Mathematical problems of nonlinear dynamics: A tutorial

~ ) Pergamon

J. Franklin Inst. Vol. 334B, No. 5/6, pp. 793 864, 1997

PII:

S0016q1032(97)00039-2

~ 1997 The Franklin Institute PubJished by Elsevier Science Ltd Printed in Great Britain 00164)032/97 $17.00+0.00

Mathematical Problems ofNonlinear Dynamics: A Tutorial by LEONIDSHILNIKOV Research Institute for Applied Mathematics and Cybernetics, 10 Ulyanov Str., Nizhny Novgorod, 603005 Russia (Received in final f o r m 1 March 1997; accepted20 March 1997)

ABSTRACT: We review the theory of nonlinear systems, especially that of strange attractors, and give its perspectives. Special attention is given to the recent results concerning hyperbolic attractors andfeatures of high-dimensional systems in the Newhouse regions. We present an example of a 'wild' strange attractor of the topological dimension three. © 1997 The Franklin Institute. Published by Elsevier Science Ltd

L Introduction

The early 1960s is the beginning of the intensive development of the theory of highdimensional dynamical systems. Within a short period of time, Smale (66, 67) had established the basics of the theory of structurally stable systems with the complex behaviour of trajectories, the theory which we now know as the hyperbolic theory. In essence, a new mathematical discipline with its own terminology, notions, etc., has been created, which at the same time interacts actively with other mathematical disciplines. Here we must emphasize the role of the qualitative theory of differential equation (QTODEs). In fact, this theory provides a foundation for investigating many problems of natural sciences and engineering which have a nonlinear dynamics origin. On the other hand, the qualitative theory of differential equations itself takes new ideas from nonlinear dynamics. The usefulness and the necessity of such a synthesis were clear for scientists with a broad vision on science, such as Poincar6 and Andronov. All this makes the QTODEs especially attractive and practical. Its achievements have led to one of the brightest scientific discoveries of the twentieth c e n t u r y ~ y n a m i c a l chaos. Since the moment of the discovery of dynamical chaos along with such customary dynamical regimes as stationary states, self-oscillations and modulations, chaotic oscillations entered science. If the mathematical images of the former are equilibrium states, periodic orbits and tori with quasiperiodic trajectories, then an appropriate image of dynamical chaos is a strange attractor, that is, an attractive limiting set with the unstable behaviour of its trajectories. Those attractors that retain this property under small smooth perturbations will interest us in this review. Such attractors have been predicted by the hyperbolic theory of high-dimensional dynamical systems. 793

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However, the role and the significance of strange attractors were not accepted by researchers in certain scientific fields, in particular turbulence, for a long time. There were a few reasons for that. The hyperbolic theory had examples of strange attractors but their structures were so topologically complex that they did not allow one to imagine simple scenarios for their origin; this is very important for nonlinear dynamics, which deals with models described by differential equations. On the other hand, those 'strange attractors' observed in concrete models were not hyperbolic attractors in the strict meaning of these words. Most of them, possessing all the properties of a 'genuine' strange attractor, had stable periodic orbits. This gave a chance to argue that the observable chaotic behaviour is intermittent. Here, we have to bear in mind that when speaking about dynamical systems we are interested not in the character of a solution over some bounded period of time but in the information on its limiting behaviour when time increases to infinity. We note also strange attractors that have hyperbolic subsets coexisting with stable long periodic orbits of a very narrow and tortuous attraction basin, the so-called quasi-attractors (stochastic attractors) (10). The breakthrough came in the mid-1970s with the appearance of a 'simple" lowdimensional model = - ~ ( x - y) = rx-y-xz 2 = -bz+xy

in which Lorenz had discovered numerically in 1962 a chaotic behaviour in the trajectories. A detailed analysis carried out by mathematicians revealed the existence of a strange attractor, not hyperbolic but non-structurally stable. Nevertheless, the main feature--the instability of the behaviour of trajectories under small smooth perturbations of the system--of this attractor persists. Such attractors, which contain a single equilibrium state of the saddle type, will be henceforth be called 'Lorenz(ian) attractors'. The second remarkable fact related to these attractors is that the Lorenz attractor may be generated on the route of a finite number of rather simple observable bifurcations from systems with trivial dynamics. Since that time the phenomenon of dynamical chaos was 'almost legislated'. In this breakthrough, the fact that the Lorenz model came from hydrodynamics played a primary role. The topological dimension of the Lorenz attractor, regardless of the dimension of the associated concrete system, is always two and its fractal dimension is less than three. At the same time, researchers who deal with extended systems often observe chaotic regimes of presumably much higher dimensions. It is customary then to say that hyperchaos occurs. But which attractors describe hyperchaos? Are they strange attractors or quasi-attractors? In principle, the hyperbolic theory predicts the possibility of the existence of strange attractors of any finite dimensions. The tragedy is that nobody has observed known hyperbolic attractors in nonlinear dynamics since the moment of their invention. There has been some progress recently: the author and Turaev (79) have proved that a number of hyperbolic attractors (structurally like the Smale Williams solenoids and the Anosov tori) may be obtained through one global bifurcation of the disappearance of a stable periodic orbit or of an invariant torus with

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a quasi-periodic trajectory on it. They have also discovered a principally new type of strange attractor--the so-called wild strange attractors. Their distinction from the known attractors is that they contain an equilibrium state of the saddle-focus type as well as saddle periodic trajectories of various types, namely, dimensions of unstable invariant manifolds of the coexisting trajectories may be equal to both two and three. Moreover, the region of the existence of such an attractor is a region of everywhere dense structural instability owing to homoclinic tangencies. Thus, the above phenomenon poses principally new problems for ergodic theory. Because of the coexistence of the trajectories of various types, the wild attractors, as well as Lorenz-like attractors, are pseudo-hyperbolic attractors. As the notion of pseudo-hyperbolicity, which plays a dominant role in the theory of structurally unstable strange attractors, will be used below, we will now discuss it in detail. We consider a smooth n-dimensional dynamical system = X(x)

in a bounded region D which satisfies the following conditions: 1. on its boundary CD the vector flow goes inward into D. This implies that for any point x~CD either an entire trajectory or a semitrajectory is defined that passes through the point x. 2. A pseudo-hyperbolicity takes place in D. This implies that, at each point xeD, the tangent space, invariant with respect to the associated linearized flow, may be decomposed as a direct sum of subspaces N~ and N2, depending continuously on the point x so that the maximal Lyapunov exponent, corresponding to N~, is much less than any Lyapunov exponent corresponding to N> In other words, the associated variational equation can be represented in the form = A~(t)~,

r7 = A 2 ( t ) , 7

where the contraction in ~ is stronger than the contraction in q. 3. The linearized semiflow is volume-expanding in N2 V,>ce~'V0,

a>0,

c > 0.

It should be noted that the property of pseudo-hyperbolicity persists under small smooth perturbations, as does the property of the exponential expansion of volumes in N2. Because of the above requirements there will exist at least one strange attractor in the region D. We note that this suggested criterion for the existence of a pseudohyperbolic attractor in D is formulated relatively simply, but, similarly to the principle of contraction mappings, its verification in concrete systems will not be trivial. Let us return to the problem of quasi-attractors. In the case of both quasi-attractors and wild strange attractors, the reason for complexity is the presence of structurally unstable Poincar6 homoclinic curves, that is, bi-asymptotic trajectories to a saddle periodic orbit, along which its stable and unstable manifolds have a non-transverse contact. This type of homoclinic trajectory is also responsible for the existence in the space of dynamical systems of regions of everywhere dense structural instability--the so-called Newhouse regions in which systems with homoclinic tangencies are dense.

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Under certain conditions, systems with infinitely many stable periodic orbits are also dense in the Newhouse regions. At least this is always true for three-dimensional systems with negative divergence. The peculiarity of such a set of stable periodic orbits is that it cannot be separated in a quasi-attractor from the coexisting hyperbolic subset to which these periodic orbits accumulate. In three-dimensional systems with signalternating divergences, for example, in the Chua circuit (22, 45), the situation is even more complicated (77): totally unstable (repelling) periodic orbits may coexist with the hyperbolic subset and the set of stable periodic orbits with very long periods. A similar hierarchy will occur in high-dimensional quasi-attractors in which stable periodic orbits, invariant tori as well as stable strange attractors of various topological dimensions may be embedded in an extremely non-trivial way. Apart from very well developed stability windows where these stable objects reveal themselves, they are practically invisible in numerical experiments because they have very long transients and weak attraction basins as they are closely mixed with the hyperbolic structure. The above observations show that a careful and correct interpretation of many applied studies on dynamical chaos is essential. In particular, this includes the activity related to the calculation of Lyapunov exponents on a finite interval of time. The justification of this calculation is based on Oseledec's theorem (52) on the majority of the Lyapunov-right trajectories in the sense of a suitable measure. Apparently, one needs to account for the effect of small probabilistic noise, which can blur the subtle (delicate) structure of quasi-attractors. Andronov had proposed a recipe for the analysis of concrete models. The basic idea is the following: (1) the parameter space is partitioned into regions of structural stability and the bifurcation set is identified; (2) the bifurcation set is divided into connected components corresponding to qualitatively similar, in the sense of topological equivalence, phase portraits. Nowadays, Andronov's scheme has not lost its practicality except for some important corrections; namely, in situations when a system admits the existence of non-transverse homoclinic curves typical for quasi-attractors, wild attractors, etc., as structurally unstable systems may fill out entire regions in the space of dynamical systems. As established by Gonchenko et al. (33, 34), the following Cr-smooth (r > 3) systems are everywhere dense in the Newhouse regions: (1) systems having infinitely many periodic orbits of any order of degeneracy; (2) systems with homoclinic tangencies of any order. Consequently, we can reach the following important conclusion: the goal of a complete investigation of systems with complex dynamics of above types is unrealistic. It appears that one should abandon the idea of a complete description and turn to the study of some special but distinctive properties of the system. The properties which are worth studying must essentially depend on the nature of the problem.

IL Basic Notions of the Theory of Dynamical Systems Three components are employed in the definition of a dynamical system: (1) a metric space E called the phase space; (2) a time variable t which may be either continuous (i.e. te~ l) or discrete (i.e. t~7/); (3) an evolution law, that is, a mapping of any given point x in E and any t to a uniquely defined state ~p(t,x)~E. It is also assumed that the following conditions hold:

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(I) ~0(0,x) = x (2) ~o(tl,q)(t2,x)) = ~o(t, + t2,x) (3) ~o(t,x)eC ° with respect to (t,x).

(1)

In the case where t is continuous the above conditions define a continuous dynamical system, or flow. In other words, a flow is a one-parameter group of homeomorphisms (i.e. one-to-one, continuous mappings with a continuous inverse; this follows directly from the group property Eq (1)) of the phase space E. Fixing x and varying t from - ~ to + ov we obtain an orientable curve (which defines the direction of motion), which, as before, we will call a phase trajectory. The following sub-division of phase trajectories is natural: equilibrium states, periodic trajectories and unclosed trajectories. We will call a positive semi-trajectory {x;q)(t,x),t> 0}, and a negative semi-trajectory {x;q)(t,x),t
V(x)

-

dqKt,x) , = dt 0

(2)

The simplest, but principal, case of a smooth dynamical system is the flow determined by the vector field

:c = X(x),

xe~"

where XeC r, r > 1. Discrete dynamical systems are, more briefly, called 'cascades'. A cascade possesses the following remarkable feature. Let us select a h o m e o m o r p h i s m ~o(l,x) and denote it by ~0(x). It is obvious that q)(t,x) = q)'(x), where

:' = v,!:(...y(x//). t times Hence, to define a cascade it is sufficient only to point out the homeomorphism

q~:E~---~E. In the case of a discrete dynamical system a sequence {xk} +~: where xk+, = (p(x~), is called the 'trajectory' of a point x0. Trajectories may be of three types: l. A point x0. The point is a fixed point of the h o m e o m o r p h i s m q~(x), that is, it is mapped by q~(x) onto itself. 2. A cycle (Xo," • ' , x k ,), where x~+, = ~o(xi) and where, moreover, x~¢:c/for iCj. The number k is called the 'period' of the cycle, and the point x~ is called a 'periodic point of period k'. We observe that a fixed point is a periodic point of unit period.

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3. A bi-infinite (i.e. infinite in b o t h directions) sequence t5x ks"~+~ . . . . where xi:#xj for i#j. In this case, as in the case o f flows, we will say that such a trajectory is unclosed. W h e n ~0(x) is a diffeomorphism, the cascade is a s m o o t h dynamical system. Examples o f cascades o f this type are n o n - a u t o n o m o u s periodic systems. Let us consider the system .~ = X ( x , t )

where X(x,t) is defined and continuous with respect to all variables in I~n x IW, is s m o o t h with respect to x and periodic of period ~ with respect to t, and has solutions which m a y be continued over the interval 0 < t < r . Given a solution x = qg(t,Xo), where ~o(0,x0) = x0, we m a y define the m a p p i n g x~ = ~0(~,x)

(3)

o f the hyper-plane t = 0 onto the hyper-plane t = ~. It follows from the periodicity o f X(x,t) that (X,tl) and (x, t2) must be identified if ( t 2 - t~) is divisible by ~. Thus, Eq (3) m a y be regarded as a diffeomorphism ~p:~" ~ ~". (We observe that, in this case, system Eq (3) m a y be written as an a u t o n o m o u s system .~ = X ( x , O ) ,

0 = 1

where 0 is taken in m o d u l o ~.) A m o n g cascades o f special interest are the so-called topological M a r k o v chains (TMCs). Let us define a T M C . Let G be an orientable multigraph such that each vertex is the beginning o f a certain edge, and at the same time, the end o f the same edge or a n o t h e r edge. Let us enumerate the edges o f G by symbols 1,2, • • - ,m and construct an (m × m)-dimensional matrix A(aij) according to the following rule: aij = 1 if the end of edge i is the beginning o f edge j; a~j = 0 otherwise. F o r example, for G in the form

the matrix A is

A =

"1

1

0

0

0

0

0

1

1

1

1

1

0

0

0

0

0

1

1

1

0

0

1

1

i

Further, let us consider a set f~ o f sequences, infinite in b o t h directions, c o m p o s e d of symbols 1, - - • ,m,

Mathematical Problems of Nonlinear Dynamics =

{("

•

• ~,_~,~0~o,

" •

'o~k"

799

• • )}

with a fixed position o f the zero coordinate such that the symbol ~o~+ l m a y follow the symbol ~k if and only if a~kOgk+1 = 1. In other words, f~ can be identified by a set o f paths, infinite in b o t h directions, along the g r a p h G. Let us define a distance dist on ~:

dist(c~,fl) =

~,

21il

for~=(• .~_~0- " "),fl=(" • 'fl-~fl0" " " ). It is a simple matter to verify that f~ with the distance d is a complete metric space. It can also be shown that if f~ has the cardinality o f continuum, it is h o m e o m o r p h i c to a reference C a n t o r set on the segment [O,l]. Let us define a t r a n s f o r m a t i o n tr on f~: crco = • ~o' if ~o =

(.

•

. ~ .

~'=(-''~o mk' = mk+l,

• • ~O_lm0'

• ")

,'m0"'')

k~Z

that is, in the sequence o) the position o f the zero coordinate is translated (shifted) by one symbol to the right: ( • - • ~o_~m0co~ • • • ). The t r a n s f o r m a t i o n a is called a shift. It can be easily verified that a is single-valued and continuous together with the inverse m a p p i n g o - i . It is the cascade (ak,t)) which is called a topological M a r k o v chain (TMC), and this will be denoted hereafter as (G,fLa). If there exists a path f r o m any vertex o f the graph G to any other one, olo is transitive, and if, moreover, h = In 2 , o- has a countable set o f periodic points and they are everywhere dense in ~. Here 2~ is the maximal eigenvalue o f the matrix A, and h is the topological e n t r o p y o f the m a p p i n g a. We introduce also the notion o f a suspension over the T M C . In the direct p r o d u c t x L w h e r e / i s the segment [0,1], we identify the points (e~,l) and (a~o,1) for all ~ef~. The topological space thus obtained is the phase space o f such a system and is denoted by Y (E m a y be provided with a metric). Let us define a flow T' on E by the following relation:

T'(og,s) =

(m,t+s), ((aco,t+s--1),

if if

--s
and for the rest, t e n j, the flow is defined by the condition that {T'} forms a g r o u p (e.g. for - 1 < t < --s,T'(o~,s) = (o l~o,1 +t+s), etc.). It can be seen that the m a p p i n g T' preserves the set o f points {(~o,0)} = f~ invariant and acts on it exactly as a does. In other words, a is the Poincar6 m a p o f the transversal ~ o f the phase space Z for the flow T'. Before proceeding further, we need to introduce some notions. A set A is said to be invariant with respect to a h o m e o m o r p h i s m ~o i f A = ~o(t,A)for any t. Here, ¢p(t,A)denotes the set ~ q~(t,x). It follows from this definition that ifxeA, then the trajectory tp(t,x) lies in A .

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We call a point x0 wandering if there exists an open neighbourhood U(xo) of x0 and a positive integer T such that

U(xo)n~o(t,U(xo)) = ~

fort>T.

(4)

Applying the transformation q~(- t,x) to Eq (4) we obtain

q~(-t,U(xo))nU(xo) = ~

fort
Hence, the definition of a wandering point is symmetric with respect to the positive and negative values of t. Let us denote by ~ the set of wandering points. The set ' ~ is open and invariant. Openness follows from the fact that, together with x0, any point in U(xo) is wandering. The invariance of #~ follows from the fact that if x0 is a wandering point, then the point ~O(to,Xo)is also a wandering point for any to. To show this, let us choose ~p(t0,U(xo)) to be the neighbourhood of the point ~O(to,Xo). Then

q~(to,U(xo))n~o(t,q~(to, U(xo)) = ~

for t > T.

Hence, the set of non-wandering points ~ ' = D. # is closed and invariant. The set of non-wandering points may be empty. To illustrate the latter, we consider a dynamical system defined by the autonomous system

.~ = X(x,O),

0= 1

with phase space ~'+ ~, x = (xl, • • • ,xn). It is clear that equilibrium states, as well as all points on periodic trajectories, are non-wandering. All points on bi-asymptotic trajectories which tend to equilibrium states and periodic orbits as t ~ _+ ~ are also non-wandering. Bi-asymptotic trajectories are unclosed and are called homoclinic trajectories. The points on Poisson-stable trajectories are also non-wandering points.

Definition A point x0 is said to be positive Poisson-stable if given any neighbourhood U(xo) and any T > 0 there exists t > T such that q~(t,x0) c U(xo).

(5)

If there exists t such that t < -- T and Eq (5) holds, then the point xo is called a negative Poisson-stable point. If point x0 is both positive and negative Poisson-stable it is said to be Poisson-stable (see Fig. 1). We observe that if the point x0 is positive (negative) Poisson-stable, then any point of the trajectory ~o(t,Xo) is also positive (negative) Poisson-stable. Thus, we may introduce the notion of a P+-trajectory for a positive Poisson-stable trajectory, a P - trajectory for a negative Poisson-stable trajectory and merely a P-trajectory for a Poisson-stable trajectory. It follows directly from Eq (5) that P+-, P - - and P-trajectories consist of non-wandering points. It is obvious that equilibrium states and periodic orbits are closed P-trajectories. We denote by E the closure of the P+(P-,P)-trajectory.

