Mathematical Problems of Nonlinear Dynamics: a Tutorial

Home , Bernoulli scheme, Hyperbolic set, Structural stability

Mathematical problems of nonlinear dynamics: A tutorial

Leonid Shilnikov

Abstract We review the theory of nonlinear systems, especially that of strange attractors, and give its perspectives. Of a special attention are the recent results concerning hyperbolic attractors and features of high-dimensional systems in the Newhouse regions. We present an example of a “wild” strange attractor of the topological dimension three.

1 Content

1. Introduction

2. Basic notions of the theory of dynamical systems

3. The Andronov-Pontryagin theorem. Morse-Smale systems

4. Poincar´ehomoclinic structures

5. Structurally stable systems

6. The bifurcation theory. Appearance of hyperbolic attractors

7. Structurally unstable systems. Wild hyperbolic sets. Newhouse regions

8. Lorenz attractor

9. Quasiattractors. Non-transverse homoclinic curves

10. Example of wild strange attractor

2 1 Introduction

The early 60th is the beginning of the intensive development of the theory of high-dimensional dynamical systems. Within a short period of time Smale [66, 67] had established the basics of the theory of structurally unstable 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 such scientists with the broad vision on science as Poincaréand Andronov. All this makes the QTODEs especially attractive and practical. Its achievements have led to one of the brightest scientific discoveries of the XX century — dynamical 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 the science. If the mathematical images of the formers are equilibrium states, periodic orbits and tori with quasiperiodic trajectories, then the adequate image of dynamical chaos is a strange attractor, i.e., an attractive limiting set with the unstable behaviour of its trajectories. Those attractors that persist this property under small smooth perturbations will interest us in this review. Namely, such attractors have been predicted by the hyperbolic theory of high-dimensional dynamical systems. However, the role and the significance of strange attractors were not accepted by researchers of certain scientific directions, in particular of turbulence, for a sufficiently long time. There were a few reasons for that. The hyperbolic theory had examples of strange attractors but their structure were so topologically

3 complex that did not allow one to imagine rather simple scenarios of their origination which 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 “qenuine” 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. Note also strange attractors that have hyperbolic subsets co-existing with stable long periodic orbits of very narrow and tortious attraction basin, the so-called quasi- 1attractors [10]. The breakthrough came in the mid 70-ths with the appearance of a “simple” low-dimensional model x˙ = σ(x y), y˙ = rx− y− xz, z˙ = 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 not the least but a primary role. The topological dimension of the Lorenz attractor, regardless of the dimen-

1stochastic

4 sion 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 quasiattractors? In principal, the hyperbolic theory predicts the possibility of the existence of strange attractor 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 a quasi-periodic trajectory on it. They have also discovered a principally new type of strange attractor — the so-called wild strange attractors. Their distinc- tion 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 invariant manifolds of the co-existing trajectories may be equal both two and three. Moreover, the region of the existence of such an attractor is a region of everywhere dense structural instability due to homoclinic tangencies. Thus, the above phenomenon poses principally new problems for ergodic theory. Due to the co-existence of the trajectories of various types the wild attractors, as well as Lorenz-like attractors, are the pseudo-hyperbolic attractors. Since the notion of pseidohyperbolicity, which plays a dominating role in the theory of structurally unstable strange attractors, will be used below. So let us stop here and discuss it in detail. Consider a smooth n-dimensional dynamical system

x˙ = X(x), in a bounded region D which satisﬁes the following conditions:

1. On its boundary ∂D the vector ﬂow goes inward D. This implies that for any point x ∂D either an entire trajectory or a semitrajectory is deﬁned ∈ that passes through the point x.

5 2. A pseudo-hyperbolicity takes place in D. This implies that at each point x D the tangent space, invariant with respect to the associated lin- ∈ earized ﬂow, may be decomposed as a direct sum of subspaces N1 and N2, depending continuously on the point x so that the maximal Lyapunov ex-

ponent, corresponding to N1, is strongly less than any Lyapunov exponent

corresponding to N2. In other words, the associated variational equation can be represented in the form

ξ˙ = A1(t)ξ, η˙ = A2(t)η,

where the contraction in ξ is stronger than the contraction in η.

3. The linearized semiﬂow is volume-expanding

V const eσtV , σ> 0. t ≥ 0

Note that the property of the pseudo-hyperbolicity persists under small smooth perturbations, as does the property of the exponential expansion of volumes in

N2. Due to the above requirements there will exist at least one strange attractor in the region D. Note that this suggested criterion on the existence of a pseudohyperbolic attractor in D is formulated relatively simply, but, similarly the principle of the contraction mappings, its real verification in concrete systems will not be trivial. Let us return to the problem of quasi-attractors. In both situation, in the case of quasi-attractors and in the case of wild strange attractors, the reason of complexity is the presence of structurally unstable Poincaréhomoclinic curves, i.e., bi-asymptotic trajectories to a saddle periodic orbit, along which its stable and unstable manifolds have a non-transverse contact. This type of homoclinic trajectories 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. Un- der 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

6 stable periodic orbits is that it cannot be separated in a quasiattractor from the co-existing hyperbolic subset to which these periodic orbits accumulate. In three-dimensional systems with sigh-alternating divergences, for example in the Chua circuit [22, 45] , the situation is even more sophisticated [77]. Namely, 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 imbedded 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 since 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 on the analysis of concrete models. The basic idea is the following:

1. Partitioning the parameter space into regions of structural stability and identifying the bifurcation set;

2. Dividing the bifurcation set into connected components corresponding to qualitatively similar, in the sense of topological equivalence, phase portraits.

Nowadays, Andronov’s program 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., since structurally unstable systems may ﬁll out entire regions in the space

7 of dynamical systems. As established by Gonchenko, Shilnikov and Turaev [33, 34] the following C r-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 make the following important conclusion; namely the goal of a complete investigation of systems with complex dynamics of above types is unrealistic. It appears that one should abandon the ideology 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.

2 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., t R1, or discrete, i.e., t Z. 3) An evolution law, ∈ ∈ i.e., a mapping of any given point x in E and any t to a uniquely deﬁned state ϕ(t, x) E. It is also assumed that the following conditions hold: ∈ 1. ϕ(0, x)= x.

2. ϕ(t1, ϕ(t2, x)) = ϕ(t1 + t2, x), (2.1)

3. ϕ(t, x) C0 with respect to (t, x). ∈ 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 2 of the phase space E. Fixing x and varying t from −∞ 2i.e., one-to-one, continuous mappings with a continuous inverse. This follows directly from the group property (2.1).

