while other points Dk and Ek of the invariant circle with LC and belonging to the same periodi e trajeetory lie the higher rank critical curves will then appear at outside a(r). the images of the intersection points Do and Eo. The noninvariance of a(r), and the "oscilla­ Since the invariant circle cannot "dip" into the zero­ ~

preimage region, "oscillations" in its shape result from these tangencies. In this case, when rnLC_ l -=I (i.e. r n Hl -=I 0) and r n r -l c LC_ l , one can make the following observations ([Gumowski & Mira, 1980a]), for a particular exampIe , and [Mira et al., 1996a] for the generaI case of Zo - Z2 maps):

o

-~

tions" of rare, of course different aspects of a unique mechanism (the intersection with LC_ l ). Indeed, the area (hJ) C Z2nR2, bounded by the r segment (DI, Ed and an LC segment, has first rank preim­ ages (ho) = (hÒ) U (h5)· Rere (hfi) C Hl is bounded by the r segment (Do, Eo) and an LC_ l segment, and (h5) C H2 is bounded by the r - l segment (Do, Eo) and the same LC_ l segment (Fig. 14). On the pIane, a point belonging to (hÒ) or (h5) (inter­ naI to a(r)) is mapped into a point belonging to (hl) (external to a(r)).

r and r - l intersect at the points Do, Eo of LC_ l . The forward image of the segment of r in Rl gives rise to tangencies and osciltations in the shape of r along LC. (b) The interior area a(r) is no longer forward 8.2. Coexistence oJ a periodic invariant under T : Some internal points solution with an attracting are mapped outside and vice versa. invariant circle (c) The inverse of the restriction Tr of T to r , must be defined for baekward invari­ As indicated above, for noninvertible maps it is ance purposes using both T l- l and T 2­ 1 possible that some of the k points of a periodic (each on different segments of the curve). solution P k may Iie inside a(r) while the rest are

P3 (a) In the simplest case

1184

C. E. Fro'Uzakis et al.

,

"

LC •• ••

\,f_ ,, 1

,, , •

',

,, •

\ a(f_ 1): ,

,

LC_

"

1

f f

,

,

' •• _.,~

,~

(a) I

Fig. 15. Illustration of (a) interior points mapped outside the invariant circle and (b) exterior points mapped in si de the invariant circle .

outside. This is essentially due to the folding of the piane under T along the first rank criticai curve Le . For the schematic in Fig. 15 this folding implies the following: (i) A point p E [a(r) n a(r -1)] (i.e. in the area inside the invariant cirele defined by the inter­ section of a(r) and a(r -d) is mapped into a point Pl, outside the invariant circle, Pl tt a(r). (ii) A point p' E [a(r) U a(r -d] \ [a(r) n a(r -d] is mapped into a point p~ E a(r) (Le. a point p' inside a(r) but not in the common area of the invariant curve and its additional preimage stays inside, and a point p' outside a(r), but inside a(r -d with the exception of the com~ mon area again, comes inside). (iii) Any other point p", (p" tt [a(r) U a(r -d]), is mapped outside a(r).

Considering a period (or order) k cyele , the points of which are denoted 1, 2, . .. , k, the points p == 3 and Pl == 4 of Fig. I6(a) illustrate property (i). The points p' == 3 and p~ == 4 of Fig. I6(b) illustrate property (ii). The points p" == 6 and p'; == 7 of Fig. I6(c) illustrate property (iii) . Actually, the above three cases are of more in­ terest when looking for parts of phase space that "switch sides" with respect to the invariant cìrele when operated on by the map, i.e . regions which become "internai" or "external" to a closed curve. Since we are discussing the relative location of peri­ odic points, it suffices to check whether any one of the periodic points falls into the region a(r) n R 1 . Then, we will necessarily have some periodic points inside a(r) and some outside. Another property concerns the relative position of the k-cycle points with respect to the regions

Properties of Planar Endomorphisms 1185

R2 (i~') ...... '

(b)

(a)

8

X

(c) Fig. 16. circle.