Theorem 2.1 (Birkhoff) If a P+ (P-,P)-trajectory is unclosed, then Z contains a continuum of unclosed P-trajectories.

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X0

Fig. 1. An unclosed Poisson stable trajectory passing through the point x0 strikes the shaded cross-section inside the E-neighbourhood of the point x0 infinitely many times. Let us choose a positive sequence o f {T,} where T , ~ + ~ as n ~ + oo. It follows from the definition o f a P+-trajectory that there exists a sequence { t , } ~ + ~ as n--* + oo such that q~(tn,Xo)C U(xo). An analogous statement holds in the case of a F trajectory. This implies that a P-trajectory successively intersects any e-neighbourhood UE(xo) of the point x0 infinitely m a n y times. (In the case of flows the P-trajectory intersects U,(xo) for values of t in an infinite set o f intervals In(E) where t~(E) is one of the times in I,(E).)We let { t,(E)} + ~ be such that t,(e) < t, + 1(•) and let ~o(t,(E),Xo)~ Ue(x)o). We introduce the notion o f the Poincar6 return times

~n(E) = t°+ ~(E)- t°(E). T w o essentially different cases are possible for an unclosed P-trajectory: 1. The sequence {r,(O} is bounded, that is, there exists a n u m b e r L(O such that r , ( O < L ( O for any n. We observe that L ( O ~ + ~ as E~0. 2. The sequence {r,(E)} is u n b o u n d e d . In the first case the P-trajectory is called recurrent. F o r such a trajectory all trajectories in its closure Z are also recurrent, and the closure itself is a minimal set. (A set is called minimal if it is n o n - e m p t y , invariant, closed and contains no p r o p e r subsets possessing these three properties.) The principal p r o p e r t y of a recurrent trajectory is that it returns to an E-neighbourhood of the point x0 within a time not greater than L(E). However, in contrast to periodic orbits, whose return times are known, the return time for a recurrent trajectory is not constrained. A m o n g recurrent trajectories we m a y also select

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a more narrow class of motions, the so-called almost-periodic trajectories. There is no uncertainty in the Poincar6 return times of such a trajectory, as it comes back to its neighbourhood over its almost-period z(0. Almost-periodic trajectories may be subdivided into the two sub-classes: quasi-periodic and limit-periodic trajectories. In the case of smooth flows the minimal set of the quasi-periodic trajectories is a torus, whereas the minimal set of the limit-periodic trajectories is a rather non-trivial set, called a solenoid. The local structure of a solenoid may be defined by a direct product ~'~ x IK, where ~ is the Cantor discontinuum. In the second case, the closure E of the P-trajectory is called a quasi-minimal set. There always exist in Z other invariant closed subsets which may be equilibrium states, periodic trajectories or invariant tori, etc. As the P-trajectory m a y approach such subsets arbitrarily closely, the Poincar6 return times can be arbitrarily large. Let us suppose that for a trajectory L given by the equation x = ~0(t) the closure of the semitrajectory L ÷ ( L - ) for t > to ( t < to) is a compact set.

Definition The point x0 is called an ~o-limiting point of the trajectory L if there exists a sequence {t~} where tk-~ + ~ as k ~ such that lira q~(tk) = x0. A similar definition for an s-limiting point t k ~ - 0o as k ~ ~ . We denote the set of all ~o-limiting points of the trajectory L by I)L, and that of the s-limit points we denote by AL. We observe that an equilibrium state has a unique limit point, namely, itself. In the case where the trajectory L is periodic all of its points are ~- and oJ-limit points (i.e. L = ~L = ~'L). In the case where L is an unclosed Poisson-stable trajectory the sets ~L and ~'L coincide with its closure L. This L is either a minimal set if L is a recurrent trajectory, or a quasi-minimal set if the return Poincar6 times of L are unbounded. All equilibrium states, periodic trajectories and Poisson-stable trajectories are said to be self-limit trajectories. The structure of the sets ff~L (alL) has been almost completely studied for twodimensional dynamical systems on the plane where the trajectory remains in some bounded domain of the plane as t-~ + ~ ( t - ~ - ~ ) . Both sets ~L and /~L are well known to be invariant and closed. In the case when the system under consideration is a flow, ~L (~L) is then a connected set. Poincar6 and Bendixson established that the set ~L m a y be of one of the following topological types: (I) equilibrium states; (II) periodic trajectories; (III) contours composed of equilibrium states and connecting trajectories tending to these equilibrium states as t ~ + ~ . Figure 2 represents examples of limit sets of the third type where we label the equilibrium states by O. Using the general classification above we may enumerate all types of positive semi-trajectories of planar systems: (1) equilibrium states; (2) periodic trajectories; (3) semi-trajectories tending to an equilibrium state; (4) semi-trajectories tending to a periodic trajectory; (5) semi-trajectories tending to a limit set of Type III. We observe that an analogous situation occurs in the case of negative semitrajectories. In the general case, besides the above types of limiting sets the closure of trajectories of flows on two-dimensional compact surfaces may also be a minimal set as well as a quasi-minimal set.

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803

(a)

(b)

Fig. 2. Examples of the heteroclinic contours. Let us discuss the second problem concerning the study of the totality of trajectories. In fact, determining a dynamical system means topological (or qualitative) partitioning of the structure of the phase space by trajectories of different topological types, or in other words, finding its phase portrait (1). This poses the question of when two phase portraits are similar. In terms of the qualitative theory of dynamical systems we can answer this question by introducing the notion of to the so-called topological equivalence.

Definition Two systems are said to be topologically equivalent if there exists a homeomorphism of the phase spaces which maps trajectories (semi-trajectories, intervals of a trajectory) of one system into trajectories (semi-trajectories, intervals of a trajectory) of the second. This implies that equilibrium states are mapped into equilibrium states, periodic trajectories and unclosed trajectories of one system are mapped into periodic trajectories and unclosed trajectories of another system. The topological equivalence of two systems

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in some sub-regions of the phase space is defined in a similar manner. The latter is used in studying local problems, for example, in studying the equivalence of structures of trajectories in a neighbourhood of an equilibrium state, or near periodic or homoclinic trajectories. This definition of two topologically equivalent dynamical systems is an indirect definition of the notion of the topological or qualitative structure of the partition of the phase space into trajectories. We may say that such a structure preserves all properties of the partition, which remains invariant with respect to all possible homeomorphisms applied to the phase space. Let G be a bounded sub-region of the phase space and let H = {hi} be a set of homeomorphisms of G mapping trajectory into trajectory of the same topological types. Then we can introduce a metric distance as follows: dist(h~,h2) = sup Ilhlx-h2xll. xEG

Definition

We call a trajectory LEG particular if there exists E> 0 such that for all h satisfying dist > (h,/)< E, where I is the identity homeomorphism, the following condition holds: h L = L.

It is clear that all equilibrium states and periodic orbits are particular trajectories. Unclosed trajectories may be particular also, for example, particular trajectories of a two-dimensional system which tend to equilibrium states both as t ~ + ~ and as t--* -~. As such trajectories separate certain regions in the plane they are called separatrices. (See samples of separatrices in Fig. 2.) The definition of particular semitrajectories m a y be introduced in an analogous manner. Definition

Two trajectories L1 and L2 are said to be topologically equivalent if for given e > 0 there exist homeomorphisms h~,h2, • " " ,hm(Q, satisfying dist(hk,/)< E, such that L2

=

hm(E)

"

'

'

haLl

where k = (1,2, - - - ,re(E)) and I i s the identity homeomorphism. We will call the set of equivalent trajectories a cellar. We observe that all trajectories in a cellar are of the same topological type. In particular, if a cellar is composed of unclosed trajectories, then all of them have equal co-limiting and a-limiting sets. The roles of particular trajectories and cellars are especially important for twodimensional systems. In this case, we may compose a set S consisting of the particular trajectories and of one trajectory from each cellar. We will call this set S a scheme. (Indeed, the set S is a factor-system over a given relationship of equivalence.) Let us suppose that S consists of a finite number of trajectories. (The condition of finiteness of S is rather c o m m o n for a wide class of planar systems.) Theorem 2.2

The scheme is a topological invariant. This theorem together with its p r o o f occupies the significant part of the book The Theory o f D y n a m i c a l S y s t e m s on a Plane by Andronov et al. (2). This is not the case when we examine systems of higher dimensions. The set of

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805

particular trajectories of a three-dimensional system already m a y be countable or a continuum even. The same situation applies to the cellars. All trajectories of topological M a r k o v chains and those of suspensions over a T M C are also particular.

Definition An attractor is a closed invariant set A which possesses a neighbourhood (an absorbing area) U(A) such that the trajectory ~o(t,x) of any point x in U(A) satisfies the condition

dist((~o(t,x),A)-~O as

t--* + ~

(6)

where dist(x,A) = inf IIx,x011. Xo~,~

The simplest examples of attractors are stable equilibrium states, stable periodic trajectories and stable invariant tori containing quasi-periodic trajectories, which satisfy Eq (6). This definition of an attractor does not preclude the possibility that it may contain other attractors. It is reasonable to restrict the notion of an attractor by requiring a quasi-minimality condition. The essence of this requirement is that A is to be a transitive set. A set M is said to be transitive if it contains an everywhere dense trajectory L (e.g. L = M). The most interesting attractors are the so-called strange attractors, which are invariant, closed sets composed of only unstable trajectories, which are, in fact, the particular trajectories. The theme of strange attractors and of mechanisms of their appearance is of particular interest in this paper.

IlL Andronov-Pontryagin Theorem;Morse-Smale Systems We will assume that all dynamical systems under consideration are in the form

.~c= X(x) where xeC ~ and are defined in some closed, bounded region t ~ c ~". For such vector fields we can introduce a norm as follows:

I)

IlXllc, = ~p (llX(x)ll+ ~X(X)ox

Having introduced this norm, the set of dynamical system becomes a Banach space of dynamical systems.

Definition The system X(x) is called rough in G if for any E> 0 there exists 6 > 0 such that if ll)?-X[I < 6, then X and a neighbouring system ~ are topologically equivalent. Moreover, the conjugating h o m e o m o r p h i s m h is close to the identity h o m e o m o r p h i s m I (i.e. dist(h,/) < e). This definition was first introduced by Andronov and Pontryagin (3) under the additional requirement that the boundary ~(~ of the region ~ is a surface without a contact (e.g. the vector field is transverse to ~(~) and, moreover, is directed inward G.

806

LeonidShilnikov

The problem of defining the region G does not exist if we consider vector fields on compact, smooth manifolds. In this situation the notion of the roughness may be introduced in a similar manner with the only difference that the set of associated vector fields is a Banach manifold. Another notion close to the notion of the roughness is that of structural stability.

Definition The system X(x) is called structurally stable in (~ if there exists an e > 0 such that if I l k - Xll < E, then X and k are topologically equivalent. It follows immediately from this definition that structurally stable systems form an open set in the Banach space of dynamical systems, whereas in the case of rough systems this is not so obvious. Nevertheless, the notion of a rough system is more 'physical' in the sense that it reflects the fact that small perturbations of the original vector field cause small changes of the associated phase portrait. However, it follows from recent studies on structurally stable systems that there exist always 6-homeomorphisms. For a two-dimensional system of the form

Yc = P(x,y)

(7)

= Q(x,y) Andronov and Pontryagin proved the following theorem.

Theorem 3.1 The system given by Eq (7) is rough if and only if 1. All equilibrium states of Eq (7) are simple, that is, if none of the roots of the characteristic equation det L Q'~(xo,yo)

l = 0

Qy(xo,yo)- 2_]

of the associated linearized system at each equilibrium state lies in the imaginary axis in the complex plane. 2. All periodic trajectories of Eq (7) are simple, that is, i f x = ~0(t),y = ~k(t) is a periodic solution of period z, then

l Px'(qg(t),~(t)) + Qy'(~o(t),t~(t)) dt ~ O. 3. There exist no trajectories which are bi-asymptotical to a saddle, as well as those going from one saddle to another saddle as t ~ ___oo, for example, the system has neither homoclinic nor heteroclinic trajectories. We call such equilibrium states and periodic trajectories rough or structurally stable. In the Banach space of two-dimensional dynamical systems, structurally stable (or rough) systems form an open, everywhere dense set. It follows from this theorem that the set of non-wandering trajectories of structurally stable systems on the plane is composed of only equilibrium states and periodic tra-

M a t h e m a t i c a l Problems o f Nonlinear D y n a m i c s

807

jectories. Moreover, a structurally stable systems may possess only a finite number of such particular trajectories. This is one reason why the scheme is a topological invariant of a structurally stable dynamical system. Peixoto (56) has shown that the analogous situation occurs in the case of smooth flows on two-dimensional surfaces. The principal feature in his p r o o f is that structurally stable flows on two-dimensional surfaces possess neither minimal nor quasi-minimal sets. Let us consider next some features of high-dimensional dynamical systems. The equilibrium state O: x = 0 of an n-dimensional systems of differential equations in ~n (for simplicity) Jc=X(x),

XEC k,

k
(8)

is called structurally stable if the roots (21, • • " ,2n) of the characteristic equation (which are called characteristic exponents of the equilibrium state), det OX(0) - 2 E

= 0

(9)

do not lie on the imaginary axis. A structurally stable equilibrium state will be assigned a topological type (m,p) where m is the number of roots in the open left half-plane, and p is the number of the roots in the open right half-plane such that m + p = n. If m = n (m = 0), the equilibrium state is called a stable (unstable) node. When m # n and m # 0, the equilibrium state is called a saddle. If, say, for example, n = 3, m = 3 and the roots with negative parts are complex-conjugate, O is called a saddle-focus. The set of all points of the phase space such that trajectories passing through these points tend to O as t ~ + ~ (t--, - ~ ) is called a stable (unstable) manifold W5 (WS) of the equilibrium state O. It is known that if O is of (m,p)-type, the Ck-smooth manifolds W5 and W5 have dimensions m and p, respectively, and are each submanifolds of ~", diffeomorphic to W" and ~P near O. Let us assume that the system of Eq (8) has a periodic trajectory L: x = q~(t) of period r. Let us write for L the equation in variations: _ OX(~o(t)) Ox

= B(t)~.

(10)

To study the behaviour of trajectories in the neighbourhood of L it is frequently more convenient to study their traces on a transversal. We let S be a smooth, ( n - 1)dimensional disc orthogonal to L at the point of intersection L u S . Let us introduce on S Euclidean coordinates s = (s~, - • • ,s,_ 1) such that the point 0 of the intersection L u S has the coordinates 0 = (0, • • • ,0). The mapping T:S~-~ S along trajectories passing in the neighbourhood of L, which relates the point s o t S to the point of the first intersection with the transversal S, is called the Poincar6 mapping. It can be written in the form ~ = A u + • . . , where A is an ( n - - 1 ) × (n--1)-dimensional matrix whose eigenvalues are multipliers of the periodic trajectory L. A set of all points of the phase space, such that trajectories passing through these points tend to L as t ~ + ~ ( t ~ - ~ ) is called a stable (unstable) manifold W~_ (W~,) of the periodic orbit L. It is clear that L belongs simultaneously both to W~ and WT.

808

Leonid Shilnikov

The manifolds W~ and W~ are each smooth submanifolds of ~" in the neighbourhood of their points, and have the dimensions m and p, respectively. For n = 3 and m = p = 2, W~ (W~_) is homeomorphic to a M6bius band, if the multiplier of L which is less (greater) than unity, in modulus, is negative, and to a cylinder if it is positive. In the latter case, the periodic orbit divides WE (W~) into two connected pieces (see Fig. 3). Let us now define the transverse intersection of stable and unstable manifolds. Let W sL I and W u be stable and unstable manifolds of periodic trajectories or equilibrium L 2 states, and W ~ n W~2 = ~ . W~, and W~, are said to intersect each other transversally if

(a)

~.s~

~

~

(b)

) Fig. 3. The stable and the unstable manifolds of a saddle periodic orbit may be homeomorphic to either a cylinder (a) or to a M6bius band (b).

Mathematical Problems of Nonlinear Dynamics

809

dim T~W~L,+ d i m TxW~2-n = dim (T~W~LwT~W~2)

(11)

where T~W[, (T~W~) denotes a tangent to W~_, ( W ~ ) at the point x. The property of intersection transversality does not change under small perturbations of a system and is, in this case, a structurally stable one. We note that a saddle periodic trajectory L belongs to the transverse intersection of its stable and unstable manifolds. All other trajectories belonging to W~c~ W~. are homoclinic curves (or trajectories). Those of the curves along which W ~ and W" intersect transversally are called structurally stable homoclinic curves. We select one more type of trajectories which connect saddle equilibrium states or (and) saddle periodic trajectories such that dim T~W~, + d i m T~W~ = n

(12)

where L~ :~ L2. We will call such trajectories heteroclinic. We observe that all mentioned trajectories are particular. Structurally stable diffeomorphisms are introduced in an analogous manner to structurally stable vector fields. Usually, it is added that the following diagram is cornmutative: G

h

G

h,

G

+x G

It is clear that in the case where a diffeomorphism is defined on a compact phase space the question of the correspondence of the image and its pre-image is resolved at the initial stage, that is, such a phase space is invariant under the action of the diffeomorphism. We note also that in the case of diffeomorphisms instead of the notion of the topological equivalence we use the notion of the topological conjugacy, or simply, the conjugacy. We consider next a diffeomorphism defined on ~" or on some subregion of ~n. We let O:x = 0 be a fixed point of the diffeomorphism .~=X(x),

X~C r,

r>l

(13)

such that 0 = X(0). The point 0 is called structurally stable if none of the roots of the characteristic equation d . IOX(O) et ~ - - p z ~

~I

= 0

of O lies on the unit circle in the complex plane. A structurally stable fixed point will be assigned a topological type (m,p) where m is the number of roots inside the unit circle, and p is the number of the roots outside of it. If m = n (m = 0), the fixed point state is called a stable (unstable) node. If m :~n, 0, O is called a saddle. The set of all points of the phase space such that trajectories passing through these points tend to O as t-~ + ~ ( t ~ - ~ ) is called a stable (unstable) manifold W~ (W~) of the equilibrium state O. It is known that if O is of type (m,p), the Ck-smooth manifolds Wg and W~ have dimensions m and p, respectively, and are each submanifolds of ~ , diffeomorphic to R '~ and ~P near O.

810

Leonid Shilnikov

Let C = (x0,xl, • • " ,Xq_ 1) be a periodic trajectory o f period q, that is, X 1 =

X(x0)

,

x2

=

X(Xl)

,

•

.

. ,

x0

~-- ~ ] l " ( X q _ 1 ) .