8 to + we obtain an orientable curve 3 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; ϕ(t, x),t 0 and x; ϕ(t, x),t 0 a negative n ≥ o n ≤ o semi-trajectory. Observe that in the case of an unclosed trajectory any point of the trajectory partitions the trajectory into two parts: a positive semi-trajectory and a negative semi-trajectory. In the case where the mapping ϕ(t, x) is a diffeomorphism 4 the flow is a smooth dynamical system In this case the phase space E must be endowed with some additional structures. The phase space E is usually chosen to be either Rn, or Rn−k Tk, where Tk = S1 S1 S1 may be a k-dimensional torus, × × ×···× a smooth surface, or a manifold. This allows us to set up a correspondence between smooth flow and associated vector fields by defining a velocity field

dϕ(t, x) V (x)= . (2.2) dt t=0

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

x˙ = X(x), x Rn, ∈ where X Cr, r 1. ∈ ≥ Discrete dynamical systems are, more brieﬂy, called cascades. A cascade possesses the following remarkable feature. Let us select a homeomorphism ϕ(1, x) and denote it by ϕ(x). It is obvious that ϕ(t, x)= ϕt(x), where

ϕt = ϕ (ϕ(...ϕ(x))) .

t times | {z } Hence, in order to deﬁne a cascade it is suﬃcient only to point out the homeomorphism ϕ : E E. 7→ +∞ In the case of a discrete dynamical system a sequence xk where xk+1 = n o−∞ ϕ(xk), is called the trajectory of a point x0. Trajectories may be of three types:

3which defines the direction of motion 4a one-to-one, differentiable mapping with differentiable inverse.

9 1. A point x0. The point is a ﬁxed point of the homeomorphism ϕ(x), i.e., it is mapped by ϕ(x) onto itself.

2. A cycle (x ,...,x ), where x = ϕk(x ), i = (0,...,k 1) and where, 0 k−1 i i − moreover, x = x for i = j. The number k is called the period of the i 6 j 6 cycle, and the point xi is called a periodic point of period k. Observe that a ﬁxed point is a periodic point of period 1.

+∞ 3. A bi-infinite (i.e., infinite in both directions) sequence xk , where n o−∞ x = x for i = j. In this case, as in the case of flows, we will say that i 6 j 6 such a trajectory is unclosed.

When ϕ(x) is a diﬀeomorphism, the cascade is a smooth dynamical system. Examples of cascades of this type are non-autonomous periodic systems. Consider the system

x˙ = X(x, t), where X(x, t) is deﬁned and continuous with respect to all variables in Rn R1, × is smooth with respect to x and periodic of period τ with respect to t, and has solutions which may be continued over the interval 0 t τ. Given a solution ≤ ≤ x = ϕ(t, x0), where ϕ(0, x0)= x0 we may deﬁne the mapping

x1 = ϕ(τ, x) (2.3) of the hyper-plane t = 0 into the hyper-plane t = τ. It follows from the period- icity of X(x, t) that (x, t ) and (x, t ) must be identified if (t t ) is divisible 1 2 2 − 1 by τ. Thus, (2.3) may be regarded as a diffeomorphism ϕ : Rn Rn. 5 7→ Among cascades of special interest are the so-called topological Markov chains (TMC). Let us define a TMC. Let G be an orientable multigraph such that each vertex is the beginning of a certain edge, and at the same time, the end of the same edge or another edge. Let us enumerate the edges of G by symbols

5Observe that in this case system (2.3) may be written as an autonomous system x˙ = X(x,θ), θ˙ = 1, where θ is taken in modulo τ.

10 1, 2,...,m and construct an (m m)-dimensional matrix A(a ) according to × ij the following rule: aij = 1 if the end of edge i is the beginning of edge j; aij =0 otherwise. For example, for G in the form

✲ ❄ ✬✩❄ ❄ ✓✏✓✏✛✓✏ r r ✒✑✒✑✒✑ ✫✪ the matrix A is 11100  11100  00011 A =   .  11100     11100     00011    Further, consider a set Ω of sequences, inﬁnite in both directions, composed of symbols 1,...,m,

Ω= (...ω−1,ω0 ω1 ...ωk ...) , n ↑ o with a fixed position of the zero coordinate such that the symbol ωk+1 may follow the symbol ωk if and only if aωkωk+1 = 1. In other words, Ω can be identified by a set of paths, infinite in both directions, along the graph G. Let us define a distance dist on Ω:

+∞ α β dist(α, β)= | i − i | 2|i| i=X−∞ for α = (α−1α0 ...), β = (...β−1β0 ...). It is a simple matter to verify that Ω with the distance d is a complete metric space. It can also be shown that if Ω has the cardinality of continuum, it is homeomorphic to a reference Cantor set on the segment [0, 1]. Let is deﬁne a transformation σ on Ω: σω = ω′ if

ω = (...ω−kωω−1ω0 ...), ′ ′ ′ ω = (...ωω−1ω0 ...),

ω′ = ω , k Z, k k+1 ∈

11 i.e., in the sequence ω the position of the zero coordinate is translated (shifted) by one symbol to the right: (...ω−1ω0ω1 ...). The transformation σ is called a shift. It can be easily verified that σ is single-valued and continuous together with the inverse mapping σ−1. It is the cascade (σk, Ω) which is called TMC, and this will be denoted hereafter as (G, Ω, σ). If there exists a path from any vertex of the graph G to any other one, σ Ω is transitive, and if , moreover, h = ln λ , σ has a countable set of | 1 periodic points and they are everywhere dense in Ω. Here λ1 is the maximal eigenvalues of the matrix A, and h is the topological entropy of the mapping σ. Introduce also the notion of a suspension over TMC. In the direct product Ω I, where I is the segment [0, 1], we identify the points (ω, 1) and (σω, 1) × for all ω Ω. The topological space thus obtained is the phase space of such a ∈ system and is denoted by Σ. 6 Let us define a flow T t on Σ by the following relation:

(ω,t + s), if s

U(x ) ϕ(t,U(x )) = for t > T. (2.4) 0 ∩ 0 ∅ Applying the transformation ϕ( t, x) to (2.4) we obtain − 6Σ may be provided with a metric.