A schematic period-k trajectory, illustrating the relative positions of its points with respect to an attracting invariant

R 1 (where the Jacobian determinant IJ(X)I < O, X E Rd and R 2 (where IJ(X)I > O). A particular overall feature of the relative arrangement of the cy­ cle points with respect to the invariant circle Xi, i = 1, 2, . . . k can be inferred from the sign of the prod­ uct of their eigenvalues 5 1 5 2 = J[Tk(Xi)], since IJ[Tk(Xi)]1 = IJ(Xdl . IJ(X2 1) ·· . IJ(Xk)l· Then, there is an even (odd) number of points of a given cycle in R 1 , if and only if 5 1 5 2 > O (5 1 5 2 < O). So , a fo cus cycle (5 1 5 2 > O) has always zero, or an even number of cycle points in RI'

8.3. A first glimpse of Arnol'd horns and the associated bifurcations The last few cases illustrated some of the dynamic complexity associated with the coexistence of at­ tracting invariant circles and periodic solutions in their neighborhood. A natural context for the study of the local and global bifurcations involving such interactions is the study of Arnol'd horns in the neighborhood of (as well as further away from) Hopf bifurcations. In the neighborhood of a Hopf bifur­ cation curve in two-parameter space, one expects periodic solutions of various periods (and rotation numbers) to appear on the invariant circle. These

"lockings" exist in the interior of the Arnol'd horns, whose boundary in two-parameter space is a locus of saddle-node bifurcations [Arnol'd, 1983]. As we discussed above, LC_ 1 must be located "far away" in the phase pIane from the Hopf bifurcation locus, and so the entire Arnol'd horn scenario should be observed in noninvertible maps in complete anal­ ogy with the invertible case. As one moves away, however , in phase and parameter space from the Hopf bifurcation" the invariant circles as well as the "locked" periodic solutions may start to interact with the criticaL lines. A wealth of interesting bifur­ cation scenarios then develops as the complicated interior structure of the Arnol'd horns far away from the Hopf curve (involving breakup of the invariant circles and several types of global bifurcations) be­ gin to interact with nonivertibility. Figure 17 presents the first glimpse of the Hopf bifurcation curve (defined by c = 1) of the fixed point (x, y) = (O, O) as well as some of the resonance horns in its neighborhood for the map

a two-parameter extension of the map G.

1186

C. E. Fro'Uzakis et al.

3.9

(b) 1.5

2.9 1.3

c 1.9

1.1 \

0.9 -2.5

"

/ -----~~~--------

-0.5

-1.5

0.5

1.5

0.9

0.7

0.9

1.1

b

1.3

b

Fig. 17. (a) Period-6 and period-7 resonance hom boundaries (SN6 , and SN7 curves, respectively) bom from a supercritical Hopf bifurcation (curve H BI) of a fixed point of the map G l. Some additionallocal bifurcation curves (period-doubling curves, P D6 and P D7), associated with the homs are included as an indication of the complexity of the full two-parameter bifurcation diagram. (b) Blow up of the region around the tip of the period-7 hom indicating the location of the three one-parameter cuts studied in the text.

(a)

0.4

r

0.2

y

0.0

LC_ 1

-0.2

LC

-0.4

0.6 (d)

0.4

yO.2

0.0

S4 ...0.2 -0.4 -0.4

0.0

x

0.2

0.4

0.6

--0.2

0.2

0.0

0.4

0.6

x

Fig. 18. Phase portraits for the one-parameter cut across the period-7 hom close to its tip at c = 1.15: (a) b = 1.187, (b) b = 1.1874, (c) b = 1.1881, (d) b = 1.1882 (Period-7 saddles, Si, are indicated by crosses, period-7 sinks, A, by red circles, the two sìdes of the saddle global unstable set are plotted in blue and green, respectively).

- - - - - - - - - --_.-

..

- -_ .. ­

Properties oJ Planar Endomorphisms 1187

0.6 (a) :==-----~

0.4

1.00

_.­

>

- --- -=- .:

x

---.-------- -.- ­ -

0.60

0.2

r

-­

~---------------- --e __

0.20

0.0

'­ -0.2

---~...-..:.-

>~----------~

-0.4 1.1872

1.1876

1.1880

b

-0.20

-0.60 0.9950

1.0050

1.0150

1.0250

b

Fig. 19. One-parameter bifurcation diagrams: (a) c = 1.15, (b) c = 1.3784; the filled circles mark the approximate location , of the points having one eigenvalue equal to zero (saddle-type solutions are indicated by dashed lines) .