It is evident that each point x~, i = O, • • • , q - 1 is a fixed point o f the m a p .~ = X q ( x )

where

= x((. q-1 t i m e s If the roots o f the characteristic equation det o X q ( x ° )

pE

~x

= 0

or, w h a t is the same, the roots o f the equation det ~ X ( x q 8x

1) O X ( x q 8x

2) " " "

OX(xo) 8x

pE = 0

do not lie on the unit circle, then such a periodic trajectory is called structurally stable. If all roots lie inside (outside) the unit cycle, C is a stable (unstable) node. If m roots lie inside and p = n - m roots lie outside the unit circle, C is o f the saddle type. Its stable (unstable) set consisting o f m Ck-smooth manifolds W sX o ~ W x. .I.,. , W "X q _ I (W~ o, W~,,, - • • , W Xq u ,) are m a p p e d consecutively to each other under the action o f the diffeomorphism. Just as for vector fields, we m a y introduce the condition o f transverse intersection o f stable and unstable manifolds o f fixed points and periodic trajectories. We notice also that if CI(Xo,Xl, ~ ~ • • • ,Xq~ ~ , ) and C 2 ( x o2, x l2, • • • ,Xq2 2 ) are two periodic trajectories, then in Eq (12), W~ m u s t be replaced by one o f W~'o,W~',," " • , W ~ , q T , , u u u2 . . . u and WL by one o f Wx0~, W w , Wx~-,. We assume that O is a fixed point o f the saddle type. Its stable and unstable manifolds m a y intersect each other along a trajectory other than O as shown in Fig. 4 for ~2. Such a trajectory is homoclinic. Here, W~ and W~ are one-dimensional. We observe that if there is a structurally stable homoclinic point in W S ~ W~, there is at least one more. The same situation applies to homoclinic trajectories o f a saddle periodic trajectory. Let L1 and L2 be saddle periodic trajectories o f the same type. Then W sL I and W"L 2 m a y intersect each other transversally along only the isolated trajectories. These trajectories are called heteroclinic trajectories. It is clear that both homoclinic and heteroclinic trajectories along which stable and unstable manifolds intersect transversally are structurally stable. Let us n o w consider the class o f C~-smooth dynamical systems satisfying the following conditions: (1) the n o n - w a n d e r i n g set consists o f a finite n u m b e r o f structurally stable equilibrium states and periodic orbits; (2) the stable and unstable manifolds o f equilibrium states, and periodic trajectories, intersect transversally. Such systems are called M o r s e - S m a l e systems. They are structurally stable, as was established by Palis (55) and Smale (66). In a certain sense, the M o r s e - S m a l e systems are a high-dimensional

Mathematical Problems of Nonlinear Dynamics

811

Fig. 4. A transverse homoclinic orbit. generalization of rough systems of Andronov and Pontryagin. In contrast to the Andronov-Pontryagin systems, which have only a finite number of particular trajectories, the set of such trajectories, in particular, homoclinic and heteroclinic trajectories, in the Morse-Smale system may be countable. Afraimovich and Shilnikov (8, 9) have shown that, as the existence of heteroclinic trajectories is unclosed, particular trajectories of the Morse-Smale systems may, in particular, lead to a complex structure of the wandering set. Loci of heteroclinic trajectories may be described in the language of the suspensions over topological Markov chains. We see that the Morse-Smale systems comprise a simplest class amongst all highdimensional dynamical systems. However, it appears that we can only point out one complete topological invariant for the Morse-Smale systems. The only known result deals with three-dimensional systems with a finite number of particular trajectories (82). (The problem of finding a complete topological invariant for systems which nowadays are called Morse-Smale systems was posed by Andronov.) Let us discuss next the Morse-Smale diffeomorphisms. A diffeomorphism is called a Morse-Smale diffeomorphism if it satisfies the following conditions: (1) its nonwandering set consists of a finite number of structurally stable periodic trajectories (a fixed point is a periodic trajectory of period one); (2) stable and unstable manifolds of periodic trajectories intersect transversally. There exists also the problem of finding a complete topological invariant. This problem has been studied only for diffeomorphisms on closed, two-dimensional surfaces provided that the diffeomorphism is gradient-like, or has an orientable heteroclinic set, or has a finite set of heteroclinic trajectories (12, 14). A common feature of both MorseSmale flows and cascades is the absence of cycles.

Definition Let LI • • Lq be either equilibrium states or periodic trajectories, and let FI, - • • , Fq be trajectories satisfying the following conditions: ~ ( F k ) = Lk+l and ~¢(Fk)= Lk (k = 1, q-- 1), fft(Fq) = L~ and ~ ( F q ) = Lq. Here we denote by ~) and ~ the ~o-limiting •

812

L e o n i d Shilnikov

set and the a-limiting set, respectively. The collection (L~,F1, - • ,Lq,Fq) is then called a cycle. Furthermore, the Morse-Smale system cannot have cycles which contain equilibrium states and periodic trajectories both of different topological types, as this violates the condition of transversality for some homoclinic or heteroclinic trajectories. Otherwise, if we suppose the Morse-Smale system has such a cycle, it is only possible when all Lis are periodic trajectories, and of the same type. We can show then that each Li has a structurally stable homoclinic trajectory and, as a result, a countable number of saddle periodic trajectories in its neighbourhood, which contradicts the definition of the Morse-Smale systems. •

IV.

Poincarb Homoclinic Structures

While studying the bounded problem of three bodies in Cartesian mechanics, Poincar6 discovered homoclinic structures (57). He established that if the stable and unstable manifolds of a saddle periodic trajectory intersect transversally along one homoclinic curve, then there are also infinitely many of such curves. The study of Poincar~ homoclinic curves was continued by Birkhoff, who proved the existence of a countable set of periodic trajectories in the neighbourhood of a homoclinic curve of a volumepreserving diffeomorphism. Andronov was the next who posed the problem of constructing structurally stable three-dimensional flows and two-dimensional diffeomorphisms possessing homoclinic structures, or, in other words, having a countable set of periodic trajectories. The essence of the problem is the following. Let us consider a periodically forced non-autonomous system :c = P ( x , y ) + #p(t) = Q ( x , y ) + I~q(t)

(14)

where p(t) and q(t) are periodic functions of period z. We assume that when p = 0 (Eq (14)) has a homoclinic loop F(0) to a saddle point at the origin O(0,0). When # ¢ 0, the system of Eq (15) may be represented as an autonomous system of the form 5c = P ( x , y ) + pp(O) = Q ( x , y ) + #q(O)

0= 1

(15)

in the phase space ~2 X ~1. In fact, for all # sufficiently small we may reduce the study of Eq (15) to the consideration of the Poincar6 map on the cross-section 0 = 0. For small/~ the diffeomorphism X = F(x,y) f = G(x,y)+#q(O),

0mod2n

(16)

has a fixed point O(/z) of the saddle type. Moreover, the stable W~ and unstable W~ manifolds of the point O(p) may transversally intersect each other along the homoclinic curve F(#) for all Ilp[I < #0. (Such a situation occurs, for instance, when the Melnikov

Mathematical Problems of Nonlinear Dynamics

813

function used for estimating the splitting of separatrices has simple zeros (47).) One can show that within the interval II#H
As periodic, homoclinic and Poisson-stable trajectories are everywhere dense in the Bernoulli subshift, the same occurs in f~. The construction proposed by Smale is called 'a Smale horseshoe'. His idea is purely geometrical. In the simplest case for an analytically defined diffeomorphism, this idea is realized in the H6non map (42) re=y,

~ = l +ay2-bx

(17)

provided that a > [(5 + 2~/5)/4](1 + Jbl)2. Considering the horseshoe, Smale showed that in the neighbourhood of a structurally stable homoclinic curve of a saddle fixed point of a diffeomorphism T, there exists an invariant set whose image under T m for sufficiently large m is homeomorphic to the Bernoulli subshift on two symbols, under the assumption that T may be linearized near the saddle point. In fact, this explains why there are no cycles in the Morse-Smale systems. The study of homoclinic curves has induced another problem which goes back to Birkhoff (17), namely, the problem of complete description of all trajectories lying entirely in a sufficiently small neighbourhood of the saddle fixed point and its structurally stable homoclinic trajectory. The answer was given by the author as follows. Let us suppose we have a saddle periodic orbit L with a homoclinic trajectory F (Fig. 5). We surround L and F by a neighbourhood U which has the shape of a solidtorus to which a handle containing F is glued (Fig. 6). We shall code a trajectory lying in U using the following rule: if a trajectory makes a complete circuit within the solidtorus, we write a '0', and if it goes along the handle, we write a '1'. Thus, the sequence {0}~; + ~

corresponds to the periodic orbit L. The sequence {...

0,0,L0,0,...}

corresponds to the homoclinic orbit F. The sequence {J.},~= +2,

j . = {0, 1}

corresponds to an arbitrary trajectory within U. Moreover, each 1 is followed by zeros

814

Leonid Shilnikov r

L Fig. 5. A homoclinic curve F to the saddle periodic orbit L.

// I "

u Fig. 6. The neighbourhood of L and F has the shape of a solid-torus with a handle.

whose number is not fewer than/~, where/~ depends on the size of the neighbourhood the narrower the neighbourhood U we choose the bigger/~ will be. There is another more suitable algorithm of coding. We introduce a symbol 1 =

i,'o,

...,

6

Then we obtain a new truncated sequence {j,}~=_+~,

j , = { 0 , i}

.

Mathematical Problems o f Nonlinear Dynamics

815

for such a trajectory, wherejn can be followed by either 1 or 0. In other words, the set o f trajectories lying in U is in correspondence with the Bernoulli scheme on two symbols. (To be m o r e precise, this set is h o m e o m o r p h i c to a suspension over the Bernoulli scheme.) Moreover, in this case the converse statement is valid also. Let us illustrate this statement with an example o f a diffeomorphism T

.re = 2x + P(x,y) Y = 7Y+ Q(x,y)

(18)

where 0 < 2 < 1, 7 > 1, and P(x,y) and Q(x,y) vanish at the origin O(0,0) along with their first derivatives. In this case, O is a fixed point o f the saddle type. We assume that its manifolds W s and W u intersect transversally along a homoclinic trajectory F. Let us choose a pair o f homoclinic points M + and M - in a small n e i g h b o u r h o o d o f O such that M+~W~o~ and M-~W~oc. We s u r r o u n d the point M + by a small rectangle H + and the point M - by a small rectangle I I - as shown in Fig. 7. The condition o f smallness o f both rectangles is that they do not intersect with their images under the m a p T. Then, inside 1-I+ and I I - we m a y define a countable set o f 'strips' {a °} and {a),}, where k
M1 17I T'I(II" ) {r o

K ~.~..___ II +

T-K (I"/- ) r

--

r

/r-(K+1)(n- ) Fig. 7. Illustration of the method of a construction of strips a °, k =/~,E+ 1 . • . which lie in H + and the domains of the maps T0kH+~--~H-. The strips ~ accumulate to the segment Ws~FI + as k~oo.

816

Leonid Shilnikov ]

•

•

•

M

I

iinl

r l l i l l ~ i i ~ f

i

l i l l ~

o? {~j

<

0

M°

Fig. 8. Under the action of the global part of the Poincar6 map, an image of an initial strip intersects the initial strip forming a horseshoe. infinite sequence t~k ct)~+ o~ ~ where k < k. In this case, the invariant set f~ lying entirely in a n e i g h b o u r h o o d o f O w F is h o m e o m o r p h i c to a topological M a r k o v chain with the g r a p h d r a w n in Fig. 9. We note that, in agreement with (15), the suspension over this g r a p h is h o m e o m o r p h i c to the suspension over a Bernoulli subshift on two symbols. It is i m p o r t a n t that here ~ is a hyperbolic set. V. Structurally Stable Systems With Complex Dynamics; Hyperbolic Attractors The n o t i o n o f a hyperbolic set is m o r e conveniently introduced first o f all with the example o f cascades ~p(k,x). We recall that a diffeomorphism in ~n inducing as a cascade is denoted by .~ = X(x).

(19)

We let {,}+~_ ~ be a trajectory. Then we m a y define an infinite-dimensional system o f linear mappings Yk+l = A~yk,

--~
+~

(20)

where Ak = OX(Xk)/OX. Eq (20) is an analogue o f the variational equation for flows. t+~ Defining Eq (20) implies that we have an infinite sequence ts E kSk = - ~ o f linear n+oo dimensional spaces and operators {Ak}k = - ~ such that Ak •

"

"

~

E

k

--+

Ek+,o

•

"

"

•

(21)

Mathematical Problems o f Nonlinear Dynamics

817

1

-k+m Fig. 9. The graph of a TMC for f~ set of the trajectories lying entirely in the neighbourhood of OwF.

We

assume that each E~ admits the representation Ek = E ~ ® E ~ such that E~+~ = AkE~, E~+~ = AkE~, and dim E~, is the same for all k. F o r a trajectory to be

hyperbolic, it is required that in the sequence Ak •

.

._,~,+1_o.~

, ~

.

.

AA L

• • • *--ET,*-

E~+x*--

• - •

(22)

there exists an uniform contraction of the exponential type. In the general case, the formalization of hyperbolicity is the following• A trajectory ~o(k,x) is called hyperbolic if the following conditions hold: IIDqg(k,x)~ 1]< a 1]~ IIe

,k

I[Oq~( -- k,x)~ II > b II¢ IIe ck IlO~o(-k,x)qll >bHr/lle c~

tlO ~ 0 (

- k,x)r/II < b

IIrt rle - c ~

(23)

where ~eE~ and qeE~, and a, b and c are positive constants independent of ~, r/and k. Here we denote the differential of the mapping by D (e.g. D~o(k) = A~_ 1A~ 2 • • " Ao). The hyperbolic set of the cascade ¢p(k,x) is an invariant set o,# such that at each point x e ~ / t h e tangent space E~ may be decomposed into the direct sum s

u

Ex = E x ® E ~

of two subspaces, stable (contracting) E~ and unstable (expanding) E~ such that for all ~eEi~ and qeE~, k > 0 estimates Eq (23) hold where a, b and c do not depend on xeo¢/. The fact that E~ is a tangent subspace is needed to define the notion of a hyperbolic set for cascades on manifolds. In this case, the norm of the tangent vectors is taken with respect to a Riemannian metric on the manifold. Moreover, choosing a proper metrics in [W as well as a Riemannian metric we can equate the constants a and b in Eq (23) to unity. Then, inequalities in Eq (23) can be recast as

818

Leonid ShiInikov IIDcp(l ,x)~ I1< 2 Ih~ II 1

IlOq~(- 1,x)~ll > ~ 11411 1

IlOq~(1,x)r/ll >_ ~ IIr/ll IIO~o(- 1,x)q II < '~ IIrt II

(24)

where 0 < 2 < 1. We consider next the case of the flow q/determined by the vector field V(x) on a manifold. The hyperbolic set of such a flow is a compact, invariant set ~ such that 1. if a point of J¢/is an equilibrium state, then the equilibrium state is hyperbolic, that is, its eigenvalues do not lie on the imaginary axis in the complex plane. 2. The set ~ ' is closed and at its each point x the tangent space Tx is decomposed into the direct sum E~OE~@E ° of its linear subspaces, the third of which is generated by the vector of the phase velocity, and the first two properties are analogous to the discrete time case, that is, E~, and E~ satisfy condition Eq (23). Examples of hyperbolic sets are equilibrium states, periodic and heteroclinic trajectories (including their closures) of the Morse Smale systems, as well as a set consisting of the points lying entirely in a neighbourhood of a structurally stable homoclinic curve. It is natural that in the general case a hyperbolic set is foliated into non-intersecting, closed, invariant sets J / j = { x e ~ , dim E~: = j}. Of special interest here are the systems satisfying the condition of hyperbolicity in the entire phase space. Such flows and cascades are called Anosov systems. Anosov proved that hyperbolic systems are rough or structurally stable. The peculiarity of Anosov systems is that all of their trajectories are particular. This is why, in the case of the Anosov cascades, the homeomorphism ofconjugacy of two close diffeomorphisms is unique. Examples of Anosov systems are geodesic flows on compact, smooth manifolds of a negative curvature (11). It is well known that such a flow is conservative and its set of non-wandering trajectories coincides with the phase space. An example of the Anosov diffeomorphism is a mapping of an n-dimensional torus 0 = AO+f(O),

mod 1

(25)

where A is a matrix with integer elements other than unity such that detlA[ = 1, and f(0) is a periodic function of period unity. The condition of hyperbolicity of Eq (25) may be easily verified for one pure case of diffeomorphisms of the type O=AO,

modl

(26)

Mathematical Problems of Nonlinear Dynamics

819

which are the algebraic hyperbolic automorphism of a torus. Automorphisms Eq (26) are conservative systems whose set f~ of non-wandering trajectories coincides with the torus Y" itself. Nevertheless, we must remark that there are Anosov flows whose f2 does not coincide with the associated phase space. Conditions of structural stability of high-dimensional systems were formulated by Smale. These conditions are in the following: a system must satisfy (1) Axiom A and (2) a strong condition of transversality. Axiom A requires that: (1A) the non-wandering set ~ be hyperbolic; (1 B) f~ = Per. Here Per denotes the set of periodic points. Under the assumption of Axiom A, the set ~ can be represented by a finite union of non-intersecting, closed, invariant, transitive sets ~1, " " • ~p. In the case of cascades, any such ~ can be represented by a finite number of sets having these properties which are mapped to each other under the action of the diffeomorphism. The sets ~ , - • • ~p are called basis sets. A condition of strong transversality is the following: the location of the manifolds W sLI and W ~ of any two trajectories L~ and L2 is in the general position, that is, either they do not intersect each other or they intersect each other transversally. We remark that by virtue of the H a d a m a r d - P e r r o n theorem each hyperbolic trajectory has its stable and unstable manifolds whose smoothness is equal to the smoothness of the system.