12 ϕ( t,U(x )) U(x )= for t < T. − 0 ∩ 0 ∅ 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 non-wandering points. The set is open W W and invariant. Openness follows from the fact that together with x0 any point in U(x ) is wandering. The invariance of follows from the fact that if x is 0 W 0 a wandering point, then the point ϕ(t0, x0) is also a wandering point for any t0. To show this let us choose ϕ(t0,U(x0)) to be the neighborhood of the point

ϕ(t0, x0). Then

ϕ(t ,U(x )) ϕ(t, ϕ(t ,U(x )) = for t > T. 0 0 ∩ 0 0 ∅ Hence, the set of non-wandering points = D is closed and invariant. The M \W set of non-wandering points may be empty. To illustrate the latter consider a dynamical system deﬁned by the autonomous system

x˙ = X(x, θ), θ˙ =1,

n+1 with phase space R , x = (x1,...,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 neighborhood U(x0) and any T > 0 there exists t>T such that

ϕ(t, x ) U(x ). (2.5) 0 ⊂ 0 If there exists t such that t< T and (2.5) holds, then the point x is called a − 0 negative Poisson-stable point. If point x0 is both positive and negative Poisson stable it is said to be Poisson-stable, (see Fig. 2.1).

13 Observe that if the point x0 is positive (negative) Poisson-stable, then any point of the trajectory ϕ(t, x0) 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 (2.5) that P +, P − and P -trajectories consist of non-wandering points. It is obvious that equilibrium states and periodic orbits are closed P -trajectories. Denote by Σ the closure of P + (P −, P )-trajectory.

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

Let us choose a positive sequence of Tn where Tn + as n + . n o → ∞ → ∞ It follows from the deﬁnition of a P +-trajectory that there exists a sequence

tn + as n + such that ϕ(tn, x0) U(x0). An analogous statement n o → ∞ → ∞ ⊂ holds in the case of a P −-trajectory. This implies that a P -trajectory succes- sively intersects any ǫ-neighborhood Uǫ(x0) of the point x0 inﬁnitely many times. +∞ 7 Let tn(ǫ) be such that tn(ǫ) < tn+1(ǫ) and let ϕ(tn(ǫ), x0) Uǫ(x0). n o−∞ ⊂ We introduce the notion of the Poincar´ereturn times

τ (ǫ)= t (ǫ) t (ǫ). n n+1 − n Two essentially diﬀerent cases are possible for an unclosed P -trajectory:

1. the sequence τ (ǫ) is bounded, i.e., there exists a number L(ǫ) such n that τ (ǫ)

2. The sequence τ (ǫ) is unbounded. n In the ﬁrst case the P -trajectory is called recurrent. For such a trajectory all trajectories in its Σ closure are also recurrent, and the closure itself is a minimal set 8. The principal property of a recurrent trajectory is that it returns to an ǫ-neighborhood of the point x0 within a time not greater than L(ǫ).

7 In the case of ﬂows the P -trajectory intersects Uǫ(x0) for values of t in an inﬁnite set of intervals In(ǫ) where tn(ǫ) is one of the times in In(ǫ). 8a set is called minimal if it is non-empty, invariant, closed and contains no proper subsets possessing these three properties.

14 However, in contrast to periodic orbits, whose return times are known, the return time for a recurrent trajectory is not constrained. Among recurrent trajectories we may also select a more narrow class of motions, the so-called almost-periodic trajectories. There is no uncertainty in the Poincaréreturn times of such a trajectory since it comes back to its neighbourhood over its almost-period τ(ǫ). Almost-periodic trajectories may be sub-divided into the two sub-classes: quasiperiodic and limit-quasiperiodic 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 Rm K where K is the Cantor discontinuum. × In the second case, the closure Σ of the P -trajectory is called a quasi-minimal set. There always exist in Σ other invariant closed subsets which may be equilibrium states, periodic trajectories or invariant tori, etc. Since the P -trajectory may approach such subsets arbitrarily closely, the Poincaréreturn times can be arbitrarily large. Suppose that for a trajectory L given by the equation x = ϕ(t) the closure of the semi-trajectory L+ (L−) for t t (t t ) is a compact set. ≥ 0 ≤ 0

Deﬁnition The point x0 is called an ω-limiting point of the trajectory L if there exists a sequence tk where tk + as k such that n o → ∞ → ∞ ∗ lim ϕ(tk)= x . l→∞

A similar deﬁnition for an α-limiting point t as k . We denote k → −∞ → ∞ the set of all ω-limiting points of the trajectory L by ΩL and that of the α- limit points we denote by . Observe that an equilibrium state has a unique AL limit point, namely, itself. In the case where the trajectory L is periodic all of its points are α and ω-limit points, i.e., L = Ω = . In the case where L AL L is an unclosed Poisson-stable trajectory the sets Ω and coincide with L AL its closure L¯. This L¯ is either a minimal set if L is a recurrent trajectory, or a quasi-minimal set if the return Poincar´etimes of L are unbounded. All equilibrium states, periodic trajectories and Poisson-stable trajectories are said to be self-limit trajectories.

15 The structure of the sets Ω ( ) has been almost completely studied for L AL two-dimensional dynamical systems on the plane where the trajectory remains in some bounded domain of the plane as t + (t ). → ∞ → −∞ Both sets Ω and are well know to be invariant and closed. In the case L AL when the system under consideration is a ﬂow, Ω ( ) is then a connected set. L AL Poincar´eand Bendixson established that the set ΩL may 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 . → ±∞ Fig. 2.2 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.

Observe that an analogous situation occurs in the case of negative semi-trajectories. 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. 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 the structure of the phase space by trajectories of different topological types, or in other words, ﬁnding its phase portrait. This poses the question of when two phase portraits are similar. In terms of the qualitative

16 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 equilibrium states, periodic trajectories and unclosed trajectories of another system. The topological equivalence of two systems in some sub-regions of the phase space is defined in the similar manner. The latter is used while studying local problems, for example, while studying the equivalence of structures of trajectories in a neighborhood 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 remain 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 = h be a { i} set of homeomorphisms of G mapping trajectory into trajectory of the same topological types. Then we can introduce a metric distance as follows

dist(h1,h2) = sup h1x h2x . x∈G k − k Deﬁnition We call a trajectory L G particular if there exists ǫ> 0 such ∈ that for all h satisfying dist(h, I) <ǫ, where I is the identity homeomorphism, the following condition holds

hL = L.

It is clear that all equilibrium states and periodic orbits are particular trajectories. Unclosed trajectories may be particular also. For example, particular

17 trajectories of a two-dimensional system which tend to equilibrium states both as t + and as t . Since such trajectories separate certain regions in → ∞ → −∞ the plane they are called separatrices. (See samples of separatrices in Fig. 2.2. The deﬁnition of particular semi-trajectories may be introduced in an analogous manner.