As the eigenvalues ofthe period-l solution move along the unit cirde crossing through various com­ plex roots of unity, the corresponding Arnol'd horns are born. In this case the Ropf bifurcation is su­ percritical, and the computationally located bound­ aries of several horns (a period-6 born at the sixth root of unity, and a period-7 horn born at the sev­ enth root of unity) lie on the side of the Ropf curve where the period-l is unstable. As can be seen in Fig. 17( a), there is a wealth of local transition and bifurcation curves associated with this horn (com­ puted with the software program AUTO [Doedel, 1981] and a graphical interface using the GL library developed by the Princeton group [Taylor & Kevrekidis, 1990]). We only show these as a pre­ liminary indication of the richness of the bifurcation scenario in the interior of the horni indeed, there is a much richer skein of global bifurcations missing from this picture, which is the subject of intense current research. We will concentrate on the period-7 horn ema­ nating from the point b = 1.246938, c = 1.0. The three horizontal lines in Fig. 17(b) show the loca­ tion of three representative one-parameter "cuts" across the resonance horn which we have induded to demonstrate some of the most fundamental differences of the Arnol'd scenario induced by noninvertibility. Figure 18 shows a sequence of phase portraits obtained for c = 1.15. In this case, the entire in­ variant circle lies far away (in phase space) from

--

-- - -------

the criticai curves, and the local picture is therefore indistinguishable from the one for diffeomorphisms: The period-7 solutions are born , via a saddle-node bifurcation on the invariant circle at b = 1.18728; the period-7 saddles Sk and the period-7 nodes Ak "separate" from each other [Fig. 18(b)], rotate along the invariant circle, me et again [but now with dif­ ferent pairings, Fig. 18(c)] and disappear at an­ other saddle-node bifurcation, on the "other side" of the resonance horn at b = 1.18810 [Fig . 18(d)]. Figure 19( a) also shows the one-parameter bifurca­ tion diagram of these period-7 solutions; at no point along this diagram does an eigenvalue of either the nodes or the saddles become zero - we are far away from noninvertibility. Figure 19(b) shows the one-parameter bifurca­ tion diagram for the period-7 solutions along the one-parameter cut for c = 1.3784. At first view this is indistinguishable from the one in Fig. 19(a); but now the situation is vastly different. To begin with , there exists multistabilitYi the periodic solutions at the saddle-node bifurcation are not born on the in­ variant cirde, but away from it. In Fig. 21(a), two attractors coexist: the stable period-7, and the sta­ ble invariant circle. This is clearly seen as we follow the global unstable set of the saddle-period-7 solu­ tions: one side asymptotically approaches the stable invariant circle, while the other side approaches the stable period-7. This phenomenon (the "decollation" of the res­ onant periodic solutions from the invariant cirde

1188

C. E. Frouzakis et al.

+

it-

----e

\-- 0

a

~

~r:

-....

t-- G/

+--­

+--­

--~!- ~~l_

t

~

_j~i-: 1

~

~

G/

--+~

.-t

t

+-­

t

d

(;I

+/

/~

~

rr­

t

!~ 0

o-

o

~

".

~I 0 ~ e

----.....~

t

~

+­

~



~

{r~

+/

G/

/+~- /~ Fig. 20. Schematic of (some of) the global bifurcation and transition curves inside a resonance horn for a planar diffeomor­ phism and corresponding phase portraits .

as we move higher up along the boundaries of the Arnol'd horn) is also observed in diffeomorphisms; a schematic of some of the global bifurcations and transitions expected in the interior of an "invert­ ible" Arnol 'd horn can be seen in Fig. 20 (see also [Frouzakis et al., 1990; Aronson et al. , 1982]). In that case, however, either all of the periodic points born at the saddle-node will be inside the invariapt circle, or else aH of them will be located outside. Rere, because of the noninvertibility, we see that one of the periodic pairs is inside the invariant cir­ cle, while the remaining six are outside. Figure 21(b) shows the result of a global bi­ furcation as we proceed towards the interior of the

resonance horn: the invariant circle "collides" with the saddle-period-7 points and disappears. This is a global bifurcation involving a homoclinic tan­ gency and subsequent crossing (homoclinic tangle) between the stable and global unstable sets of the saddle- period- 7. Figure 21 (c) shows the relative location of the saddle invariant "manifolds" after this bifurcation: both sides approach stable period­ 7 points. As we approach the saddle-node bifurca­ tion a t the other side of the resonance horn the sit u­ ation is reversed. Another global bifurcation occurs (again involving the stable and unstable sets of the saddle-period-7 solutions, and the stable invariant circle is reconstituted [see Fig. 21(d)]; only now 6 of

Properties oJ Planar Endomorphisms

1189

1.3 ,-----------r-----,-------,

(a)

A S.