Theorem 5.1 (Robinson) If a dynamical system satisfies Axiom A and the condition of strong transversality, then the system is structurally stable. For necessary conditions we have the following theorem: Theorem 5.2 (Mane (46)) A structurally stable cascade satisfies Axiom A and the condition of strong transversality. In the case of flows there are no explicit statements of the same kind. Nevertheless, it follows, more or less, from a series of recent results that the conditions formulated by Smale are necessary for flows also. The basis sets of Smale systems (satisfying the enumerated conditions) m a y be of the following three types: attractors, repellers and saddles. Repellers are the basis sets which becomes attractors in backward time. Saddle basis sets are such that may both attract and repel outside trajectories. The most studied saddle basis sets are onedimensional in the case of flows and null-dimensional in the case of cascades. The former are homeomorphic to the suspension over topological M a r k o v chains; the latter are homeomorphic to simple topological M a r k o v chains (18). For other basis sets the situation is more difficult. Currently, we do not have any proper classification for them. It is known only that some m a y be obtained from M a r k o v systems under suitable gluing and change of time (19). Attractors of Smale systems are called hyperbolic. The trajectories passing sufficiently close to an attractor of a Smale system satisfy the condition dist(q~(t,x),A)
t>0

where k and 2 are some positive constants. As we have said earlier, these attractors are

820

Leonid Shilnikov

transitive. Periodic, homoclinic and heteroclinic trajectories as well as Poisson-stable ones are everywhere dense in them. In particular, we can note one more of their peculiarities: the unstable manifolds of all points of such an attractor lie within it, that is, W~eA where xeA. Hyperbolic attractors may be smooth or non-smooth manifolds, have a fractal structure, and are not locally homeomorphic to a direct product of a disc and a Cantor set. Below we will discuss a few hyperbolic attractors which might be curious for nonlinear dynamics. The first example of such a hyperbolic attractor is the Anosov torus q]-" with a hyperbolic structure on it. The next example of a hyperbolic attractor was designed by Smale on a two-dimensional torus by means of a 'surgery' operation over the a u t o m o r p h i s m of this torus with a hyperbolic structure. This is the so-called D A (derived from Anosov) diffeomorphism. An original two-dimensional diffeomorphism is taken in the form

01 = allOl +a1202,

(mod 1)

02 = a2101-k-a2202,

(rood 1)

(27)

This diffeomorphism possesses two invariant foliations given by equations 01=2101+c,,

(modl)

02 = 2202q-¢2,

(rood 1)

(28)

where 21 and 22 are the roots of the characteristic equation det a l l - - 2 a21

a12 2 = 0

(29)

a22 --

and c~ and c2 are constant, 0 < c,.2 < 1. Both roots, because of assumptions of hyperbolicity, are always irrational. The stable foliation corresponds to the root 1211< 1, the unstable foliation corresponds to the second root 1221 > 1. The leaves of the stable (unstable) foliation are the stable (unstable) manifolds of the points of the torus. The operation over the linear automorphism is as follow: a small rectangle is chosen in a neighbourhood of the origin O. Within this rectangle the diffeomorphism is modified so that the stable foliation of the altered diffeomorphism coincides with that of the old diffeomorphism with the only difference that one leaf passes through the point O. The point O breaks it into two parts. Moreover, we choose the perturbation such that on this particular layer two new hyperbolic fixed points appear to the left and to the right of O whereas the point O becomes totally unstable (i.e. a repeller). The closure of unstable manifolds of one of the two new fixed points is an attractor (see Fig. 10). Thus, the new diffeomorphism possesses an attractor, a repeller and a set consisting of wandering trajectories. It is interesting to note that the construction of such attractors is designed in the same way as that of minimal sets known from the Poincar6-Donjoy theory in the case of Cl-smooth vector fields on a two-dimensional torus (24). Let us consider a solid torus H~eE", that is, FI~ = D -~× ~ ' , where ~ 2 is a disc and 5 ~ is a circumference. We now expand q]-2 m times (m is an integer) along the cyclic coordinate on ~ and shrink it q times along the diameter of D 2 where q< i/m. We then embed this deformed torus H e into the original one so that its intersection with ID2 consists of m smaller discs as shown in Fig. 11. We repeat this routine with H 2 and

Mathematical Problems of Nonlinear Dynamics

(a)

82 ]

(b)

\

/

/

\

Fig. 10. Geometry of DA-diffeomorphism (a) before and (b) after the surgery.

Fig. 1 I. A Witorius-Van Danzig solenoid. so on. The set Z = A?_' ,Fli so obtained is called a W i t o r i u s - V a n D a n z i g solenoid. Its local structure m a y be represented as the direct p r o d u c t o f an interval and a C a n t o r set. Smale also observed that W i t o r i u s - V a n Danzig solenoids m a y have hyperbolic structures, that is, they m a y be hyperbolic attractors o f diffeomorphisms on solid tori.

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Leonid Shilnikov

Moreover, similar attractors can be realized as a limit of the inverse spectrum of the expanding cycle map (84)

ff = mO,

modl.

The peculiarity of such solenoids is that they are expanding solenoids. Generally speaking, an expanding solenoid is called a hyperbolic attractor such that its dimension coincides with the dimension of the unstable manifolds of the points of the attractor. Expanding solenoids were studied by Williams (85), who showed that they are generalized (extended) solenoids. The construction of generalized solenoids is similar to that of minimal sets of limit-quasi-periodic trajectories. We note that in the theory of sets of limit-quasi-periodic functions the Witorius-Van Danzig solenoids are quasiminimal sets. Hyperbolic solenoids are called the Smale Williams solenoids. We return to them in Section VII. We remark also on an example of a hyperbolic attractor of a diffeomorphism on a two-dimensional sphere, and, consequently, on a plane, which was built by Plykin. In fact, this is a diffeomorphism of a two-dimensional torus projected onto a two-dimensional sphere. Such a diffeomorphism, in the simplest case, possesses not three fixed points as in Smale's example, but four fixed points, all of them repelling. The question of finding complete topological invariants of structurally stable systems with non-trivial dynamics concerns mainly Anosov diffeomorphisms and two-dimensional diffeomorphisms on surfaces. We refer the reader to (13, 14). To conclude this section we remark that structurally stable high-dimensional systems are not dense in the space of all dynamical systems.

VI. Everywhere Dense Structurally Unstable Systems; Newhouse Regions As we have noticed above, amongst all two-dimensional flows the structurally stable ones form an open and dense set in the space of dynamical systems. This is not the case for multi-dimensional flows. Smale was the first who noted this. Let us consider a three-dimensional diffeomorphism having a saddle fixed point of the type (2,1) and the Anosov attractor T 2. We assume that the points of W~ tend to the torus q]-2 as k ~ + oe. It is evident that W s, where x~qr 2, is a family of two-dimensional manifolds and locally looks like a family of parallel planes such that in a neighbourhood of T 2 they constitute a solid 'fence'. The location of W~9 with respect to this family may be as schematically shown in Fig. 12. In the first case, W~ intersects transversally the leaves of WsT2; in the second case, there is a point of tangency. Moreover, we cannot get rid of this point by means of small perturbations. Furthermore, if the trace of W~ goes through the point of tangency and the co-limit set of W~ is a periodic point, then the original diffeomorphism may be perturbed so that the co-limit set of W~ will be, for instance, an unclosed Poisson-stable trajectory. This simple example demonstrates that structurally unstable systems can form an open set in the space of dynamical systems. However, in such a case there is structural instability on the non-wandering set. Then, it is reasonable to weaken the requirement of structural stability up to that of f~-stability, where f~ denotes the set of non-wandering points.

Mathematical Problems o f Nonlinear Dynamics

(a)

/

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/

Fig. 12. A transverse (a) and a non-transverse (b) intersection of the unstable manifold I~o and the stable leaves of an invariant torus. Definition A dynamical system X is called ~-stable (~-rough) if given E there exists 6 > 0 such that for any system )( E-close in C~-metrics to X the set 1)()() is homeomorphic to ~()~); moreover, the h o m e o m o r p h i s m is E-close to the identity homeomorphism. In other words, the non-wandering set is preserved under small perturbations; it is only shifted slightly. We assume that our flow satisfies Axiom A. Then, as we know, f~ = A~w • . • w A k . Let us introduce the following relationship Ai > Aj, which means that in the phase space there exists a point x such that the ~o-limit set of its trajectory q~(t,x) lies in Aj and the corresponding a-limit set lies in A~. It may appear that there is a chain Ai 1>

• .

. >Ai~>Ai

1

that is, it is not a partially ordered set and, therefore, there are cycles in it. If there are no cycles, we say that acyclicity takes place. Theorem 6.1 (Smale) I f a dynamical system satisfies Axiom A and the condition of acyclicity, then this system is ~)-stable.

Leonid Shilnikov

824

The condition of acyclicity is essential. Although A i persists under small perturbations they m a y cause the so-called f~-explosion. Nevertheless, f~-stable systems are also not dense in the space of dynamical systems. Example of such systems are strange attractors of the Lorenz type (which we will discuss below) and 'wild' hyperbolic attractors. Let us consider a diffeomorphism possessing an f~-stable null-dimensional hyperbolic set E which is homeomorphic to a transitive M a r k o v chain, for example, to the nonwandering set of the Smale horseshoe. As this set is of the saddle type, we may define W~: and W~. W~: (W~) is a union of stable (unstable) manifolds of the points of E. As both W~: and W~: are "hole'-like sets, each can be represented by a direct product of an interval and a Cantor set. We assume that W~- and W~: behave as shown in Fig. 13. Newhouse noted that in the case of C2-smooth diffeomorphisms if there are points xl and x 2 in 5: such that W~, and W~2 have a quadratic tangency, then under certain conditions all C2-close diffeomorphisms preserve this tangency. Such E-sets are called wild hyperbolic sets. In essence, this problem is reduced to the problem of the existence of regions of everywhere dense structural instability of systems close to a system with a structurally unstable homoclinic trajectory. Here, we have a result proved by Newhouse.

Theorem 6.2 (Newhouse) In any neighbourhood of a Cr-smooth (r > 2) two-dimensional diffeomorphism having a saddle fixed point with a structurally unstable homoclinic trajectory there exist regions where systems with structurally unstable homoclinic trajectories are dense everywhere. These regions are called Newhouse regions. Let us denote by B ~the set of diffeomorphisms which possess a saddle point O whose stable and unstable manifolds have a quadratic tangency along a homoclinic curve. We consider a one-parameter family X, of two-dimensional Cr-smooth diffeomorphisms such that X~, is transverse to B ~ at p = 0.

W

u

y

0

s

q

Fig. 13. A homoclinic tangency.

W

Mathematical Problems of Nonlinear Dynamics

825

Theorem 6.3 (Newhouse) Given sufficiently small #0, the interval (-#0,/%) contains a countable set of Newhouse intervals. This theorem is especially important for m a n y problems of nonlinear dynamics, as it guarantees the existence of the Newhouse regions in finite-parameter models. The Newhouse results were generalized for the multi-dimensional case by Gonchenko et al. (33). The study of systems with a structurally unstable homoclinic trajectory was started by Gavrilov and Shilnikov (26). It was established that the typical systems with quadratic homoclinic tangencies may be classified with respect to the character of the sets N of trajectories lying entirely in a small but fixed neighbourhood of the homoclinic trajectory I~: 1. N = O ~ F . This situation occurs, for example, in the case of a two-dimensional diffeomorphism with 1271< 1 at 0 (see Fig. 14(a)). Under small perturbations, which lead to the appearance of infinitely m a n y structurally stable homoclinic trajectories, the structure of a neighbourhood of 1~ is non-trivial. This is the f~-explosion. 2. N admits a precise description. For flows, N is homeomorphic to a suspension over a Bernoulli shift on three symbols in which two trajectories are glued. An example is shown in Fig. 14(b). 3, N contains a hyperbolic set but does not admit a precise description. In the bifurcation set B ~of such systems, those systems that have a countable set of structurally unstable periodic trajectories of any degree of degeneracy, as well as systems having a countable set of structurally unstable homoclinic trajectories of any order of tangency, are everywhere dense. A possible situation is shown in Fig. 14(c). We observe that homoclinic tangencies of the third type are everywhere dense in the Newhouse regions. The 'wild' hyperbolicity accomplishes one more principal case. Let us consider a C r, (r > 4)-smooth system. We assume that the following conditions are fulfilled: 1. at the origin the system has an equilibrium state of the saddle-focus type, that is, the root 2 , • • • ,2, of the characteristic equation at O are such that

21,2 = --p+_iog, Re2i<-p,

(a)

o)¢0

i=3,.

• .,n-1

(b)

Fig. 14. Three main types of homoclinic tangencies.

(e)

826

Leonid Shilnikov

W u

°1 Fig. 15. A homoclinic loop to a saddle-focus.

a=-p+An>0. In this case, dim W~ = 1 and dim W~ = n - 1. 2. One of two separatrices of the saddle-focus comes back to O as t ~ + oo (see Fig. 15). 3. As t ~ + ~ F tends to O tangentially to a two-dimensional hyper-plane corresponding to the eigenvalues 2, and 22. 4. A certain value, called a separatrix value, which characterizes the global behaviour of F is non-zero. The systems with the above properties form a subset B ~ of codimension one in the space of dynamical systems with C 3 metrics. Shilnikov (71, 73) established that the behaviour of the trajectories in a neighbourhood of the separatrix F is rather complicated. In those works he focused on the separation of the set of hyperbolic trajectories and its description via the language of symbolic dynamics. Ovsyannikov and Shilnikov (53, 54) established that everywhere dense structural instability takes place on B I. In particular, the following theorem was proven. Theorem 6.4 In B ~the system with (1) structurally unstable periodic trajectories and (2) structurally unstable Poincar6 homoclinic curves is everywhere dense. Further, in Section X, we present an example of an n, (n_> 4)-dimensional system possessing a wild attractor including a saddle-focus.

VII. Bifurcations of Dynamical Systems; Blue Sky Catastrophe; Appearance of Hyperbolic Attractors We will discuss here the border bifurcations which lead us out of the class of MorseSmale systems to systems with complex dynamics. Such bifurcations interest us in the

Mathematical Problems of Nonlinear Dynamics

827

context of scenarios of the appearance of strange attractors, that is, the routes to dynamical chaos. The violation of the condition of the structural stability of Morse-Smale systems may occur in the following three cases: (1) the loss of the condition of hyperbolicity of equilibrium states or periodic trajectories; (2) the loss of the condition of intersection transversality of stable and unstable manifolds of equilibrium states and periodic trajectories, both of the saddle type; (3) the formation of cycles (see the definition in Section III). Owing to the huge volume of existing materials, an attempt to account completely for all aspects of this problem is unrealistic (10). Nevertheless, based on his personal preferences, the author will try to reflect certain principal moments. In essence, the key aspects have been connected to the requirements (inquiries) of nonlinear dynamics, or to be more precise, to bifurcations of periodic orbits which are, when stable, the mathematical image of self-oscillations. The principal cases of the birth of limit cycles of planar systems were already studied by Andronov and Leontovich by the end of 1930s. There are four key cases of bifurcations: (1) generation of limit cycles from complex foci; (2) bifurcation of a double (saddle-node) periodic orbit; (3) bifurcation from a separatrix loop to a saddle; (4)bifurcation from a separatrix loop of the simplest equilibria of a saddle-node. In the 1950s-1960s these cases were generalized to the high-dimensional case; moreover, new ones were added: (1) period-doubling bifurcation (known, in fact, to Poincar6); (2) birth of an invariant torus from a structurally unstable periodic orbit (see (5) for more details). To the above cases we must add one more global bifurcation of a simple nonrough periodic trajectory which leads to the appearance of either a two-dimensional torus or of a Klein bottle. It is clear that the problem of bifurcations of periodic orbits is, in part, related to the problem of the determination of the principal (codimension one) stability boundaries of stable periodic orbits. The number of distinct stability boundaries of stable limit cycles of planar systems is precisely equal to that of the above bifurcations. This is not the case for systems of dimension three and higher. In particular, we are talking about boundaries such that when a periodic trajectory approaches them both its length and period increase unboundedly even though the periodic trajectory is at a finite distance away from any equilibrium state. The existence of such a boundary was established by Shilnikov and Turaev (80). The bifurcation itself is called a blue sky catastrophe. Let us consider a two-dimensional diffeomorphism with a structurally unstable heteroclinic trajectory behaving as shown in Fig. 16. We denote by 21,2 and 7~,2 the characteristic roots at O12 such that 12~,21< 1 and 17~,21> 1. We assume that W ~O I and , W52 have a homoclinic quadratic tangency along a heteroclinic trajectory F. Let us define 0-

lnl221 lnlTd

which we call a modulus. Palis (55) had established the following result: two diffeomorphisms X and k both belonging to the boundary B 1 (see the previous section) are homeomorphic if their moduli are equal. Such diffeomorphisms form a bifurcation surface B ~ of codimension one. Thus, the set of such systems may be subdivided into a

828

Leonid Shilnikov

Fig. 16. A structurally unstable heteroclinic connection with a tangency. continuum of classes of topological equivalence distinguished by their moduli. As the notion of the modulus is important in the following discussion, let us give its complete definition. Definition We say that a system X has a modulus if in the space of dynamical systems a Banach space ~ ' passes through X, and on J / / a locally non-constant continuous functional h is defined such that for two systems X and X from J / t o be equivalent it is necessary that h(X) = h(,("). We will say that X has at least m moduli in a Banach space through X on which m independent moduli are defined, and that X has a countable number of moduli if X has an arbitrarily finite number of moduli. We must also mention the work of de Melo and Van Streen (23) where, for more general requirements (but still within the class of the Morse-Smale systems), the existence of a countable set of moduli was established. We note, however, that if we require only the condition of f~-equivalence to hold for systems, then the systems of the given type are f~-rough. Next we need the notion of an f2-modulus. This may be introduced in a similar manner as above with the only difference that f~-equivalence instead of a pure equivalence is used. Let us now pause to discuss the principal aspects related to the transition from Morse-Smale systems to systems with complex dynamics• Let us assume that there is a three-dimensional smooth system = X(x,#)

(30)

having a non-rough equilibrium state of saddle-saddle type at the origin for/~ = 0. This means that in a small neighbourhood of the origin O, the system can be written in the following form:

y~

--y

Z~Z.

Mathematical Problems of Nonlinear Dynamics

829

l

Wu

02

Fig. 17. There are two saddles Oj and 02 for p < 0 .

$

w

o

u

wo

Fig. 18. A non-rough equilibrium state of the saddle-saddle type at the bifurcation moment #=0. F o r p < 0, it has two saddle equilibrium states O 1 and 02; O 1 has a two-dimensional stable invariant manifold W~, and a one-dimensional unstable invariant manifold W ~o,, whereas for 02, dim W so2 = 1 and dim Wo2U= 2 (see Fig. 17). W h e n p = 0, the point O is o f saddle-saddle type which has stable manifold W~ and unstable manifold W~ h o m e o m o r p h i c to semi-planes (Fig. 18). Let us assume that at p = 0, Wb and W~ intersect each other transversely along the trajectories F~, • • • ,Fro, m > 2.

Theorem 7.1 (Shilnikov (78)) As p increases t h r o u g h 0, the equilibrium state O disappears and an unstable set ~ is born whose trajectories are in one-to-one correspondence to the Bernoulli scheme on m symbols. As p ~ 0 +, ~ is tightened into a b o u q u e t defined b y : { O u F l • • - • wF,,}. This example demonstrates that M o r s e - S m a l e systems m a y be b o u n d e d f r o m Smale systems by a bifurcation surface o f codimension one.

Leonid Shilnikov

830 (a)

(b)

O 0 0

0

Fig. 19. (a) Geometry of the blue-sky catastrophe bifurcation. The unstable manifold, homeomorphic to a semi-cylinder, returns to the saddle-node periodic orbit tangentially to the strong stable manifold. (b) The Poincar6 map on a transversal to the saddle-node periodic orbit.