Deﬁnition Two trajectories L1 and L2 are said to be topologically equivalent if for given ǫ > 0 there exist homeomorphisms h1,h2,...,hm(ǫ), satisfying dist(hk, I) <ǫ, such that

L = h h L . 2 m(ǫ) ··· 1 1 where k = (1, 2,...,m(ǫ)) and I is the identity homeomorphism. We will call the set of equivalent trajectories a cellar. 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 ω-limiting and α-limiting sets. The role of particular trajectories and cellars are especially important for two-dimensional 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. 9 Let us suppose that S consists of a ﬁnite number of trajectories. 10

Theorem 2.2 The scheme is a topological invariant.

This theorem together with its proof occupies the signiﬁcant part of the book “Qualitative theory of dynamical systems on the plane” by Andronov, Leontovich, Gordon and Mayer [2]. This is not the case when we examine systems of higher dimensions. The set of particular trajectories of a three-dimensional system already may be inﬁnite or a continuum even. The same situation applies to the cellars. All trajectories of topological Markov chains and those of suspensions over TMC are also par-

9Indeed, the set S is a factor-system over a given relationship of equivalence. 10The condition of ﬁniteness of S is rather common for a wide class of planar systems.

18 ticular.

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

dist((ϕ(t, x), A) 0 as t + , (2.6) → → ∞ where

dist(x, A) = inf x, x0 . x0∈A k k

The simplest examples of attractors are stable equilibrium states, stable periodic trajectories and stable invariant tori containing quasi-periodic trajectories, which satisfy (2.6). 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 special attention in this paper.

3 Andronov-Pontryagin theorem. Morse-Smale systems

We will assume that all dynamical systems under consideration to be in the form

x˙ = X(x), where x C1 and are deﬁned in some closed, bounded region G¯ Rn. For such ∈ ⊂

19 vector ﬁelds we can introduce a norm as follows

∂X(x) X C1 = sup X(x) + . k k ¯ k k ∂x x∈G

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 an ǫ > 0 there exists δ > 0 such that if X˜ X < δ, then X and a neighboring system X˜ are k − k topologically equivalent. Moreover, the conjugating homeomorphism h is close to the identity homeomorphism I, i.e., dist(h, I) <ǫ. This definition was first introduced by Andronov and Pontryagin under the additional requirement that the boundary ∂G¯ of the region G¯ is a surface without a contact, e.g., the vector field is transverse to ∂G¯ and, moreover is directed inward G¯. 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 associated vector field 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 G¯ if there exists an ǫ> 0 such that if X˜ X <ǫ, then X and X˜ are topologically equivalent. k − k 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 on structurally stable systems that there exist always δ-homeomorphisms. For two-dimensional system of the form

x˙ = P (x, y), (3.1) y˙ = Q(x, y)

Andronov and Pontryagin proved the following theorem

20 Theorem 3.1 System (3.1) is rough if and only if

1. all equilibrium states of (3.1) are simple, i.e., if none of the roots of the characteristic equation

′ ′ Px(x0,y0) λ Py(x0,y0) det ′ − ′ =0 Q (x0,y0) Q (x0,y0) λ x y − of the associated linearized system at each equilibrium state lies in the imaginary axis in the complex plane.

2. all periodic trajectories of (3.1) are simple, i.e., if x = ϕ(t), y = ψ(t) is a periodic solution of period τ, then

τ P ′ (ϕ(t), ψ(t)) + Q′ (ϕ(t), ψ(t)) =0. Z x y 6 0 3. There exists no trajectories which are bi-asymptotical to a saddle, as well as those going from one saddle to another saddle as t , e.g. 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 trajectories. 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 takes place in the case of smooth flows on two-dimensional surfaces. The principal moment in his proof 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 Rn (for simplicity)

21 x˙ = X(x), X Ck, k 1, (3.2) ∈ ≤ is called structurally stable if the roots (λ1,...,λn) of the characteristic equation (which are called characteristic exponents of the equilibrium state),

∂X(0) det λE = 0 (3.3) ∂x − 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 p = 0, the equilibrium state is called a 6 6 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 W s (W u of the equilibrium → −∞ O O Ck s state O. It is known that if O is of (m,p)-type, the -smooth manifolds WO u and WO have dimensions m and p, respectively, and are each submanifolds of Rn, diﬀeomorphic to Rm and Rp near O. Assume that system (3.2) has a periodic trajectory L : x = ϕ(t) of period τ. Let us write for L the equation in variations:

∂X(ϕ(t)) ξ˙ = ξ = B(t)ξ. (3.4) ∂x In order to study the behaviour of trajectories in the neighbourhood of L it is frequently more convenient to study their traces on a transversal. Let S be a smooth, (n 1)-dimensional disk orthogonal to L at the point of intersection − L S. Let us introduce on S Euclidean coordinates s = (s ,...,s ) such ∪ 1 n−1 that the point O of the intersection L S has the coordinates O = (0,..., 0). ∪ The mapping T : S S along trajectories passing in the neighbourhood of 7→ L, which relates the point s S to the point of the ﬁrst intersection (for 0 ∈ t> 0) of the trajectory passing through s0 with the transversal S, is called the Poincar´emapping. It can be written in the formu ¯ = Au + , where A is an ···

22 (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) mani- → ∞ → −∞ s u fold WL (WL ) of the periodic orbit L. It is clear that L belongs simultaneously s u s u both to WL and WL . The manifolds WL and WL are each smooth submanifolds of Rn in the neighborhood of their points, and have the dimensions m and p, s u respectively. For n = 2 and m = p = 2, WL (WL ) is homeomorphic to a M¨obius band, if the multiplier which is less (greater) than 1, in modulus, is negative, s u and to a cylinder if it is positive. In the latter case divides WL (WL ) into two unconnected pieces, see Fig. 3.1. Let us now deﬁne the transverse intersection of stable and unstable manifolds. Let W sL1 and W uL2 be stable and unstable manifolds of periodic trajectories or equilibrium states, and W s W u = . W s and W u are said to L1 ∩ L2 6 ∅ L1 L1 intersect each other transversally if

dim T W s + dim T W u n = dim(T W s T W u ), (3.5) x L1 x L2 − x L1 ∪ x L2 s u s u where TxWL1 (TxWL2 ) denotes a tangent to WL1 (WL2 ) 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. Let us note that a saddle periodic trajectory L belongs to the transverse intersection of its stable and unstable manifolds. All other trajectories belonging to W s W u L ∩ L are homoclinic curves (or trajectories). Those of the curves along which W s and W u intersect transversally are called structurally stable homoclinic curves. Select one more type of trajectories which connect saddle equilibrium states or (and) saddle periodic trajectories such that

s u dim TxWL1 + dim TxWL2 = n, (3.6) where L = L . We will call such trajectories heteroclinic. Observe that all 1 6 2 mentioned trajectories are particular. Structurally stable diﬀeomorphisms are introduced in an analogous manner as structurally stable vector ﬁelds. Usually, it is added that the following