0.8

y

Le

0.3

nB

-0.2

~

1.3 , . . - - - - - - - , - - - - - , - - . . . . . . , - - - - - y - - - - - - ,

(c)

0.8

y

0.3

A 5

-0.2

-0.7 '----_~~'__I...__C_~._ -0.6 -0.1

_____"c........_____'__'__

_____"_ _~_____"

0.4

-0.6

0.9

-0.1

0.4

1.4

x

x

Fig. 21. Phase portraits for the one-parameter cut across the period-7 horn at c = 1.3784: (a) b = 1.0: invariant circle (black) coexisting with the period-7 saddles, Si, (crosses) and the sinks , Ai, (red circles) (one pair "in", six pairs "out"), (b) b = 1.0004: invariant circle disappears after a global bifurcation; only the saddle-sink pairs remain , (c) b = 1.01: nodes turn to foci , (d) b = 1.0225: a stable invariant circle reconstituted via a reverse global bifurcation involving th e saddle stable and unstable sets coexists with the p eriod-7 saddles and sinks (one pair "out", six pairs "in"). The color scheme is the same as in the previous figure.

the period-7 pairs are "inside" it, and one is outside. A sequence of global bifurcations involving the same basic phenomena (the formation and destruction of an invariant circle through global (homoclinic) bifurcations of the saddle-type resonant periodic so­ lutions) is also part ofthe generic internaI structure of the Arnol'd horns for diffeomorphisms. Rere, the "twist" provided by noninvertibility has to do with the relative location of the attracting invariant cir­ cle and the resonant periodic solution pairs.

- - - - - - - - - - _._-. - ­

_

_

__ o

Figure 22 , the "higher" one-parameter cut at c = 1.5, offers a glimpse of something truly differ­ ent: noninvertibility now plays a major new role in determining the bifurcations and dynamics in the interior of a resonance horn. In this case, it is better to start the description from the interior of the horn. Figure 22(b) shows the saddles and nodes period- 7 points connected in a chain: one side of the global unstable set of each of the saddles asymptotically approaches the period- T node it was

_

. _

.. _ _ • _

_ __

_

_

_ __

1190

C. E. Frouzakis et al.

"born with", while the other side asymptotically ap­ proaches the one it will collide with and disappear on the "other side" of the resonance horn. This pic­ ture is very reminiscent of a "locked" invariant cir­ cle [see Fig. 18(c)] with two obvious noninvertible "twists": the first is that the "chain" of periodic points connected by saddle global unstable sets does intersect Le-l' The second, and more important one, however, is that the global unstable sets of dif­ ferent saddles intersect each other; since they inter­ sect once they actually have to intersect infiniteLy many times, even though only one of them is visi­ ble at the scale of magnification of the inset. As the parameter b is decreased, the saddles and the nodes

move closer to each other and the self-intersections of the global unstable sets become more pronounced [Fig. 22(b)]. In addition to this, the global unsta­ ble set intersects the global stable set, thus creat­ ing a homoclinic tangle . When the period-7 pair disappears at the saddle-node at the "left" bound­ ary of the period-7 horn, a self-intersecting chaotic attractor ("weakly chaotic ring" [Mira et al., 1996a] [Fig. 22(a)]) appearsi very close to the saddle-node bifurcation trajectories on it are intermittent. As the parameter b is increased from that ofFig. 22(c), first one of the stable eigendirections of the nodes [Fig. 22(c)]' and then the stable eigendirection of the saddle period-7 points [Fig. 22(d)] become

(a') y

r ·

( \. ' -~~.~!'-_.

0.8

0.63

A,;;.

(b)

(b')

A, 0. 8

A S,

lì

S,

~

'--

LC 0.0

LC_ I

0.8

0.60 0.70

0.72

Fig. 22. Representative phase portraits (a)-(e) and local blow ups (a')-(e') for the one-parameter cut across the period-7 hom at c = 1.5: (a), (a') b = 0.83 (b), (b') b = 0.8311, (c), (c') b = 0.84, (d), (d') b = 0.88, (e), (e') b = 0.89. (a) and (e) show weakly chaotic rings just outside the resonance homo (b)-(d) show different types of self-intersections of the global unstable sets of the period-7 saddles. The color scheme is the same with the previous figure.

Properties 01 Planar Endomorphisms

0.8

r----,

(c)

(c')

y 0.7 0,8

A,~ __---M:;"!t

S 0.6

LC 0.0

SI

0.5

S.

-0.8 L-~~--'''-'----L_~~_~...I..J...._~~_~--' 0.4 L.....L...~~---' -0;8 0.0 0.8 0.72 0.75

ti

(d')

0.85

0.8 0.80 AI

LC

~ 7

0.75

LC_I -0.8 -0.8

0.0

(e)

y /

/

0.8

I

0.70 0.8

0.8

~

.~

1.0

1.2

(e')

0.88

/

0.78

i i : I

0.0

1'\,\ ii

\\

LC_ 1 -0.8 -0.8

0.0

~

0.8 x

Fig. 22.