The next example demonstrates the transition through a codimension-one bifurcation surface from a Morse Smale system to a system with a Smale-Williams solenoid. The key moment here is a global bifurcation of disappearance of a periodic trajectory. Let us discuss this bifurcation in detail following the paper of Shilnikov and Turaev (80). Let us consider a Cr-smooth one-parameter family of dynamical systems X, in R=. We suppose that the flow has a periodic orbit L0 of the saddle-node type at # = 0. We choose a neighbourhood U0 of L0 which is a solid torus partitioned by the ( n - 1)dimensional strongly stable manifold W ~= L0 into two regions: the node region U + where all trajectories tend to L0 as t ~ + ~ , and the saddle region U- where the two-dimensional unstable manifold W u L0 bounded by L0 lies. We suppose that all of the trajectories of W~0 return to L0 from the node region U + as t ~ + ~ and do not lie in W ==, as shown Fig. 19. Moreover, as any trajectory of W" is bi-asymptotic to Lo, [~uL0 is compact. We observe that systems close to X0 and having a simple saddle-node periodic trajectories close L0 form a surface B of codimension one in the space of dynamical systems. We assume also that the family X, is transverse to B. Thus, w h e n / t < 0, the orbit L0 is split into two periodic orbits, namely: L~- of the saddle type and stable L +. When # > 0 L0 disappears. It is clear that X u is a Morse-Smale system in a small neighbourhood U of the set W" for all small/~ < 0. The non-wandering set here consists of the two periodic orbits L + and L 7. All trajectories of U. W~L; tend to L + as t ~ + ~ . At/~ = 0 all trajectories on U tend to Lo.

Mathematical Problems o f Nonlinear Dynamics

831

The situation is more complicated when ~t>0. It was established by Afraimovich and Shilnikov (8) that if W u is a smooth submanifold of ~", then a two-dimensional stable invariant surface (a torus) m a y be generated from W u. In the case where W u is a non-smooth manifold, the disappearance of the saddle-node periodic orbit may lead to a rather non-trivial behaviour of the trajectories; in particular, to the appearance of Poincar6 homoclinic orbits. We note that if X0 has a global cross-section, W u is always homeomorphic to a torus. The Poincar6 m a p to which the problem under consideration is reduced m a y be written in the form X = f(x,O,p) 0 = mO+g(O)+og+h(x,O,#),

(mod 1)

(31)

where f, 9 and h are periodic functions of 0. Moreover, ]lf[[C 1--*0 and []hlfC 1~ 0 as/~--, 0, m is an integer and ~o is a parameter defined in the set [0,1). Diffeomorphism Eq (31) is defined in a solid-torus ~)" 2 x ~1, where ~ , - 2 is a disc [ix[] < r , r > 0 . We observe that Eq (31) is a strong contraction along x. Therefore, mapping Eq (31) is close to the degenerate m a p 2=0 0 = mO+g(O)+og,

(mod 1).

(32)

This implies that its dynamics is determined by the circle m a p 0 = mO+g(O)+~o,

(mod 1)

(33)

where 0 < ~o < 1. We note that in the case of the flow in p3, the integer m in Eqs (31 (33) may be 0 , 1 , - 1. Theorem 7.2 (Shiinikov-Turaev) If m = 0 and if maxllg'(0)ll < 1, then for sufficiently small p > 0, the original flow has a periodic orbit of which both length and period tend to infinity as p ~ 0 . This is the blue sky catastrophe we mentioned above. In the case where m = 1, the closure g/~0 is a two-dimensional torus. Moreover, it is smooth provided that Eq (31) is a diffeomorphism. In the case where m = - 1 IJ,'~L0 is a Klein bottle, also smooth if Eq (31) is a diffeomorphism. In the case of the last theorem W~0 is not a manifold. We must say that in the general case, if the diffeomorphism Eq (33) has an interval [0~,02] where Im +9'(0)[ < 1, then the original system possesses a countable set of intervals [#~n,/~2n]accumulating to zero where the system X, has stable periodic orbits. In the case of ~n (n_> 4) the constant m m a y be any integer. Theorem 7.3 (Shilnikov-Turaev) Let [m[_>2 and let [m+#'(0)[ > 1. Then for all p > 0 suffÉciently small, the Poincar6 m a p Eq (31) has a hyperbolic attractor homeomorphic to the Smale-Williams solenoid, whereas the original family has a hyperbolic attractor homeomorphic to a suspension over the Smale-Williams solenoid. The idea of the use of the saddle-node bifurcation to produce hyperbolic attractors may be extended to that of employing the bifurcations of an invariant torus. We are

832

Leonid Shilnikov

not developing here the theory o f such bifurcations but restrict ourself by consideration o f a modelling situation. Let us consider a one-parameter family o f s m o o t h dynamical systems sc = X ( x , # )

which possesses an invariant m-dimensional torus 3-" with a quasi-periodic trajectory at # = 0. We assume that the vector field m a y be recast as = C(~)y 2 = ,u+z 2

0 = f~(#)

(34)

in a n e i g h b o u r h o o d o f t m. Here, z e ~ ~, y e ~ "-m 1, 00-m and f~(0) = (f2~, • • • ,f~m). The matrix COO is stable, that is, its eigenvalues lie to the left o f the imaginary axis in the complex plane. At # = 0 the equation o f the t o m s is y = 0, the equation o f the unstable manifold W u is y = 0, z > 0, and that o f the strongly stable manifold W ~s partitioning the n e i g h b o u r h o o d o f ~-m into a node and a saddle region is z = 0. We assume also that all o f the trajectories o f the unstable manifold W" o f the torus come back to it as t--, + ~ . Moreover, they do not lie in W ss. O n a cross-sections transverse to z = 0 the associated Poincar6 m a p m a y be written in the f o r m y =f(y,O,#) 0 = AO+y(O)+og+h(x,O,#),

(mod 1)

(35)

where A is an integer matrix, and f, 9 and h are periodic functions o f 0. Moreover, 10qlC1 ~ 0 and tlhllC1 ~ 0 as # ~ 0 , ~o = (oh, " • • ,~Om)where 0 < ~ k < 1. We denote G(O) = AO+9(O).

I f IG'(0)I < 1 for all 0 (e.g. if A = 0 and i f g ( 0 ) is small), then the shortened m a p is contracting for all/~ > 0 as well as the original m a p Eq (35). As a contracting m a p p i n g has only one fixed point, we arrive at the following statement. Proposition 7.1

If IIG'(0)II < 1 for all small 0, then the flow X, has a unique attracting periodic orbit for all # > 0 sufficiently small. This result is analogous to the result o f T h e o r e m 7.2. It yields us a new example o f the blue sky catastrophe. We observe that the restriction o f the Poincar6 m a p on the invariant torus is close to the shortened m a p 0 = AO+9(O) = co,

( m o d 1)

(36)

This implies, in particular, that if Eq (36) is an A n o s o v m a p for all ~o (e.g. when the eigenvalues o f the matrix A do not lie on the unit circle o f the complex plane, and 9(0) is small), then the restriction o f the Poincar6 m a p is also an A n o s o v m a p for all # > 0. Hence, we arrive at the following statement.

Mathematical Problems of Nonlinear Dynamics

833

Proposition 7.2 If the shortened map is an Anosov map for all small o9, then for all # > 0 sufficiently small, the original flow possesses a hyperbolic attractor which is topologically conjugate to the suspension over the Anosov diffeomorphism. The birth of hyperbolic attractors may be proven not only in the case where the shortened map is a diffeomorphism. Namely, this result holds true if the shortened map is expanding. A map is called expanding in length if any tangent vector field grows exponentially under the action of the differential of the map. An example is the algebraic map O = AO,

(modl)

such that the spectrum of the integer matrix A lies strictly outside the unit circle, and any neighbouring map is also expanding. If If(G'(0))-lll < 1, then the shortened map

0 = og+G(O) = o9+A0+9(0),

(mod 1)

(37)

is expanding for all/~ > 0. Shub (81) has established that expanding maps are structurally stable. The study of expanding maps and their connection to smooth diffeomorphisms was continued by Williams (83). Using the result of his work, we come to the following result, which is analogous to our Theorem 7.2.

Proposition 7.3 If II(G'(0))- lip < 1, then for all small/~ > 0, the Poincar6 map possesses a hyperbolic attractor locally homeomorphic to a direct product of •"+ ~ and a Cantor set. An endomorphism of a torus is called an Anosov coveting if there exists a continuous decomposition of the tangent space into the direct sum of stable and unstable submanifolds as in the case of the Anosov map (the difference is that the Anosov covering is not a one-to-one map, therefore it is not a diffeomorphism). The map Eq (36) is an Anosov covering if, we assume, IdetAI > 1 and if 9(0) is sufficiently small. Thus, the following result is similar to the previous proposition. Proposition 7.4 If the shortened map Eq (36) is an Anosov covering for all eg, then for all small/~ > 0 the original Poincar6 map possesses a hyperbolic attractor locally homeomorphic to a direct product of R "~+' and a Cantor set. In connection with the above discussion we can ask what other hyperbolic attractors may be generated from Morse-Smale systems? Of course, there are other scenarios of the transition from a Morse-Smale system to a system with complex dynamics, for example, through O-explosion, period-doubling cascade, etc., but these bifurcations do not lead explicitly to the appearance of strange attractors. In a more interesting case, they generate the so-called quasi-attractors, which we will consider in Section IX. VIII. Strange Attractors of the Lorenz Type In 1963, Lorenz (44) suggested the model = -- ~(x--y)

834

Leonid Shilnikov 9 = rx-y-xz

(38)

= -bz+xy

in which he discovered numerically a vividly chaotic behaviour of the trajectories when a = 10, b = 8/3 and r = 28. This model is obtained by applying the Galerkin approximations to the problem of convection of a planar infinite layer. F o r a number of years this work of Lorenz did not attract special attention. The breakthrough occurred in a mid-1970s, when this model became the centre of attention of mathematicians as well as researchers from other fields such as non-linear optics, magneto-hydrodynamics, etc. We note that, from the viewpoint of mathematics, the Lorenz model, as well as its extension 2=y ¢ = x(1 -- z) -- 2y -- x 3 2 =

--~,7+X

(39)

2

where 2 > 0 , ~ > 0 and B > 0 m a y also be considered as principal normal forms for symmetrical flows with triple degenerate singularities in the form of equilibria or periodic orbits (70). As a result of mathematical studies of the Lorenz model we have achieved an important conclusion: simple models of nonlinear dynamics may have strange attractors. Similar to hyperbolic attractors, periodic as well as homoclinic orbits are everywhere dense in the Lorenz attractor, but the Lorenz attractor is structurally unstable. This is due to the embedding of a saddle equilibrium state with a one-dimensional unstable manifold into the attractor. Nevertheless, under small smooth perturbations stable periodic orbits do not arise. Moreover, it became obvious that such strange attractors m a y be obtained through a finite number of bifurcations. In particular, in the Lorenz model (owing to its specific feature: it has the symmetry group ( x , y , z ) , - - ~ ( - x , - y , z ) ) such a route consists of three steps only. Below we present a few statements concerning the description of the structure of the Lorenz attractor as it was given in (6, 7). The fact that we are considering only threedimensional systems is not important, in principle, because the general case where only one characteristic value is positive for the saddle whereas the others have negative real parts, and the least value with the modulus is real, the result is completely similar to the three-dimensional case. We let B denote the Banach space of Cr-smooth dynamical systems (r > 1) with the Cr-topology, which are specified on a smooth three-dimensional manifold M. We suppose that in the domain U c B each system X has an equilibrium state O of the saddle type. In this case, the inequalities 21 < 22 < 23 hold for the roots 2i = 2i(X), i = 1,2,3 of the characteristic equation at O, and the saddle value or(X) = 22 + 23 > 0. A stable two-dimensional manifold of the saddle will be denoted by W s = W s ( x ) and the unstable one, consisting of O, and two trajectories F~.2 = FI,2(X) originating from it by W u = Wu(X). It is known that both W ~and W u depend smoothly on X on each compact subset. Here it is assumed that in a certain local m a p V = {(Xl, x2, x3)}, containing O, X can be written in the form 2i = 2 i x i + P i ( x l , x 2 , x 3 ) ,

i = 1,2,3.

(40)

Mathematical Problems of Nonlinear Dynamics

835

Fig. 20. Two one-dimensional separatrices F, and F2 form a homoclinic butterfly. Let us suppose that the following conditions are satisfied for the system X0 c U (see Fig. 20): 1. Fi(X0)c WS(Xo), i = 1,2 (i.e. ri(x'0) is doubly asymptotic to O). 2. F,(X0) and F2(X0) approach O tangentially to each other. Further considerations will require some concepts and facts presented in (71, 73). The condition 2~ < 22 implies that the non-leading manifold W~ of W~, consisting of O and the two trajectories tangential to the axis Xl at the point O, divides W~, into two open domains: W% and W~_.Without loss of generality we may assume that F~(X0)c W%(Xo), and hence F~ is tangent to the positive semiaxis x2. We let v2 and v2 be sufficiently small neighbourhoods of the separatrix 'butterfly' F = F~uOuF2. We let ~ > stand for the connection component of the intersection of W%(Xo)with vi, which contains F,(X0). In the general case Mr,.is a two = dimensional C°-smooth manifold homeomorphic either to a cylinder or to a M6bius band. The general condition lies in the fact that certain values A~(Xo) and A2(Xo), called the separatrix values, should not be equal to zero. It follows from the above assumptions that X0 belongs to the bifurcation set B 2 of codimension two, and B~ is the intersection of two bifurcation surfaces BI and B 1 each of codimension one, where B~ corresponds to the separatrix loop r i = Ok-.)ri. In such a situation it is natural to consider a two-parameter family of dynamical systems X(#), = ( # . m ) , I~1 < m , x ( 0 ) = x0, such that X(/~) intersects with B~ only along X0 and only for/~ = 0. It is also convenient to assume that the family X(/~) is transverse to B 2. By transversality we mean that for the system X(#) the loop F,(X(#)) 'deviates' from W% (X(/0) by a value of the order o f #l, and the loop F2(X(/~)) 'deviates' from W%(X(12)) by a value of the order of #2It is known from (73) that the above assumptions imply that in the transition to a system close to Xo the separatrix loop can generate only one periodic orbit which is of the saddle type. Let us assume, for certainty, that the loop F~(X0)uO generates a periodic orbit L~ for/~t > 0 and Fz(Xo)uO generates the periodic orbit L2 for/~2 > 0. The corresponding domain in U, which is the intersection of the stability regions for L~ and L2, that is, the domain in which the periodic orbits L~ and L2 are structurally

836

Leonid Shilnikov

stable, will be denoted by U0. A stable manifold of Lg for the system X c U0 will be denoted by W~ and the unstable one by W~. If the separatrix value A i ( X o ) > O, W 7 is a cylinder; if A,(X0) < 0, W~' is a M6bius band. We note that, in the case where J//is an orientable manifold, W~ will also be a cylinder if A i ( X o ) > O . Otherwise it will be a M6bius band. However, in the forthcoming analysis the signs of the separatrix values will play an important role (21). Therefore, it is natural to distinguish the following three main cases: Case A (orientable): A I ( X o ) > 0, A d X o ) > 0; Case B (semiorientable): A l(X0) > 0, AR(X0)< 0; Case C(nonorientable): A l(X0) < 0, A d X o ) < O. In each of these cases the domain U0 also contains two bifurcation surfaces B~ and 1. In Case A, B~ corresponds to the inclusion F l c W~ and B4~ corresponds to the inclusion F2 c W~. 2. In Case B, B~ corresponds to the inclusion F1 c W~ and B ] corresponds to the inclusion F2 c W]. 3. In Case C, along with the above-mentioned generated orbits L1 and L2, there also arises a saddle periodic orbit L3 which makes one revolution 'along' Fl(X0) and F2(X0), and if both W7 are M6bius bands, i = 1,2, the unstable manifold Wg of the periodic orbit L3 is a cylinder. In this case, the inclusions F i c W~ and F2= W~ correspond to the surfaces B~ and B ], respectively. Let us suppose that B 1 and B ] intersect transversally over the bifurcational set B~ (see Fig. 21). In a two-parameter family X(/~) this means that the curves B 1 and B4~intersect at some point #1 = (~11,#12)" In a small neighbourhood of the origin of the parameter plane, equations of B~ will be of the form Pl = alp~/~(1 - F ' ' " ) and #1 = bl#~/~(1 + . • • ) in cases A and B, and of the form p~ = c 1 # ~ / ~ 2 ( 1 q - • • • ) in case C. Equations of B~ will be of the form #2 = a2Pl/~( 1 + " " " ) in case A, and of the form y 2 = b 2 p ] / ~ ( l + . . - ) and ~ t 2 = c 2 p 1 1 / ~ ( l + . . . ) in cases B and C. Here,

r

P-1 Fig. 21. A structure of the bifurcation set on the plane (#~,g2). T

Mathematical Problems of Nonlinear Dynamics

837

ct = -,~2/J.3,al, • • • ,c 2 are positive, and the ellipsis stand for terms which tend to zero when the argument tends to zero. Let us denote a domain lying between B~ and B4~ by U~. It can be shown (this can be deduced from the results of Shilnikov (78)) that there exists in U a sufficiently small E-neighbourhood UE of the system X0 such that a onedimensional limiting set, homeomorphic to the suspension over the Bernoulli subshift on two symbols, will exists for each X E U E m U~. This statement m a y be proved if one considers the mapping T of a certain transversal to F~(X0) and F2(X0) along the trajectories of the system X. The generalization of this situation requires the existence of a global transversal. Let us assume, therefore, that for each X ~ U there exists a transversal D (see Fig. 22) with the following properties:

1. the

Euclidean

coordinates (x,y) can be introduced on D such that D = {(x,y):lxl < 1,1Yl<2}. 2. The equation y = 0 describes a connection component S of the intersection W~oc~D such that no ~o-semitrajectory that begins on S possesses any point of intersection with D for t > 0. 3. The mapping TI(X):D~--~D and T~(X):D2w-~D are defined along the trajectories of the system X, where 01 = { ( x , y ) : l x l < l ,

0
0 2 = {(x,y):lxl < 1,

- 1
and T,(J0 is written in the form x = f~(x,y)

Fig. 22. The images of the two halves of the return plane under the action of the Poincar6 map. Trajectories started within the plane next strike it within the shaded areas. The points M~ and M: are the first points of the intersection of the separatrices and the cross-section.

Leonid Shilnikov

838

(41)

y = g,(x,y) wheref,g,eC% i = 1,2. 4. f and gi admit continuous extensions on S, and

~im f ( x , y ) = x**,

lim° g,(x,y) = y**,

i = 1,2.

TIDIEPfi = ((x,y):~1 < x_< 1,]y] < 2 }

.

T2D2eP, =

{

'

( x , y ) : - 1 < x < - ~,lyl < 2

}

.

Let

T ( X ) - T~(X)ID,,

(f,g) = -=(f,gi)onDi,

i = 1,2.