23 diagram is commutative:

G h G −→ X X˜ ↓ ↓ G h 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, i.e., such a phase space is invariant under the action of the diffeomorphism. Remark 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. Consider next a diffeomorphism defined on Rn or on some subregion of Rn. Let O : x = 0 be a fixed point of the diffeomorphism

x¯ = X(x), X Cr, r 1, (3.7) ∈ ≥ such that 0 = X(0), The point 0 is called structurally stable if none of the roots of the characteristic equation

∂X(0) ρE =0 ∂x −

of O lies on the unit circle of 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 6 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 → ∞ → −∞ s u (unstable) manifold WO (WO of the equilibrium state O. It is known that if O Ck s u is of type (m,p), the -smooth manifolds WO and WO have dimensions m and p, respectively, and are each submanifolds of Rn, diffeomorphic to Rm and Rp near O.

Let C = (x0, x1,...,xq−1) be a periodic trajectory of period q, i.e.,

x1 = X(x0), x2 = X(x1), ...,x0 = X(xq−1).

24 It is evident that each point x , i =0,...,q 1 is a ﬁxed point of the map i −

x¯ = Xq(x), where Xq(x)= X((...X(x))) .

q−1 times If the roots of the characteristic equation| {z }

∂Xq(x ) 0 ρE =0, ∂x −

or, what is the same, the roots of the equation below

∂X(x ) ∂X(x ) ∂X(x ) q−1 q−2 0 ρE =0, ∂x ∂x ··· ∂x −

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 of the − unit circle, C is of the saddle type. Its stable (unstable) set consisting of m Ck s s s u u u -smooth manifolds Wx0 , Wx1 ,...,Wxq−1 (Wx0 , Wx1 ,...,Wxq−1 ) are mapped consecutively to each other under the action of the diffeomorphism. Just like vector fields we may introduce the condition of transverse intersection of stable and unstable manifolds of fixed points and periodic trajectories. We notice also 1 1 1 2 2 2 that if C1(x0, x1,...,xq1−1) and C2(x0, x1,...,xq2−1) are two periodic trajecto- s s s s ries, then in formula (3.6), WL must be replaced by one of Wx1 , Wx1 ,...,Wx1 , 0 1 q1−1 u u u u and WL by one of Wx2 , Wx2 ,...,Wx2 . 0 1 q1−1 Assume that O is a fixed point of the saddle type. Its stable and unstable manifolds may intersect each other along a trajectory other than O as shown R2 s u in Fig. 3.2 for . Such a trajectory is homoclinic. Here, WO and WO are one-dimensional. Observe that if there is a structurally stable homoclinic point W s W u, there is at least one more. The same situation applies to homoclinic O ∩ O trajectories of a saddle periodic trajectory. s Let L1 and L2 be saddle periodic trajectories of the same type. Then WL1 u and WL2 may intersect transversally each other along only the isolated trajectories. These trajectories are called heteroclinic trajectories. It is clear that

25 both homoclinic and heteroclinic trajectories along which stable and unstable manifolds intersect transversally, are structurally stable. Let us now consider the class of C1-smooth dynamical systems satisfying the following conditions:

1. The non-wandering set consists of a ﬁnite number of structurally stable equilibrium states and periodic orbits.

2. The stable and unstable manifolds of equilibrium states, and periodic trajectories, intersect transversally.

Such systems are called Morse-Smale systems. They are structurally stable as it was established by Palis and Smale. In a certain sense the Morse-Smale systems are a high-dimensional 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 amd heteroclinic trajectories, in the Morse-Smale system may be countable. Afraimovich and Shilnikov [15] have shown that the existence heteroclinic trajectories being unclosed, particular trajectories of the Morse- Smale systems may, in particular, leads 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 simpler class amongst all high-dimensional 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]. 11 Let is discuss next the Morse-Smale diffeomorphisms. A diffeomorphism is called a Morse-Smale diffeomorphism if it satisfies the following conditions:

1. Its wandering set consists of a ﬁnite number of structurally stable periodic trajectories. 12

11The problem of ﬁnding a complete topological invariant for systems which nowadays are called Morse-Smale systems was posed by Andronov. 12A ﬁxed point is a periodic trajectory of period 1.

26 2. Stable and unstable manifolds of periodic trajectories intersect transversally.

There exits 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 either gradient-like, or has an orientable heteroclinic set, or has a finite set of heteroclinic trajectories [12, 14]. A common feature of both Morse-Smale flows and cascades is the absence of cycles. Definition Let L L be either an equilibrium state or a periodic tra- 1 ··· q jectory, and let Γ1,..., Γq be trajectories satisfying the following conditions: Ω(Γ ) = L and (Γ ) = L , (k = 1, q 1), Ω(Γ ) = L and (Γ ) = L . k k+1 A k k − q 1 A q q Here we denote by Ω and the ω-limiting set and the α-limiting set, respec- A tively. The collection (L1, Γ1,...,Lq, Γq) 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 since this violates the condition of transversality for some homoclinic or heteroclinic trajectories. Otherwise, if we suppose Morse-Smale system have such a cycle, it is only possible when all Li’s 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 neighborhood which contradicts the definition of the Morse-Smale systems.