( Continued)

0.68 0.85

1.10

1.35

1191

1192

C. E. Frouzakis et al.

associated with a negative eigenvalue. This results in the "oscillations" of the self-intersections seen in Fig. 22(d). Since, as in Fig. 22(b), the global un­ stable set of one saddle approaches both sides of the unstable set of another saddle, it has to intersect its local stable manifold: we have a true global bifurca­ tion creating a homoclinic tangle. After the saddle­ no de bifurcation at the "right" side of the period-7 resonance horn boundary, the periodic points dis­ appear leaving behind the remnant of this tangle: another "weakly chaotic ring" [22( e)]. This third one-parameter cut shows only a cou­ pIe of the possibilities that noninvertibility provides to enrich the scenarios of invariant circle breakup in the Arnol'd horn context. The development of chaotic attractors via self-intersections of invariant circles interacting with LC_ 1 was observed by Lorenz [1989] j a more detailed study of his origi­ naI findings can be found in [Frouzakis et al., 1997]. The bifurcations resulting from the interaction of invariant circles with noninvertibility are a frontier of current "noninvertible" dynamical systems re­ searchj what we presented here only shows a glimpse of this interplay in the context of Arnol'd horns, a study which we are currently pursuing (see for example [Gicquel et al., 1996]).

9. Summary and Conclusions Noninvertibility in planar endomorphisms gives rise to a wealth of dynamics and bifurcations that are "new" in the sense that they are qualitatively dif­ ferent from those generic for diffeomorphisms. This "new" type of phenomena have been the subject of research since the 1960's, and many researchers over the world have rediscovered them over the years. In more recent years, the systematic theoretical and computational study of such systems has been rapidly progressing through the research of many groups and across many disciplinesj experimental realizations of higher-dimensional noninvertible sys­ tems are also being studied. This paper presents an attempt at systematically studying the hierarchy of such "truly noninvertible" phenomena in planar endomorphisms. The criticaI lines (and more gen­ erally the criticaI set) of the map is used as the basis of the study: the types of interactions that global unstable sets and attracting invariant cir­ cles can have with these curyes are probed first by examining them locally, and then examining their global implications. This "program" of study of 10­ cal and global elements of the ex and w limit sets of

endomorphisms with the criticaI set is being pur­ sued through both computer-assisted exploration and analysis. It must be stressed that this study poses a very strong computational challenge even on the pIane: the arborescent structure of preimages of local stable- and unstable manifolds, as well as that of attracting invariant circles, gives very quickly rise to unmanageable data structures. While our group has already implemented "reasonable" com­ putational approaches with graphical interfaces for the numerical approximation of such objects, "rea­ sonable" is far from being satisfactorYj the complex­ ity of the structures requires much better database programming to be of help in computer-assisted ex­ perimentation. Current directions of research in the case of planar endomorphisms include the study of piecewise linear maps, the study of generic 10­ cal and global bifurcations of diffeomorphisms "in the neighborhood" of noninvertibility [e.g. Nien & Wicklin, 1995] the study of global implications that folding of the pIane imposes on the structure of so­ lutions (such as the relative arrangement or peri­ odic solutions and invariant circles probed here), and the study of chaotic noninvertible dynamics . Many other obvious paths are being pursued: the extension to three dimensionsj the extension to re­ lations, where both forward and backward trajec­ tories are non-uniquej and -last but not least- the experimental verification of many of these dynamic predictions is still lacking.

Acknowledgment This work was partially supported by the NSF. We are also grateful to Nathalie Gicquel for many discussions on the manuscript and to the NSF Geometry Center in Minneapolis and the Center for Nonlinear Studies at Los Alamos National Labora­ tory for their gracious hospitality.

References Adomaitis, R. A. & Kevrekidis, I. G. [1991] "Noninvert­ ibiIity and the structure of basins of attraction in a model adaptive control system ," 1. Nonlin. Sci. 1, 95-105.

Amoroso, S. & Cooper, G. [1970] "The Garden-of-Eden theorem for finite configurations," Frac . Amer. Math. Soc. 26, 158-164. Arnol'd, V. L [1983] "Geometrical methods in the theory of ordinary differential equations," (Springer-Ve rl ag , New York). Aronson, D. G. , Chory, M. A. , Hall, G. R. & McGe­ hee, R. P. (1982) "Bifurcations from an invariant circle

Properties 01 Planar Endomorphisms

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