6. Let us impose the following restrictions on T(X): (a)

II(fdll < 1

(b)

1-11(gy)-lll - IlLll >2x/ll(gy)-lll • II(gx)ll Il(g~) -~ %11

(c)

II(gy)-lll < 1

(d)

tl(9,) -~ "Lll Ilgxll <(1 --Ibrxll)(1 -II(0y)-'ll)

"1 (42)

Hereafter,

1111 =

sup I'[. (x,y)ED/S

It follows from the analysis of the behaviour of trajectories near W~ that in a small n e i g h b o u r h o o d o f S the following representation is valid: f l = X**-t-q)l(x,y)Y ~, gl

=

Y~*'Jf-I~I(x,y)Y ~t

fz = x** + ~o2(x,y)(--y)% g2 = Y** + 02(x,Y)(--Y)"

(43)

where qh, • ' • ,02 are smooth with respect to x,y for y ¢ 0, and Ti(x) satisfies estimates Eq (39) for sufficiently small y. Moreover, the limit of ~o~ will be denoted by A~(X) and that o f 02 by A2(X). The functionals A~(X) and A2(X) will be also called the separatrix values in analogy with A~(Xo) and A2(Xo) which were introduced above. We note that for a system lying in a small n e i g h b o u r h o o d of the system X all the conditions ( 1 ) ( 6 ) are satisfied near S. Moreover, the concept of orientable, semiorientable and nonorientable cases can be extended to any system XeU. It is convenient to assume, for simplicity, that AI,2(X) do not vanish. It should be also noted that the point Pi with the coordinates (x**, y**) is the first point o f intersection o f F,(X) with D. Let us consider the constant q =

1 + I[fxltI[(gy)-' II + ~'/1 - II(gO- ~11211(fOll- 411(gy)- ill IlgJI II(g')- ~f" 211(9,)- ~ll

(44)

Conditions Eq (44) implies that q > 1 and this (together with conditions Eq (42)) implies that all periodic points are o f the saddle type.

Mathematical Problems of Nonlinear Dynamics

839

In terms of mappings, the conditions for the existence of periodic orbits L~ and L 2 can be formulated simply: it is required that P1~D2, P2~DI in Case A; PloD2, P2~D2 in Case B; P~D~, P2ED2in Case C. The point of intersection of Li with D will be denoted by Mi, and its coordinates by x* and y*. As already mentioned, the periodic orbit L3 will exist in Case C together with L1 and L 2. It intersects D at two points M3(x*,y*) and M4(x*,y*), where TM3 = M4 and TM4 = M3. The periodic orbits L1, L2 and L3 will be called basic. The conditions imposed above imply that the equations for the intersections of the connection components of the stable manifolds W s of the periodic orbits Li, i = 1,2, with D containing fixed points M~, can be represented in the form y = y~(x), Ixl < 1. In Case C the equation for the connection component D n W~, containing point Mi, can also be written in the form y = yi(x), i = 3,4. Let us define the functionals R~ and R2 as follows: in Case A R 1 = - - [ _ , V * * - - y 2 ( x * * ) ] , R2 = y * * - y ~ ( x * * ) ; in Case B g~ = --[u**--y~(x**), R2 = y * * - y l ( x * * ) ; in Case C R1 = --[Yl**--y4(x**)], R2 = - - [ Y 2 * * - - Y 3 ( X * * ) ] ; where (u**,v**) are the coordinates of TPi in case B (see Fig. 23). It can be easily seen now that R~ = 0 for the system XeB] and R2 = 0 if Xeff4, and that the domain U consists of systems X for which R~ > 0 and R2 > 0. The condition R~ > 0 means that: 1. in Case A, L~ and L2 have a heteroclinic trajectory; 2. in Case B, L~ has a structurally stable homoclinic curve; 3. In Case C, L3 also has a structurally stable homoclinic curve. I f R2 > 0, it is only L3, which has a homoclinic curve in Case C. In Cases A and B, L~ and L2 will have a heteroclinic curve for R2>0. The domain {XeUEIRIO} will be denoted by U + and the domain {XEUEIR~< 0 , R 2 < 0 } by U +. Let us first consider the limiting sets of this class of dynamical systems. Special attention will be paid to the case where a domain can be specified in the phase space into which all the trajectories come and which does not contain stable periodic orbits. If this domain contains only one limiting set, it appears to be a quasi-hyperbolic attractor. However, this domain may contain several limiting sets; all except one are unstable, and this one is og-limiting and, in the general case, where it is locally maximal, is an attractor. We let E denote the closure of the set of points of all the trajectories of the mapping T(X), which are contained entirely in D. As a result of the above considerations, Z is non-trivial and contains a countable set of periodic orbits in Uw U~-n U+ for Case C and in U - 1 u U( for Case B. F o r Case A in U + u U + and for Case B in UJ-, the abovecited situation is possible, and also a trivial one where Z consists of M1M2 and heteroclinic points. The corresponding bifurcational surface that divides these sets of systems will satisfy the following equations: In Case A: R3 = 0 and R4 = 0, where R3 = --v**+yl(Ul**),

R, = v**--yz(u**)

where [u**,v**] are coordinates of points T(X)P~, i = 1,2;

Leonid Shilnikov

840 (a)

(b)

H1

I12

111

1-12

1

CASE A

CASE B

(c) H1

112

D 1

CASE C Fig. 23. Three possible cases of the Poincar6 map: Case A, orientable; Case B, semiorientable; Case C, nonorientable. In Case B:

R 3 = 0,

where R3 = - v * * + y ~ ( u * * )

and -3"'**,~3"'**are coordinates o f the point T2(X)PI . It is obvious that in Case A the surface R 4 = 0 lies in U~-, and R 3 in Ui ~ (the same is true for Case B). Y, is described m o s t simply in the d o m a i n U~. Here the following t h e o r e m holds.

Theorem 8.1 If X~UI, T(X),[Z is topologically conjugated with the Bernoulli scheme (a,f~2) with two symbols.

Mathematical Problems of Nonlinear Dynamics

841

I f X~(Jz/U2, Y. will be a one-dimensional set. We shall not concentrate on this case. Let us only note that 5: is unstable and its closure is similar, in m a n y respects, to the structure of Z for XeU2. We confine ourselves to the following theorem.

Theorem 8.2 The system X~Uz has a two-dimensional limiting set ~, which satisfies the following conditions: 1. 2. 3. 4.

f~ is structurally unstable; [I'lwF2nO] c ~; structurally stable periodic orbits are everywhere dense in f~; under perturbations of X periodic orbits in f~ disappear as a result of matching to the saddle separatrix loops F~ and F2.

We note that in this case the basic periodic orbits will not belong to f~. In terms of mappings, the properties of f~ can be formulated in more detail. Let us first single out a d o m a i n / 5 on D as follows: we assume that in Case A

/3 = { (x,y)~D1w D2IYz(X) < y
/3 = {(x,y)eD,wD2lYlz(x) < y
D = {(x,y)eDlwD2lY3(X)

The closure of points of all the trajectories of the mapping T(X), which are entirely contained in/3, we will denote by Z.

Theorem 8.3 Let X~U2. Then: I. Z is compact, one-dimensional and consists of two connection components in Cases A and C, and of a finite number of connection components in Case B. II./3 is foliated by a continuous stable foliation H ÷ into leaves, satisfying the Lipshitz conditions, along which a point is attracted to 5~; inverse images of the discontinuity line S: y = 0 (with respect to the mapping T k, k = 1,2, • • • ) are everywhere dense in

/3. III. There exists a sequence of T(X)-invariant null-dimensional sets Ak, k~7/+, such that T(X)IAk is topically conjugated with a finite topological M a r k o v chain with a nonzero entropy, the condition Ak~Ak+ ~ being satisfied, and A k ~ E as k ~ ~ . IV. The non-wandering set EIeE is a closure of saddle periodic points of T(X) and either Z1 = E or Zt = E + u Z -, where: 1. E - is null-dimensional and is an image of the space ~ - of a certain T M C (G-,f~-,a) under the h o m e o m o r p h i s m fl:E-~--~E- which conjugates a l ~ - and T(X)IE-;

Leonid Shilnikov

842

/(x)

Z-

= mt~) 1Y~r~ ,

~(X) < oo

where Z(S)~n

= '~m,

Z ~ I O]~m 2 =

for m~ #m2 and T(X)IEm is transitive; 2. Z + is compact, one-dimensional and 3. if E + n Z - = O, Z + is an attracting set in a certain neighbourhood; 4. if Z+c~Z 5 0 , then Z + n E - = Y~m+nZm for a certain rn, and this intersection consists of periodic points of no more than two periodic orbits, and (a) if Zm is finite, Z + is ~-limiting for all the trajectories in a certain neighbourhood; (b) if Era is infinite, Z + is not locally maximal, but is ~o-limiting for all the trajectories in/~, excluding those asymptotic to Z - • Z + . We let U2' define a set of systems from U2, for which the separatrices F1 and F2 tend to O or to periodic orbits, and let U2" stand for a set of such systems from Uz', for which F~ and F2 tend to O as y ~ .

Theorem 8.4 I. U2" is dense in U2. II. F o r X~U2': (a) the mapping T(X) admits a finite M a r k o v partition of Z whose boundary belongs to a finite set of leaves from H + (X); (b) for a T M C (G,fLa) constructed according to this partition, there exists a continuous mapping/~:f2 ~--~Z,which is one-to-one on the residual set of ~ and is such that the diagram

ft\fl-'( ,ns)

a

is commutative; (c) ~2 is locally homeomorphic to the direct product of a Cantor set by a segment at each point, which does not belong to the M a r k o v partition boundary. III. E + is an attractor for X~U2". IV. T(X) does not admit a finite M a r k o v partition of Z for M a r k o v partitions).

X~UdU2' (see (64) on

We will say that the rational case takes place if X~U2". It follows from the definition of the rational case that a bifurcation set B~ of codimension two, which is an intersection of two bifurcational surfaces Bl~xand B]x of codimension one, is connected with each system XeUd. Let us call the surface B]x a rational one, if Fi tends either to O or to a periodic orbit; if Fi tends to O then BJx will be called a bifurcational surface of the first type, and if F~ tends to the periodic orbit, a bifurcational surface of the second type. Surfaces of the first type are of interest in that the bifurcation of periodic orbits is connected only with t h e m - - a periodic orbit may disappear only via matching either to

Mathematical Problems o f Nonlinear Dynamics

843

a separatrix loop or to a contour. Transition through surfaces of the second type is responsible, in general, for local rearrangements of an attractor. However, there may be exceptional rearrangements, namely those related to a global internal crisis of E which results in the appearance of lacunae within an attractor. It follows from Theorem 8.4 that two families of rational surfaces exist in the neighbourhood of any system, the rational surfaces of the first type being everywhere dense. In general, the neighbourhood of X is foliated into a family of bifurcational surfaces, including those that can be naturally found as limits of rational surfaces of the first type. Below we will give the conditions under which the existence of the Lorenz attractor are guaranteed. We consider a finite-number parameter family of vector field defined by the system of differential equations (45)

5c = X(x,#)

where x~R "+~, p~R m, and X(x,p) is a Cr-smooth function of x and p. We assume that the following two conditions hold: (A) System Eq (45) has a equilibrium state O(0,0) of the saddle type. The eigenvalues of the Jacobian at O(0,0) satisfy R e 2 , < • • - Re22<2,<20. (B) The separatrices F1 and

U2

of the saddle O(0,0) return to the origin as t-o + ~ .

Then, for p > 0 in the parameter space there exists an open set V, whose boundary contains the origin, such that in system V Eq (45) possesses the Lorenz attractor in the following three cases (75): Case 1

(A) F~ and F return to the origin tangentially to each other along the dominant direction corresponding to the eigenvalue 21;

(B) 1

<7<1,

vi>l,

7-

21 20'

v~=

Re2i 20

(C) The separatrix values AI and A2 (see above) are equal to zero. In the general case, the dimension of the parameter space is four, as we may choose /~1.2 to control the behaviour of the separatrices FI.2 and/23,4 = A3,4. In the case of the Lorenz symmetry, we need two parameters only. Case 2

(A) F, and F belong to the non-leading manifold W s ~ W S and enter the saddle along the eigendirection corresponding to the real eigenvector 22;

Leonid Shilnikov

844

(B) 1

~
v/>l,

21 7=-20,

Re2i vi=-20

In the general case, the dimension of the phase space is equal to four. Here, ]o~3,4 controls the distance between the separatrices.

Case 3 (A)

1-'1,2¢ WSS;

(B) ~ = 1;

(C) A1.2¢0, and IA1.2I<2. In this case m = 3, #3 = 7 - 1. In the case where the system is symmetric, all of these bifurcations are of codimension two. In (68, 69), it was shown that both subclasses (A) and (C) are realized in the ShimizuMarioka model, in which the appearance of the Lorenz attractor and its disappearance through bifurcations of lacunae are explained. Some systems of Type (A) were studied by Rychlik (62) and those of Type (C) by Robinson (61). The distinguishing features of strange attractors of the Lorenz type is that they have a complete topological invariant. Geometrically, we can state that two Lorenz-like attractors are topologically equivalent if the unstable manifolds of both saddles behave similarly. The formalization of'similarity' may be given in terms of kneading invariants, which were introduced by Milnor and Thurston (48) while studying continuous, monotonic mappings on an interval. This approach may be applied to certain discontinuous mappings as well. As there is a foliation (see above), we may reduce the Poincar6 map to the form

.~ = F(x,y)

)7 = G(y)

(46)

where the right-hand side is, in general, continuous, apart from the discontinuity line y = 0, and G is piecewise monotonic. Therefore, it is natural to reduce Eq (46) to a one-dimensional map

37= G(y). Williams and Guckenheimer (40), by using the technique of taking the inverse spectrum, showed that a pair of the kneading invariants is a complete topological invariant for the associated two-dimensional maps provided inf IG'I > 1. The latter is possible only when the stable foliation is smooth (for smoothness of the stable foliation in mapping of the Lorenz type, see (60) and (63)). To conclude, we remark that the topological dimension of strange attractors of the Lorenz type is equal to two. The fractal dimension does not exceed three. In Section IX we will discuss the method of constructing a strange attractor of a higher topological dimension.

Mathematical Problems of Nonlinear Dynamics

845

IX. Quasi-attractors; Dynamical Phenomena in the Newhouse Regions Hyperbolic attractors, despite their 'attraction', have never been observed in nonlinear dynamics. This is perhaps because the scenarios of their appearance are not simple from the practical point of view. It is likely that the bifurcations leading to the appearance of Smale-Williams attractors from the boundary of Morse-Smale systems may change this situation. We have mentioned already that strange attractors of the Lorenz type occur in concrete applications. This is not the case for the 'universal' Lorenz attractor, which may exist if an associated flow meets certain specific conditions, the foremost of which is the realization of a geometrical configuration called a homoclinic butterfly. The dimension of the flow is not essential in the sense that a Lorenz attractor embedded in a high-dimensional flow will still have the same topological dimension. On the other hand, there is a certain interest in non-linear dynamics to multi-dimensional dynamical chaos, or to hyper-chaos. However, the nature of strange attractors observed in numerous applications has scarcely been discussed because of either the absence of the proper mathematical fundamentals or the complexity of the phenomenon under consideration, such as well-developed turbulence. The exception includes models described by three-dimensional systems of differential equations, or by two-dimensional diffeomorphisms. Complex limiting sets of such systems are usually not 'genuine' strange attractors but quasi-attractors (10), which possess the following features: (1) they have a non-trivial hyperbolic subset; (2) either a quasi-attractor itself or a quasi-attractor of a close system has stable periodic orbits. In a more typical case this hyperbolic subset is 'wild', and the set of stable periodic orbits is countable. Moreover, it cannot be separated in the phase space from the hyperbolic subset. It is evident that this is caused by structurally unstable Poincar6 homoclinic trajectories. Let us consider first the two-dimensional case. We let O be a fixed point of the saddle type with eigenvalues 121< 1 and I~l < 1. We assume that the stable and unstable manifolds of O have a quadratic tangency along the structurally unstable homoclinic curve F.

Theorem 9.1 (Gonchenko, Turaev and Shilnikov (34)) If [2rl is a trajectory of the third type (see Section VI), then in the set B 1 of such diffeomorphisms the diffeomorphisms with a countable set of stable periodic points are everywhere dense. As diffeomorphisms with homoclinic tangency of the third type are everywhere dense in the Newhouse regions, then diffeomorphisms with a countable set of stable periodic points are also everywhere dense in the Newhouse regions close to those of the original diffeomorphism. An analogous statement holds true for the Newhouse intervals in the case of a C2-smooth family, as it follows from the results of Gavrilov and Shilnikov (26). This is especially important as it concerns three-dimensional systems with a negative divergence, for example, in the Lorenz model with large Rayleigh numbers, in systems with spiral chaos, in Chua's circuit, and in some diffeomorphisms whose Jacobians are less then one, for example, the H6non map. Let us consider the H6non map (42) for a while. This is a well-studied map, so, for example, it is known from (25) that when Ibl is small, the values of a for which the stable periodic points exist form an open, everywhere dense set. At the same time, numerical experiments exhibit a rather

846

Leonid Shilnikov

complex behaviour of the trajectories in the H6non map. It should be also mentioned that Benedicks and Carleson (16) proved the existence of a Cantor set of the values a of a positive measure for values of which the H6non map has a strange attractor. In the Benedicks-Carleson attractor, all of the periodic points are structurally stable whereas the attractor itself is not. Small perturbations of the parameter a m a y destroy such an attractor. This situation resembles the situation for the orientable C2-smooth circle diffeomorphism: quasi-periodic trajectories exist for irrational values of the Poincar6 rotation number but are structurally unstable. In the case of a one-parameter family of C2-smooth diffeomorphisms X~ with properties formulated in Theorem 9.1 when /~ = 0, the associated m a p Tk+m:a°~--~II0 (see Section VIII) after rescaling takes the form close to the H6non m a p (see Fig. 14(a)). The case where 12~1> 1 is reduced to the previous one if we consider the inverse of the original diffeomorphism. Then, the diffeomorphisms with a countable set of totally unstable (repelling) periodic trajectories will be everywhere dense in the Newhouse regions. The peculiarity of systems with homoclinic tangencies is that such systems have a countable set of moduli. The simplest one is In I,~1 0 = - -In 171"

(47)

We can go further in the case of systems with homoclinic tangencies of the third type: such systems have a countable set of f~-moduli (31). Below we consider also a high-dimensional case. We note at once that the results obtained for this case cannot always be generalized. O f special interest here are questions on the coexistence of periodic trajectories of various topological types as well as strange attractors of distinct structures. All together will allow us to understand deeply the nature of quasi-attractors. In addition, a number of results presented below are automatically properties of a new type of strange attractors, namely, a 'wild' attractor containing an equilibrium point of the saddle-focus type. The part of our discussion that concerns the strong results, m a y be found in the series of papers of Gonchenko, Shilnikov and Turaev (33-35). We let a family of vector fields X0 be Cr-smooth (r > 3) with respect to all of its arguments. We assume that the system X0 satisfies the properties (A)-(E) below. (A) X0 has a saddle periodic orbit L0 with multipliers 2i, 7j such that [Aml__< " • " <1211<1
21 is real and 2 > 1221, or 21 = 22 = 2e/~°, (q~¢0,rt)and2>1231, or 7 is real and 71 < 1721, or ~'1 = ~2 = vei~, (~k¢0,n)andT>lv3[;

(B) the saddle value a~ = 12rl# 1; x(C) the stable W s and the unstable W u manifolds of L have a quadratic tangency

Mathematical

Problems

of Nonlinear Dynamics

847

along the homoclinic curve To. In particular, dim ( pMn FM) = 2 where PM and wM are the subspaces tangential to w” and w”, respectively, at the point MEI,. The study of dynamics of such systems is usually reduced to that of the Poincare map on some cross-section S to periodic orbits L,. This map is constructed as a superposition of two maps T&L) and T,(p) such that T, is a map along the orbits close to the periodic orbit L, and T, is a global map along the orbits in a neighbourhood of I,,. The point 0 = LnS, is a saddle fixed point; its invariant manifolds we also denote by W” and W”. We let M’E W” and MpEW” be two points of intersection To with S, then we assume that the map T, is defined in a neighbourhood of a point M-, T,(M-) = M’. We denote the nonleading stable and unstable manifolds of 0 by W” and by W”“, respectively, and the subspaces tangent to the manifolds W”, w”, W”“, W”” at the point 0 by w, lV”, p, and V. Similarly, we let wS+, W”+, pi and IV”’ denote the leading directions. We can now show that W” and W”“, as associated leaves, may be uniquely embedded into invariant C-‘-smooth foliations F”” and F”” on W:, and and that there exist invariant Cl-smooth manifolds H,, and H, Kk, respectively, tangential to IV’@ V’ and IV0 p+, H,c W;b,, H,c W&,, (43). We assume also that (D) M+$W”, M-#W”“; (E) (1) T,(H,,) is transverse to F”” at M+ and (2) T;‘(H,) is transverse to F”” at M-; (F) the curve A’, in the space of C-smooth vector fields is transverse to H’. It is convenient to distinguish between the leading and non-leading coordinates. We let XJV be the leading coordinates, and let U,U be the non-leading coordinates. Then, the map To may be written in the form (&ii) = A(V)

+ (fi(X,~,Y,~,~),f*(X,~,Y,~))

(48)

C.v>fi)= w_w) + @I(X,~,Y,~,~L),92(X,U,Y,U))

where fi(-~,Y,O,O) =

0, "a0,0,Y,~) = $O,O,y,a)

g1 = (O,O,y,u) = 0,

g,(x,u,O,O) =

@(x,u,o,O) =0 aY

andf; and g - C’- ’ are smooth functions. We let II, and II, be sufficiently small neighbourhoods and Mp (O,O,y-,u-): &I = {(~w0?Yddl n, =

II(x0 - x+,w-~+)II

{~~l,~l,Ul~~l~/ll~~,,ul~ll I%,IIh

=0

of the points M+ (x+,u+,O,O)

IG3ll~o,~o>ll I%) --YEA

-I.‘-

(49)

II5%).