4 Poincar´ehomoclinic structures

While studying the bounded problem of three bodies in the Cartesian mechan- ics Poincarédiscovered homoclinic structures [57]. He established that if the stable and unstable manifolds of a saddle periodic trajectory intersect along one homoclinic curve, then there are also infinitely many of such curves. The theme of studying of Poincaréhomoclinic curves was continued by Birkhoff who proved the existence of a countable set of periodic trajectories in a neighborhood of a homoclinic curve of a volume-preserving diffeomorphism. Andronov was the next who posed the problem of constructing structurally stable three-

27 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. Consider a periodically forced non-autonomous system

x˙ = P (x, y)+ µp(t), (4.1) y˙ = Q(x, y)+ µq(t), where p(t) and q(t) are periodic functions of period τ. Assume that when µ =0 (4.1) has a homoclinic loop Γ(0) to a saddle point at the origin O(0, 0). When µ = 0 system (4.1) may be represented as an autonomous system of the form 6 x˙ = P (x, y)+ µp(θ), y˙ = Q(x, y)+ µq(θ), (4.2) θ˙ =1 in the phase space R2 S1. In fact, for all µ sufficiently small we may reduce × the study of (4.2) to the consideration of the Poincarémap on the cross-section θ = 0. For small µ the diffeomorphism

x¯ = F (x, y), (4.3) y¯ = G(x, y)+ µq(θ), θ mod 2π, s has a ﬁxed point O(µ) of the saddle type. Moreover, the stable WO and unstable u WO manifolds of the point O(µ) may transversally intersect each other along the homoclinic curve Γ(µ) for all µ <µ . 13 One can show that within the interval k k 0 µ < µ there exits a countable set of values of µ at which the intersection k k 0 s u WO and WO is no longer transverse. Therefore, apriori we cannot exclude the existence of regions of everywhere dense structural instability. This problem was resolved by Smale who presented a simple example of a two dimensional diﬀeomorphism which is structurally stable and at the same time has a non- wandering invariant set Ω homeomorphic to a Bernoulli subshift on two symbols. The Bernoulli subshift is a particular case of topological Markov chains whose graph has the following form

13such a situation occurs, for instance, when the Melnikov function used for estimating the splitting of separatrices, has simple zeros citeME63.

28 ✛ ✬✩ 0 1 ❄ ✓✏✻ ✓✏ r ✲ r ✒✑ ✒✑ ✫✪ Since periodic, homoclinic as well as Poisson-stable trajectories are everywhere dense in the Bernoulli subshift, the same occurs in Ω. The construction proposed by Smale is called “a Smale horseshoe”. His idea is purely geometrical. In a simplest case for an analytically deﬁned diﬀeomorphism this idea is realized in the Henon map [42]

x¯ = y, y¯ =1+ ax2 + by (4.4) 5+2√5 provided that a> (1 + b )2. 4 | | 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 to the saddle point. In fact, this is the answer why there are no cycles in the Morse-Smale systems. The study of homoclinic curves has induced another problem which goes back to Birkgoff [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: Suppose we have a saddle periodic orbit L with a homoclinic trajectory Γ (Fig.4.1). We surround L and Γ by a neighbourhood U which has the shape of a solid-torus to which a handle containing Γ is glued (Fig.4.2). We shall code a trajectory lying in U using the following rule: if a trajectory makes a complete circuit within the solid-torus, we write a “0”, and if it goes along the handle, we write a “1”.ˆ Thus, the sequence

n=+∞ 0 n=−∞

29 corresponds to the periodic orbit L. The sequence

... 0, 0, 1ˆ, 0, 0,... corresponds to the homoclinic orbit Γ. The sequence

n=+∞ jn , jn = 0, 1ˆ n=−∞ { } corresponds to an arbitrary trajectory within U. Moreover, each 1ˆ is followed by zeros whose number is not fewer than k¯, where k¯ depends on the size of the neighbourhood: the narrower neighbourhood U we choose the bigger k¯ will be. There is another more suitable algorithm of coding. Introduce a symbol

k¯ 1 = 1ˆ, 0,..., 0 . z }| { Then we obtain a new truncated sequence

n=+∞ jn , jn = 0, 1 n=−∞ { } for such a trajectory, where jn can be followed by either 1 or 0. In other words, the set of trajectories lying in U is in correspondence with the Bernoulli scheme on two symbols. 14 Moreover, in this case the converse statement is valid also. Let us illustrate this statement with an example of a diﬀeomorphism T

x¯ = λx + P (x, y), (4.5) y¯ = γy + Q(x, y), where 0 <λ< 1, γ > 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 of the saddle type. Assume that its manifolds W s and W u intersect transversally along a homoclinic curve Γ. Let us choose a pair of homoclinic points M + and M − in a small neighbourhood of O such that M + W s and M − W u . Surround ∈ loc ∈ loc the point M + by a small rectangle Π+ and the point M − by a small rectangle Π− as shown in Fig.4.3. The condition of smallness of both rectangles is that they do not intersect with their images under the map T . Then, inside Π+ and Π− we may define a countable set of “strips” σ0 and σ1 , where k k¯, k¯ is { k} { k} ≤ 14To be more precise, this set is homeomorphic to a suspension over the Bernoulli scheme.

30 k 0 1 + m − sufficiently large, such that T σk = σk. Since M = T M , where m is some m 1 integer, then under the action of T the strips σk’s are mapped onto Π0 (see Fig. 4.4), i.e., all T mσ1, k k¯ intersect transversally all σ0, k k¯. Itisseennow k ≤ k ≤ that each trajectory which is not asymptotical to the fixed point O, may be set +∞ in correspondence to a bi-infinite sequence kα where k k¯. In this case n o−∞ ≤ the invariant set Ω lying entirely in a neighbourhood of O Γ is homeomorphic ∪ to a topological Markov chain with the graph drawn in Fig. 4.5. Notice that accordingly with [15] the suspension over this graph is homeomorphic to the suspension over a Bernoulli subshift on two symbols. It is important that here Ω is a hyperbolic set.

5 Structurally stable systems with complex dynamics. Hyperbolic attractors

The notion of a hyperbolic set is more conveniently introduced ﬁrst of all on example of cascades ϕ(k, x). Recall that a diﬀeomorphism in Rn inducing as cascade is denoted by

x¯ = X(x). (5.1) +∞ Let xn be a trajectory. Then we may deﬁne an inﬁnite-dimensional n on=−∞ system of linear mappings

y = A y , k + (5.2) k+1 k k −∞ ≤ ≤ ∞ where Ak = ∂X(xk)/∂x. The relation (5.2) is an analogue of the variational equation for flows. Defining (5.2) implies that we have an infinite sequence +∞ +∞ Ek of linear n-dimensional spaces and operators Ak such that n ok=−∞ n ok=−∞

A E k E . (5.3) ··· → k → k+1 → · · · Assume that each E admits the representation E = Es Eu such that Es = k k k ⊗ k k+1 s u u s AkEk, Ek+1 = AkEk , and dim Ek is the same for all k. In order that a trajectory is hyperbolic it is required that in the sequences below