The map T’: l&-+lI, along the trajectories of the system in a neighbourhood of L, is defined on o. c II,, where cro consists of a countable set of non-crossing ‘strips’ al and u,“= kT$jj = Too c l-I,, where k stands for a number (maximal) of the iterations under which the map of the strips is defined; R depends on the size of the chosen neighbourhood. We denote GA= T:aE.

848

L e o n i d Shilnikov

We introduce the coordinates (xg,u~,y~,v~) in neighbourhoods Fli. The map Toff , 0n can be then written in a cross-form x1 = ATxo + ~O1(Xo,Uo,yl,vO

I

u~ = A~uo + ¢2(Xo,Uo,yl,v3

(50)

Yo = B ? " y l + ~O3(Xo,Uo,yl,vl) Vo = B z " v l +

~I4(Xo,Uo,Yl,t)I)

where N~billCr-2<2 ", i = 1,2, Ilq3jllcr-2<~ -", ~ <~
j = 3,4,

2>2>max{12Ll2M},

in the case (A1): A7 = 24; in the case (A2): At = 2"E(n~o), where (cos(n~o) - sin(nq~) E(mp) = \

sin(nq~)cos(mp) J

in the case (A3): BT" = 7i-"; /cos(n~) -sin(nq~)\ in the case (A4): B~" = 2"E(--n~O), where E ( n ¢ ) = |\ sin(n~9)cos(n~9) |J" Taking into account the conditions (C)-(F), the map T~: Sw-~S along orbits of the system Xu in a neighbourhood F0 recast as Xo- X

=

FX(xl,ul,yl -- y - , v o )

Uo--U +

=

F"(xl,ul,yl--y-,vo)

Yo

=

G(xl,ul,yl -- y - , v o )

=

H(xl,ul,yl--y-,vo),

vl--v-

(51)

where F x, F ~, G, H, C ~ are smooth functions, OF x --

(0,0,0,0,0)

= b ¢ 0,

Oyt G(xl,u~,yl - - y - , V o ) = D(x~,ul,yl - - y - , v o ) " (Yl - - Y - t ~ ) 2 + C(x~,ul,vo)

D(0,0,0,0) = d, ~k(0,0,0) = 0, C(0,0,0) = 0, O~?C0,0,0) = c ¢ 0 and y , - y -

(52)

= ~9(x~,u~,Vo)

U.~

is a solution of the equation ~3G Oy~

- - (Xl,Ul,yl -- y-,Vo) = 0.

As we stated above, the homoclinic trajectories may be three types: we partition the systems satisfying (A)-(E) into three subclasses. The systems of the first class are defined by the condition that if a < 1, then 7 is real and positive and d < 0 , and if a > 1, then 2 is real and positive and dc > 0. The second class consists of systems with real and positive 7 and 2 such that d > 0 and c < 0. All the rest go into the third class.

Mathematical Problems of Nonlinear Dynamics

849

We consider a small neighbourhood U of the contour LwF. We let N be the set of trajectories of the system X that lie entirely within N.

Theorem 9.2 For systems of the first class, N = {L,F}. For systems of the second class, N is equivalent to the suspension over the Bernoulli shift on three symbols {0,1,2} in which the trajectories{. • . 0 . • - 0 1 0 . - . 0 . • • } a n d { . - - 0 . • . 0 2 0 . • . 0 . - - } a r e identified. For systems of the third class, N also has nontrivial hyperbolic subsets, but they do not in general exhaust all of N. Moreover, under the motion along the surface B~ of systems of the third class the structure of the set N varies continuously. The main reason for this is the presence of moduli of ~-equivalence. The multipliers 21 and yj such that 12il = 2 and I~jl = 7 are said to be leading and the remainder are non-leading. We let Ps denote the number of the leading 2i, and Pu denote the number of leading 7j. We will then say that X has type (Ps,Pu). We also set 0 = - / l n 2/ln y. Theorem 9.3 Let XI,X26B~. Then for f~-equivalence (we talk about f~-equivalence via a homeomorphism that is homotopic to the identity in U) of X~ and )(2 it is necessary that 01 = 02 in the case of systems of type (1,1); that 01 = 02 and ~0~ = ~o2for systems of type (2,1); that 0~ = 02 and ff~ = if2 for systems of type (1,2); for systems of type (2,2): ~ol = ~o2 and ffl = if2, and also 01 = 02, except, perhaps, for the case q~l = ~02 = ~9, = ~p2~{2~/3,~/2,2~/5,~/3}. The quantities ~k, q~ and 0 are the fundamental moduli in the sense that they are defined everywhere on B~. If ~k/2rc, q~2rr, or 0 is irrational, then there also exist other moduli, for example, analogous to the quantity z introduced in (28--30). Moreover, we have the following. Theorem 9.4 In B~ there is a dense set B* of systems that have a countable number of saddle periodic orbits with structurally unstable homoclinic curves of the third class. F o r systems from B* the quantities 0 computed for the corresponding saddle periodic orbits are moduli of f~-equivalence (Theorem 9.3). Thus, we have a theorem which generalizes a result from (33, 34) to the higher-dimensional case. Theorem 9.5 In B] the systems with a countable number of moduli of f~-equivalence are dense. We note that if the ~-moduli are chosen as parameters, as they change, bifurcations of the non-wandering trajectories must occur--periodic orbits, homoclinic orbits, etc. Here, the presence of a countable number of moduli leads to infinitely degenerate bifurcations. Thus, we have the following. Theorem 9.6 In B~ the systems with a countable number of structurally unstable homoclinic curves of arbitrary orders of tangency and with a countable number of structurally unstable

850

Leonid Shilnikov

periodic orbits, both with multipliers p = 1 and with p = - 1, of arbitrary orders of degeneracy are dense. We let X be a system from B ~ (not necessarily of the third class). We consider a C ~ = smooth finite-parameter family X,, transversally intersecting B t at X for # = 0. We will call the open sets in the space of dynamical systems, in which the systems with structurally unstable homoclinic curves are dense, the Newhouse regions. F r o m (35, 37) we obtain the following.

Theorem 9.7 There exists a sequence of open sets A~ accumulating to/~ = 0 such that X, for/~A~ lie in the Newhouse region, and the values of kt for which the periodic orbit L, has a structurally unstable homoclinic curve of the third class are dense in A~. F r o m the two last theorems we obtain the following. Theorem 9.8 In the Newhouse regions the systems with infinitely degenerate periodic orbits and homoclinic orbits and systems with a countable number of moduli of ff2-equivalence are dense. Thus, the following theorem shows that the dynamics of the system X and all nearby ones is effectively determined only by the leading coordinates, and the nonleading coordinates lead to trivial dynamics. Theorem 9.9 For all systems close to X, the set N is contained in an invariant (Ps+Pu+ 1)dimensional CLsmooth manifold W c, depending continuously on the system. Any orbit L ~ N as ( m - P 2 + 1)-dimensional strong stable and ( n - p u + 1)-dimensional strong unstable invariant Cr-smooth manifolds W~s and W~", respectively, such that the trajectories belonging to W)~ (or W~u) tend exponentially to L as t > ~ + ~ (or t ~ - 0o) with exponent not less in modulus than 12l/T (or ~/T), where T is the period of the orbit L0, 0 < 2 < 2 , and ~>7. The intersection W C n S is defined by an equation of the form (u,v) = fc(x,y), wherefc(0,0) = 0 and 8fc(O,O)/O(x,y) = 0. The manifolds W~ and W"uL in the intersection with S are collections of leaves transversal to B~, and B], respectively, and having the form (x,y,v) = f~(u) and (x,y,u) = f~u(v). The dynamics on W c essentially depends on the modulus of the product of the leading multipliers, that is, on #2 = 2P'~¢°. Theorem 9.10 The trajectories of the intersection to W c of any system close to X have for #2 not less than ( P s - [pu/O]) < 1 negative Lyapunov exponents, and for #2 > 1 that have not less than ( P u - [pJ0]) < 1 positive Lyapunov exponents. (Here [. ] denotes the integer closest to • but strictly less than -.) For #2 < 1 we distinguish the three classes (1 +) (P~,Pu) = (1,1) or (Ps,Pu) = (2,1) for 27< 1. (2 +) (Ps,Pu) = (2,1) 27< 1, and (Ps,Pu) = (1,2) or (Ps,Pu) = (2,2) for 272< 1. (3 +) (Ps,Pu) = (2,2) for 272> 1.

Mathematical Problems of Nonlinear Dynamics

851

F o r ~2> 1 we distinguish the three classes (1-), ( 2 ) and (3-), which are obtained respectively f r o m (1 +)-(3 +) by replacing t by - t. It follows f r o m T h e o r e m 9.10 that in case ( l - ) (or (l+)) the restriction to W c o f any system close to X c a n n o t have periodic orbits with m o r e than l stable (or unstable) multipliers. F r o m this and T h e o r e m 9.9 we obtain the following.

Theorem 9.11 If ~2 > 1 or if L has nonleading unstable multipliers, then all trajectories o f the system X itself and any sufficiently close system are unstable. We have noticed already in Section VII that the n o r m a l form o f a system with a triple degenerate periodic orbit with the multipliers ( - 1 , - 1 , 1 ) m a y have the Lorenz attractor for certain parameter values. As we k n o w n o w f r o m the theorems above, such periodic orbits o f systems o f dimensions n > 5 exist in the N e w h o u s e regions. Moreover, it should be noted that, in the N e w h o u s e regions, along with stable periodic orbits there coexist a countable n u m b e r o f strange attractors. This implies that a quasiattractor m a y have a very complex hierarchy o f embedded limiting sets; in other words, accurate numerical experiments m a y reveal the existence the stability windows o f stable periodic orbits as well as those o f strange attractors.

X. Example of Wild Strange Attractor In this section, following the paper o f Shilnikov and Turaev (79), we will distinguish a class o f dynamical systems with strange attractors o f a new type. The peculiarity o f such an attractor is that it m a y contain a wild hyperbolic set. We note that such an attractor is to be u n d e r s t o o d as an almost stable, chain-transitive closed set. Let X be a s m o o t h (C r, r_>4) flow in R n (n>_4) having an equilibrium state O o f a saddle-focus type with characteristic exponents ~, - 2_+ i~o, - ~1, " ' " , - ~n- 3 where > 0, 0 < 2 < Rec~j, ~o # 0. We suppose 7 > 22

(53)

This condition was introduced in (53), where it was shown, in particular, that it is necessary so that no stable periodic orbit could appear when one o f the separatrices o f O returns to O as t ~ + ~ (i.e. when there is a homoclinic loop; see also (54)). Let us introduce coordinates (x,y,z) (x~Rl,y~R2,z~R "-3) such that the equilibrium state is at the origin, the one-dimensional unstable manifold o f O is tangent to the xaxis, and the ( n - 1)-dimensional stable manifold is tangent to {x = 0}. We also suppose that the coordinates Yl,2 correspond to the leading exponents y _+ i~o and the coordinates z correspond to the non-leading exponents ~. We suppose that the flow possesses a cross-section, say, the surface II: {flyll = 1,1[zll < 1,Ix[ < 1}. The stable manifold W s is tangent to {x = 0} at O, therefore it is locally given by an equation o f the form x = hS(y,z), where h ~is a s m o o t h function hs(O,O) -- 0, (hg'(0,0) = 0. We assume that it can be written in such f o r m at least when (IlYll < 1,1VzJI< 1) and that IhSl< 1 here. Thus, the surface I I is a cross-section for W~oc and the intersection o f WlSc with 1-I has the f o r m II0: x = ho(cp,z), where q~ is the angular coordinate: Yl = IlYIlcos ~o, Y2 = IIYllsin ~p, and h0 is a s m o o t h function - 1 < h0 < 1. One can make h 0 - 0 by a coordinate transformation and we assume that it is done.

Leonid Shilnikov

852

F+

D

F1

Fig. 24. A pseudo-projection in R3 of the neighbourhood of a homoclinic contour of a saddlefocus. We suppose that all the orbits starting on H/1-I0 return to 11, thereby defining the Poincar6 map: T+: I - I + ~ H , T_: 1-I ~I-I, where H+ = H n { x > 0 } , H_ = H n { x < 0 } . It is evident that if P is a point on H with coordinates (x,~o,z), then lira T_(P) = P~_ lim T+(P) = P~+

x~--0

x~+0

where P U 1 and P~+ are the first intersection points of the one-dimensional separatrices of O with H. We may therefore define the maps T+ and T so that T (110) = P' ,

T+(H0) = P~+

(54)

Evidently, the region @ filled by the orbits starting on H (plus the point O and its separatrices) is an absorbing domain for the system X in the sense that the orbits starting in ~ enter ~ and stay there for all positive values of time t. By construction, the region @ is the cylinder { Uyll < 1,lIzll _< l,]xl < 1} with two glued handles surrounding the separatrices (Fig. 24). We suppose that the (semi)flow is pseudo-hyperbolic in 9 . It is convenient for us to give this notion a sense more strong than is usually done (43). Namely, we propose the following.

Definition A semiflow is called pseudo-hyperbolic if the following two conditions hold: (A) At each point of the phase space, the tangent space is uniquely decomposed (and this decomposition is invariant with respect to the linearized semiflow) into a direct

Mathematical Problems of Nonlinear Dynamics

853

sum of two subspaces N~ and N2 (continuously depending on'the point) such that the maximal Lyapunov exponent in N~ is strictly less than the minimal Lyapunov exponent in N2: at each point M, for any vectors u~NI(M) and veN2(M) lim sup t1 In IIIu, ~ -lP < lim inf 1_In [Iv, II where u, and v, denote the shift of the vectors u and v by the semiflow linearized along the orbit of the point M; (B) The linearized flow restricted on N2 is volume expanding: 1I, > const • e ~' 1Io with some a > 0; here, V0 is the volume of any region in N2 and V, is the volume of the shift of this region by the linearized semiflow. The additional condition (B) is new here and it prevents of appearance of stable periodic orbits. Generally, our definition includes the case where the maximal Lyapunov exponent in N~ is non-negative everywhere. In that case, according to condition (A), the linearized semiflow is expanding in N2 and condition (B) is satisfied trivially. In the present paper, we consider the opposite case, where the linearized semiflow is exponentially contracting in N1, so condition (B) is essential here. We note that the property of pseudo-hyperbolicity is stable with respect to small smooth perturbation of the system: according to (43), the invariant decomposition of the tangent space is not destroyed by small perturbations and the spaces N~ and N2 depend continuously on the system. Therefore, the property of volume expansion in N2 is also stable with respect to small perturbations. Our definition is broad; it embraces, in particular, hyperbolic flows for which one may assume (N~,N2) = (NS,NUGNo) or (Nj,N2) = (NSONo,NU), where N s and N u are, respectively, the stable and unstable invariant subspaces and No is a one-dimensional invariant subspace spanned by the phase velocity vector. The geometrical Lorenz model from (6, 7) or (40) belongs also to this class: here N~ is tangent to the contracting invariant foliation of codimension two and the expansion of areas in a two-dimensional subspace N2 is provided by the property that the Poincar6 map is expanding in a direction transverse to the contracting foliation. In the present paper, we assume that N~ has codimension three (i.e. dim Nt = n - 3 and dim N2 = 3), and that the linearized flow (at t > 0) is exponentially contracting on N~. Condition (A) means here that if for vectors N2 there is a contraction, it has to be weaker than those on N~. To stress the last statement, we will call N~ the strong stable subspace and N2 the centre subspace, and will denote them as N ~s and N c, respectively. We also assume that the coordinates (x,y,z) in R" are such that at each point of ~ the space N ss has a non-zero projection onto the coordinate space z, and N c has a nonzero projection onto the coordinate space (x,y). We note that our pseudo-hyperbolicity conditions are satisfied at the point O from the very beginning: the space N ~ coincides here with the coordinate space z, and N ~ coincides with the space (x,y); it is condition Eq (53) which guarantees the expansion of volumes in the invariant subspace (x,y). The pseudo-hyperbolicity of the linearized flow is automatically inherited by the orbits in a small neighbourhood of O. Actually, we require that this property should extend into the non-small neighbourhood ~ of O.

Leonid Shilnikov

854

According to (43), the exponential contraction in N ss implies the existence of an invariant contracting foliation JV ss with Cr-smooth leaves which are tangent to N ~s. As in (7), one can show that the foliation is absolutely continuous. After a factorization along the leaves, the region @ becomes a branched manifold (as ~ is bounded and the quotient-semiflow expands volumes, it follows evidently that the orbits of the quotientsemiflow must be glued on some surfaces in order to be bounded; see (40)). The property of pseudo-hyperbolicity is naturally inherited by the Poincar6 m a p T-(T+,T) on the cross-section H: here, we have: (A*) There exists a foliation with smooth leaves of the form (x,(p) = h(z)l_l
L e m m a 10.1 Let us write the m a p T as (2,Cp) = g(x,q),z),

2 = f(x,(p,z)

w h e r e f a n d g are functions smooth at x # 0 and discontinuous at x = 0: lim (9,f) = ( x _ , ~ o _ , z ) - P L ,

X~

xlim0 (9,f) = (x +,(p +,z +)=- P'+

0

We let det

(ss)

c3g # O. a(x,~)

We denote A

-

c

=

{ @ ~'@

( @ '~-' D

=

"

If lim C = 0,

x~O

lim NAIIIIDII = 0

x~O

(56)

Mathematical Problems o f Nonlinear Dynamics

855

sup I[BIIx,~rt.no sup IlCll
(57)

sup ~ I I D I I

P~YI. IIO

then the map has a continuous invariant foliation with smooth leaves of the form (x,~k) = h(z)l_ l_
P E H - 17 o

(58)

P E E I - FI o

then the foliation is contracting and if, moreover, for some fl > 0 the functions Alxl -~, Dlxl a, B and C are uniformly bounded and H61der continuous, and

din det D din det D 0~ and O(x,~o~ D l x f are uniformly bounded

(59)

then the foliation is absolutely continuous. The additional condition sup x / d e t D + x / s u p P ~ H " Fl 0

IIBII sup IPCIJ
PE[I" l] 0

(60)

PEn •H 0

guarantees that the quotient map ~ expands areas. It follows from (53, 54) that,in the case where the equilibrium state is a saddle-focus, the Poincar6 map near I-I0 = 1-In W' is written in the following form under some appropriate choice of the coordinates: (.~,~) = Q+_(Y,Z),

(61)

2 = R+_(Y,Z).

Here Y = IxlP( cos(~lnlxl + q~) sin(~lnlx] + q~)'] - sin(~lnlxl + ~0) cos(f~lnlx[ + ~o)] + Wl(x,q~,z)

(62)

Z = ud2(x,~o,z )

where p = 2/7 < ½(see Eq (53)), ~ = ~o/7 and, for some q > p, {~P + IqlkIJ i

Oxp0(~p,z)q = O([x["-P),

0 < p + [ql < r-- 2

(63)

The functions Q+, R+ in Eq (61) ( ' + ' corresponds to x>0---the map T+, ' - - ' corresponds to x < 0--the map T_) are smooth functions in a neighbourhood of (Y,Z) = 0 for which the Taylor expansion can be written: Q+ = ( x + , q g + ) + a + Y + b + Z +

• •.,

R+ = z + + c + Y + d + Z +

• • •

(64)

It is seen from Eqs (61)-(64) that if O is a saddle-focus satisfying Eq (53), then if a+ 5 0 and a ¢0, the map T satisfies conditions Eq (56) and Eq (59) with fle(p,q). Furthermore, analogues of conditions Eq (55), Eq (57), Eq (58) and Eq (60) are fulfilled where the supremum should be taken not over Ix[ < 1 but it is taken over small x. The map Eq (61), Eq (62) and Eq (64) is easily continued onto the whole cross-section H

856

L e o n i d Shilnikov

so that the conditions of the lemma were fulfilled completely. An example is given by the m a p = 0.9[xlPcos(ln[xl + q~) 37 = 31xlpsin(lnlxl + ~0) z7 = (0.5 + O . l z [ x [ " ) s i g n x

(65)

where 0.4 = p < it. As stated above, the expansion of volumes by the quotient-semiflow restricts the possible types of limit behaviour of orbits. Thus, for instance, in ~ there may be no stable periodic orbits. Moreover, any orbit in @ has a positive maximal Lyapunov exponent. Therefore, one must speak about a strange attractor in this case. Beforehand, we recall some definitions and simple facts from topological dynamics. We let XtP be the time-t shift of a point P by the flow X. For given ~ > 0 and r > 0 let us define as an (E,~)-orbit a sequence of points P~,Pz, " " " ,Pk such that Pi+~ is at a distance less than e from X~Pi for some t > r. A point Q will be called (E,T)-attainable from P if there exists an (E,v)-orbit connecting P and Q; it wilt be called attainable from P if, for some r > 0, it is (E,r)-attainable from P for any E (this definition, obviously, does not depend on the choice of r > 0). A set C is attainable from P if it contains a point attainable from P. A point P is called chain-recurrent if it is attainable from X~P for any t. A compact invariant set C is called chain-transitive if for any points P E C and Q e C C and for any E> 0 and r > 0 the set C contains an (e,r)-orbit connecting P and Q. Clearly, all points of a chain-transitive set are chain-recurrent. A compact invariant set C is called orbitally stable if for any its neighbourhood U there is a neighbourhood V(C)_c U such that the orbits starting in V stay in U for all t > 0. An orbitally stable set will be called completely stable if for any its neighbourhood U(C) there exist e0 > 0, z > 0 and a neighbourhood V(C)_c U such that the (~0,z)-orbits startinog in V never leave U. It is known that a set C is orbitally stable if and only if C = n Uj, where { Uj}f;= l is a system of embedded open invariant (with respect to the forw~r~ flow) sets, and C is completely stable if the sets Uj are not just invariant but they are absorbing domains (i.e. the orbits starting on c~Uj are inside Uj for a time interval not greater than some rj; it is clear in this situation that (E,r)-orbits starting on c3Uj lie always inside Uj if E is sufficiently small and ~ > ~j). As the maximal invariant set (the maximal attractor) which lies in any absorbing domain is, evidently, asymptotically stable, it follows that any completely stable set is either asymptotically stable or is an intersection of a countable number of embedded closed invariant asymptotically stable sets. Let us construct such an attractor of the system X and give an estimate of the number of the connected components of the intersection of the attractor with the cross-section rI. Definition

We call the set d of the points attainable from the equilibrium state O the attractor of the system X. This definition is justified by the following theorem.

Mathematical Problems of Nonlinear Dynamics

857

Theorem 10.1 The set o~¢ is chain-transitive, completely stable and attainable from any point of the absorbing domain 9 . The stability of s~¢ follows immediately from the definition: it is known that for any initial point (and for the point O, in particular) the set of attainable points is completely stable (the system of absorbing domains is given by the sets of the points (Ej,z)-attainable from the initial point, with arbitrary Ej--*+ 0 and ~ > 0). To prove the rest of the theorem, we note that the following lemma is valid. Lemma 10.2 Points asymptotic to 0 as t---, + oe are dense in @ (in other words, the stable manifold of 0 is dense in 9). Indeed, let us take an arbitrary point on II and let U be a neighbourhood of II. If U would not intersect with the pre-images of the surface 110 = Wi~ocC~ri,then for all i > 0 the map Tirv is continuous and the areas of the projections of the sets T~ onto z = 0 along the leaves of the invariant foliation would increase exponentially (by virtue of property (B*)), which contradicts the boundedness of ri. Thus, the pre-images of rl o (and these are the intersections of the stable manifold of 0 with the cross-section H) are dense in II, which implies that the stable manifold of 0 is dense in 9 . Lemma 10.2 implies immediately that 0 is attainable from any point of 9 . In particular, for any two points P and Q in ~4, the point 0 is attainable from P whereas Q is attainable from 0 according to the definition of •. Hence, Q is attainable from P, and to prove the chain-transitivity of d it remains to show that the (E,z)-orbits connecting P and Q can be chosen lying in d . This, however, follows from the complete stability of s~': for any S > 0 , an (E,v)-orbit connecting P and Q does not leave the 6neighbourhood of ~ ' if E is sufficiently small, whence this (E,~)-orbit may be approximated by an ((,~)-orbit which lies in sJ where E' may be greater than E but, clearly, d--*0 as E~0, 6 ~ 0 . Theorem 10.1 implies that d is the least completely stable subset of 9 : as any point belongs to a completely stable set together with all points attainable from it, Theorem 10.1 implies that any completely stable subset of ~ must contain the point 0 and, consequently, the set s~¢. Thus, s~¢ is the intersection of all completely stable subsets of 9. The next theorem shows that sJ is a unique chain-transitive and completely stable set in 9 . Theorem 10.2 If an orbitally stable subset of ~ contains points which do not belong to d , it contains points which are not chain-recurrent. Proof As the stable manifold of 0 is dense in 9 , any orbitally stable subset of ~ contains the point O. Because any connected component of an orbitally stable set is orbitally stable itself, each of the components must contain O, whence there is only one corn-

858

Leonid Shilnikov

ponent; that is, any orbitally stable subset of ~ is connected. We let C be such a set and suppose there is a point P~C, P e a l . As d is completely stable, there exists an absorbing domain U which contains ~¢ and does not contain P. The point O belongs to both ~ ' and C: we have that C contains points both inside U and outside U. The set C is connected and, hence, it must contain at least one point on dU. As we stated above, the (E,r)-orbits starting on 0U always lie inside U, at a finite distance from 0U, if E is sufficiently small and r is sufficiently large, which means that no point of ~3U is chain-recurrent. The next theorem extends into the multi-dimensional case the result of item (1) of Theorem 1.5 in (7) and gives a description of the decomposition into connected components of the intersection of the attractor d with the cross-section H; we will use this in the p r o o f of Theorem 10.4. We let q > 1 be the area expansion factor for the quotient m a p 7": if V is a region in H not intersecting H0, then (66)

S( TV) > qS( V)

where S denotes the area of the projection of the region onto the surface {z = 0} along the leaves of the invariant foliation. According to (B'), q > 1. Also, as S(H) = 2S(H+) and T H + o H , we have S ( T H + ) / S ( H + ) < 2 , and therefore 1 < q < 2 . We denote the separatrices of O by F + and F - , and let {P+ } be the consecutive points of intersection of F -+ with l-I; these sequences may be infinite or finite, the latter if the corresponding separatrix forms a homoclinic loop (returns to O as t ~ + ~ ) .

Theorem 10.3 The number N of connected components of the set d n H

2_
is finite and

I l n ( q - 1)1 lnq

(67)

Each of the components contains at least one of the points P+. Furthermore, for some integers N+ > 1, N_ > 1, N+ -4-N_ = N satisfying the inequality q-U++q

the set d n H

U >1

(68)

is represented in the form ~¢nH = d+u

• • . wd~+wd~w

• • • UZ~'N_

(69)

where z¢ + and d 7 are the components containing the points P+ and PT, respectively. In this formula all the components d + are different, J+c~IIo = ~,

d~+nHo # ~,d~

i < N +,

~¢,-c~FIo = ~ ,

nl-lo#~

i
(70)

Mathematical Problems of Nonlinear Dynamics

859

and

~¢~ = T~-1o~¢{ a t i < N

,

~4,.+ = T ~ - ' d ? - a t i < N +

(71)

(see Fig. 25). Let us take a one-parameter family X~ of systems of the kind under consideration and assume that a homoclinic loop of the saddle-focus O exists at/~ = 0, that is, one of the separatrices (say, F+) returns to O as t--* -t- c~. In other words, we assume that the family X~, intersects, at # = 0, a bifurcational surface filled by systems with a homoclinic loop of the saddle-focus and we suppose that this intersection is transverse. The transversality means that when/~ varies, the loop splits and if M is the number of the last point of intersection of the separatrix F+ with the cross-section I-I at/a = 0 (P~t~I-I0 at/J = 0) then the distance between the point P h and FI0 changes with a 'nonzero velocity' when/~ varies. We choose the sign of/J so that P~t~rl+ w h e n / ~ > 0 (or P ~ I - I when/~ < 0).

Theorem 10.4 There exists a sequence of intervals Ai (accumulated at p = 0) such that when p~Ai, the attractor .4~, contains a wild set (non-trivial transitive closed hyperbolic invariant set whose unstable manifold has points of tangency with its stable manifold). Furthermore, for any/t*~Ag, for any system Cr-close to a system X%, its attractor o4 also contains the wild set. As far as the general case is concerned when the separatrix of the point O does not form homoclinic loops, we note that it follows from the density of the stable manifold in ~ that the separatrix of the point O has non-wandering trajectories, which we can suppose m a y be closed by small perturbations of the system. A similar problem has already arisen in the case of the Lorenz m a p but it was overcome (in cr-topology) using certain specific properties of the map. We now have a rather important lemma of Hayashi (41), which allows us to solve this problem. Hence the following statement is true.

• , °, +

AN+-I

°

I~ O

° +

AI 17 0

Fig. 25. Schematic illustration of the connection components of the attractor A.

860

Leonid Shilnikov

Statement The systems with a homoclinic loop to a saddle-focus are dense in Cr-topology in the class of systems under consideration. We have mentioned that the presence of structurally unstable (non-transverse) homoclinic trajectories leads to non-trivial dynamics. Thus, using results (33, 37) we can conclude from Theorem 10.4 that the systems whose attractors contain structurally non-transverse homoclinic trajectories as well as structurally unstable periodic orbits of high order of degeneracy are dense in the given regions in the space of dynamical systems. In particular, the values o f # are dense in the intervals At for which an attractor of the system contains a periodic orbit of the saddle-saddle type along with its threedimensional unstable manifold. For these parameter values, the topological dimension of such an attractor is already not less than three. The latter implies that the given class of systems is an example of hyperchaos. U p to now we have not yet spoken about the statistical properties of the attractors. This is necessary, as all of the trajectories of the strange attractors are unstable. Sinai (65) introduced the following notion of a stochastic attractor. A stochastic attractor is an invariant closed set A in the phase space with the following properties: 1. there exists a neighbourhood U, A c U, such that ifx~U, then dist(x(t),A)~O as t ~ +~; 2. for any initial probability distribution P0 on A, its shift as t ~ + ~ to an invariant distribution P on A, independently of P0; 3. the probability distribution P is mixing, that is, the autocorrelation function tends to zero as t ~ o o . It is known that on hyperbolic attractors (and even on non-trivial basis sets) an invariant, 'rather' physical measure, the so-called Bowen-Ruelle-Sinai measure, may be introduced. This allows one to prove that hyperbolic attractors are stochastic in the sense above. This result is true also for Lorenz-like attractors (20). Both types of such attractors have a specific feature: the dimension of the stable (unstable) manifolds of their trajectories (excluding the saddle point at the origin in the case of the Lorenz attractor) is always the same, that is, such an attractor contain the trajectories of the same topological type. This is not the case when we deal with the wild strange attractors, as they m a y contain coexisting hyperbolic trajectories of various types. This is a challenge for the ergodic theory.

Acknowledgements The author would like to thank L. Chua for useful discussions. He is also grateful to A. Shilnikov for help in preparing this manuscript. This work was supported in part by the Russian Foundation of Basic Research 96-01001135, by INTAS-93-0570, and by NATO Linkage Grant no. OUT.LG960578.

References (1) Andronov, A. A., Leontovich, E. A., Gordon, I. E. and Maier, A. G., The Theory oJ Bifurcations of Dynamical Systems on a Plane. Israel Program of Scientific Translations, Jerusalem, 1971.

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(2) Andronov, A. A., Leontovich, E. A., Gordon, I. E. and Maier, A. G., The Theory oJ Dynamical Systems on a Plane. Israel Program of Scientific Translations, Jerusalem, 1973. (3) A. A. Andronov and L. Pontryagin, "Syst6mes grossiers", Doklady Akademii Nauk SSSR, Vol. 14, pp. 247-251, 1937. (4) Andronov, A. A., Vitt, E. A. and Khaikin, S. E., Theory of Oscillations. Pergamon Press, Oxford, 1966. (5) Afraimovich, V. S., Arnold, V. I., II'yashenko, Yu. S. and Shilnikov, L. P., Bifurcation theory. In Dynamical Systems 1I, Encyclopedia of Mathematical Sciences, 5. Springer, Berlin, 1989. (6) V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov, "On the appearance and structure of Lorenz attractor", Doklady Akademii Nauk SSSR, Vol. 234, pp. 336-339, 1977. (7) V. S. Afraimovich, V. V. Bykov and L. P. Shilnikov, "On the structurally unstable attracting limit sets of Lorenz attractor type", Transactions of the Moscow Mathematical Society, Vol. 2, pp. 153-215, 1983. (8) Afraimovich, V. S. and Shilnikov, L. P., On critical sets of Morse-Smale systems. Trudy Moscovskogo Matematicheskogo Obchestva, 1973, 28, 181-214; English translation in Transactions of the Moscow Mathematical Society, 1973, 28. (9) V. S. Afraimovich and L. P. Shilnikov, "On some global bifurcations connected with disappearance of saddle-node fixed point", Doklady Akademii Nauk SSSR, Vol. 219, pp. 1981-1985, 1974. (10) Afraimovich, V. S. and Shilnikov, L. P., Strange attractors and quasi-attractors. In Nonlinear Dynamics and Turbulence, ed. G. I. Barenblatt, G. Iooss and D. D. Joseph. Pitman, New York, 1983, pp. 1-28. (11) Anosov, D., Geodesic flows on compact Riemannian manifolds of negative curvature (in Russian). Trudy Math. Inst. im. I1".A. Steklova, 1967, 90. (12) Anosov, D. V., Aronson, S. Kh., Bronshtein, I. U. and Grines, V. Z., Smooth dynamical systems, In Dynamical Systems, Encyclopedia of Mathematical Sciences, 1. Springer, Berlin, 1985. (13) Anosov, D. V. and Solodov, V. V., Hyperbolic sets. In Dynamical Systems 9, Encyclopedia of Mathematical Sciences, 66. Springer, Berlin, 1991. (14) Aronson, S. Kh. and Grines, V. Z., Cascades on surfaces. In Dynamical Systems with Hyperbolic Behaviour. Dynamical Systems 9, Encyclopedia of Mathematical Sciences, 66. Springer, Berlin, 1991. (16) M. Benedicks and L. Carleson, "The dynamics of the H6non map", Annals of Mathematics, Vol. 133(1), pp. 73-169, 1991. (17) G. D. Birkhoff, "Nouvelles recherche sur les syst6mes dynamiques", Memorie Pont Acad. Sci. Novi Lyncaei, Vol. 3(1), pp. 85 216, 1935. (18) R. Bowen, "Markov partition for Axiom A diffeomorphisms", American Journal of Mathematics, Vol. 92(3), pp. 724-747, 1970. (19) R. Bowen, "Symbolic dynamics for hyperbolic flows", American Journal of Mathematics, Vol. 95(2), pp. 429-460, 1973. (20) Bunimovich, L. A. and Sinai, Ya. G., Stochasticity of an attractor in the Lorenz model (in Russian). In Nonlinear Waves, ed. A. V. Gaponov-Grekhov. Nauka, Moscow, 1980, pp. 212-226. (21) V. V. Bykov and A. L. Shilnikov, "On boundaries of the region of existence of the Lorenz attractor", Selecta Mathematica Sovietica, Vol. 11(4), pp. 375 382, 1992. (22) O. L. Chua, "Global unfolding of Chua's circuit", IEICE Trans. Fund. ECCS, Vol. 76(SA), pp. 704-734, 1993. (23) W. De Melo and S. J. van Streen, "Diffeomorphisms on surfaces with finite number of moduli", Ergodic Theory and Dynamical Systems, Vol. 7, pp. 415-462, 1979.

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