31 A Es k Es ··· → k → k+1 → · · · (5.4) − A 1 Eu k Eu . ··· ← k ← k+1 ← · · · there exists an uniform contraction of the exponential type. In the general case, the formalization of hyperbolicity is the following. A trajectory ϕ(k, x) is called hyperbolic if the following conditions hold

D ϕ(k, x)ξ < a ξ e−ck, k k k k D ϕ( k, x)ξ > b ξ eck, k − k k k (5.5) D ϕ( k, x)η > b η eck, k − k k k D ϕ( k, x)η < b η e−ck, k − k k k where ξ Es and η Eu, a, b, c are positive constants independent of ξ, η ∈ 0 ∈ 0 and k. Here we denote the diﬀerential of the mapping by D, e.g., D ϕ(k) = A A A . k−1 k−2 ··· 0 The hyperbolic set of the cascade ϕ(k, x) is an invariant set such that at M each point x the tangent space E may be decomposed into the direct ∈ M x sum E = Es Eu x x ⊕ x s u of two subspaces, stable (contracting) Ex and unstable (expanding) Ex such that for all ξ Es and η Eu, k 0 estimates (5.5) hold where a, b, c do not ∈ x ∈ x ≥ depend on x . ∈M The fact that Ex is a tangent subspace is needed in order 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 Rn as well as a Riemannian metric we can equate the constants a and b in (5.5) to 1. Then, inequalities (5.5) can be recast as

32 D ϕ(1, x)ξ λ ξ , k k ≤ k k 1 D ϕ( 1, x)ξ > λ ξ , k − k k k (5.6) D ϕ(1, x)η 1 η , k k ≥ λ k k D ϕ( 1, x)η < λ η , k − k k k where 0 <λ< 1. Consider next the case of the flow ϕt determined by the vector field V (x) on a manifold. The hyperbolic set of such a flow is a compact, invariant set M such that

1. if a point of is an equilibrium state, then the equilibrium state is M hyperbolic, i.e., 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 T is decom- M x posed into the direct sum

Es Eu E0 x ⊕ x ⊕ 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 case s u discrete time case, i.e., Ex and Ex satisfy condition (5.5).

Examples of hyperbolic set 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

s j = x , dim Ex = j . M n ∈M o Of special interest here are the systems satisfying the condition of hyperbolicity in the entire phase space. Such ﬂows and cascades are called Anosov

33 systems. Anosov proved that hyperbolic systems are rough or structurally stable. The peculiarity of the Anosov systems is that all of their trajectories are particular. This is a reason of why in the case of the Anosov cascades the homeomorphism of conjugacy of two close diffeomorphisms is unique. Examples of the Anosov systems are geodesic flows on compact, smooth manifolds of a negative curvature [11]. It is well-know such that 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

θ¯ = Aθ + f(θ), mod 1, (5.7) where A is a matrix with integer elements other than 1 such that det A = 1, | | and f(θ) is a periodic function of period 1. The condition of hyperbolicity of (5.7) may be easily veriﬁed for one pure case of diﬀeomorphisms of the type

θ¯ = Aθ, mod 1, (5.8) which are the algebraic hyperbolic automorphism of a torus. Automorphisms (5.8) are conservative systems whose set Ω of non-wandering trajectories coincides with the torus Tn itself. Nevertheless we must remark that there are Anosov ﬂows whose Ω does not coincides with the associated phase space. Conditions of structural stability of high-dimensional systems was 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 Ω be hyperbolic;

1B Ω= Per. Here Per denotes the set of periodic points.

Under the assumption of Axiom A the set Ω can be represented by a ﬁnite union of non-intersecting, closed, invariant, transitive sets Ω1,... Ωp. In the

34 case of cascades, any such Ωi 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 Ω1,... Ωp are called basis sets. A condition of strong transversality is the following: the location of the s u manifolds WL1 and WL2 of any two trajectories L1 and L2 is in the general position, i.e., they either do not intersect each other, or intersect each other transversally. We remark that by virtue of the Hadamard-Perron 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 the strong transversality, then the system is structurally stable.

As 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. Never- theless, it follows, more or less, from a series of resent results that the conditions formulated by Smale are necessary for flows also. The basis sets of Smale systems (satisfying the enumerated conditions) may 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. A most studied saddle basis sets are one-dimensional in the case of flows and null-dimensional in the case of cascades. The former ones are homeomorphic to the suspension over topological Markov chains; the latter ones are homeomorphic to simple topological Markov chains [Bowen [18]]. As 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 may be obtained from Markov systems under suitable gluing and change of time [Bowen [19]]. Attractors of Smale systems are called hyperbolic. The trajectories passing sufficiently close to an attractor of a Smale system, satisfies the condition

35 dist(ϕ(t, x), A) < ke−λt, t 0 ≥ where k and λ are some positive constants. As we have said earlier these attractors are transitive. Periodic, homo- and heteroclinic trajectories as well as Poisson-stable ones are everywhere dense in them. In particular, we can tell one more of their peculiarity: the unstable manifolds of all points of such an attractor lie within it, i.e., W s A where x A. Hyperbolic attractors x ∈ ∈ may be smooth or non-smooth manifolds, have a fractal structure, not locally homeomorphic to a direct product of a disk 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 Tn with a hyperbolic structure on it. The next example of hyperbolic attractor was designed by Smale on a two-dimensional torus by means of a “surgery” operation over the automorphism of this torus with a hyperbolic structure. This is the so-called DA-(derived from Anosov) diffeomorphism. An original two-dimensional diffeomorphism is taken in the form

θ¯ = a θ + a θ , mod 1, 1 11 1 12 2 (5.9) θ¯2 = a21θ1 + a22θ2, mod 1 This diﬀeomorphism possesses two invariant foliations given by equations

θ = λ θ + c , mod 1, 1 1 1 1 (5.10) θ2 = λ2θ2 + c2, mod 1, where λ1 and λ2 are the roots of the characteristic equation

a λ a 11 − 12 , (5.11) a21 a22 λ − and c1 and c2 are constant, 0 c1,2 < 1. Both roots, due to assumptions ≤ of hyperbolicity, are always irrational. The stable foliation corresponds to the root λ < 1, the unstable foliation corresponds to the second root λ > 1. | 1| | 2| 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

36 altered diffeomorphism coincides with that of the old diffeomorphism with the only difference of one leaf passing through the point O. The point O breaks it into two parts. Moreover, choosing 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 one of the two new fixed points is an attractor, see Fig. 5.1, 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 as that of minimal sets known from the Poincaré-Donjoy theory in the case of C1-smooth vector fields on a two-dimensional torus [24]. Let us consider a solid torus Π Rn, i.e., T2 = D2 S1 where D2 is a ∈ × disk and S1 is a circumference. We now expand T2 m-times (m is an integer) along the cyclic coordinate on S1 and shrink it q-times along the diameter of D2 where q 1/m. We then embed this deformated torus Π into the original ≤ 1 one so that its intersection with D2 consists of m-smaller disks as shown in Fig.5.2. Repeat this routine with Π and so on. The set Σ = ∞ Π so 1 ∈i=1 i obtained is called a Witorius-Van Danzig solenoid. Its local structure may be represented as the direct product of an interval and a Cantor set. Smale also observed that Witorius-Van Danzig solenoids may have a hyperbolic structures, i.e., be hyperbolic attractors of diffeomorphisms on solid tori. Moreover, similar attractors can be realized as a limit of the inverse spectrum of the expanding cycle map [84] θ = mθ, mod 1.

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. Note that in the theory of sets of limit-quasi-periodic functions the Wictorius- Van Danzig solenoids are quasi-minimal sets. Hyperbolic solenoids are called the Smale-Williams solenoids. We return to them in section 7.

37 We remark also on an example of a hyperbolic attractor of a diffeomorphism on a two-dimensional sphere, and, consequently, on the 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. As for the question of finding complete topological invariants of structurally stable systems with non-trivial dynamics, this concerns mainly Anosov diffeomorphisms and two-dimensional diffeomorphisms on surfaces. We refer the reader to the issue [13, 14]. To conclude this section we remark that structurally stable high-dimensional systems are not dense in the space of all dynamical systems.

6 Everywhere density of structurally unstable systems. Newhouse regions

As we have noticed above that 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 it. Let us consider a three-dimensional diffeomorphism having a saddle fixed point of the type (2,1) and the Anosov attractor T2. Assume that the points of W u tend to the torus T2 as k + . It is evident that W s , where x T2 is a O → ∞ x ∈ family of two-dimensional manifolds and looks like locally as a family of parallel planes such that in a neighbourhood of T2 they constitute a solid “fence”. The u location of WO with respect to this family may be such as schematically shown s s in Fig. 6.1. In the first case WO intersects transversally the leaves of WT2 ; in the second case there is a point of tangency. Moreover, we cannot get rid of this u point by means of small perturbations. Furthermore, if the trace of WO goes u through the point of tangency and the ω-limit set of WO is a periodic point, then the original diffeomorphism may be perturbated so that the ω-limit set u of WO will be, for instance, an unclosed Poisson-stable trajectory. This simple example demonstrates that structurally unstable systems can form an open set

38 in the space of dynamical systems. But in such a case there is structural instability on the non-wandering set. Then, it is rather reasonable to weaken the requirement of structural stability up to that of Ω-stability, where Ω denotes the set of non-wandering points.

Definition. A dynamical system X is called Ω-stable (Ω-rough) if given ǫ there exists δ > 0 such that for any system X˜ ǫ-close in C1-metrics to X the set Ω(X˜) is homeomorphic to Ω(X˜); moreover the homeomorphism is ǫ-close to the identity homeomorphism. In other words, the non-wandering set is preserved under small perturbations; it is only shifted a little bit. Assume that our flow satisfies Axiom A. Then, as we know, Ω = A A . 1 ∪···∪ k Let us introduce the following relationship Ai > Aj which means that in the phase space there exists a point x such that the ω-limit set of its trajectory

ϕ(t, x) lies in Aj and the corresponding α-limit set lies in Ai. It may appear that there is a chain

A > > A > A , i1 ··· ik i1 i.e., 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) If a dynamical system satisﬁes Axiom A and the condition of acyclicity, then this system is Ω-stable.

The condition of acyclicity is essential. Although Ai persists under small perturbations they may cause the so-called Ω-explosion. Nevertheless, Ω-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 Ω-stable null-dimensional hyperbolic set Σ which is homeomorphic to a transitive Markov chain, for example, to the non-wandering set of the Smale horseshoe. Since this set is of

39 s u s s the saddle type, we may define WΣ and WΣ . WΣ (WΣ) is a union of stable s s (unstable) manifolds of the points of Σ. Since both WΣ and WΣ are “hole”-like sets, each can be represented by a direct product of an interval and a Cantor s s set. Assume that WΣ and WΣ behave as shown in Fig. 6.2. Newhouse noted 2 that in the case of C -smooth diffeomorphisms if there are points x1 and x2 s u in Σ such that Wx1 and Wx2 have a quadratic tangency, then under certain conditions all C2-close diffeomorphisms preserve this tangency. Such Σ-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 dence everywhere.

These regions are called Newhouse regions. Let us denote by B1 the set of diﬀeomorphisms which possess a saddle point O whose stable and unstable manifolds have a quadratic tangency along a homo- r clinic curve. Consider a one-parameter family Xµ of two-dimensional C -smooth 1 diﬀeomorphisms such that Xµ is transverse to B at µ = 0.

Theorem 6.3 (Newhouse) Given suﬃciently small µ , the interval ( µ ,µ ) 0 − 0 0 contains a countable set of Newhouse intervals.

This theorem is especially important for many problems of nonlinear dynamics since it guarantees the existence of the Newhouse regions in ﬁnite-parameter models. The Newhouse results were generalized for the multi-dimensional case by Gonchenko, Shilnikov and Turaev [33]. The study of systems with a structurally unstable homoclinic trajectory was started by Gavrilov and Shilnikov in [26]. It was established that the typical

40 systems with quadratic homoclinic tangencies may be classiﬁed with respect to the character of the sets N lying entirely in a small but ﬁxed neighbourhood of the homoclinic trajectory Γ:¯

1. N = O Γ. This situation occurs, for example, in the case of a two- ∩ dimensional diﬀeomorphism with λγ < 1 at 0, see Fig. 6.3a. Under small | | perturbations, which lead to the appearance of inﬁnitely many structurally stable homoclinic trajectories, the structure of a neighbourhood of Γ¯ is non-trivial. This is the Ω-explosion.

2. N admits a precise description. For ﬂows, N is homeomorphic to a suspension over a Bernoulli shift on 3 symbols in which two trajectories are glued. An example is shown in Fig. 6.3b.

3. N contains a hyperbolic set but does not admit a precise description. In the bifurcation set B1 of such systems those systems, which 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. 6.3c. Observe that homoclinic tangencies of the third type are everywhere dense in the Newhouse regions.

The “wild” hyperbolicity accomplishes one more principal case. Consider a Cr, (r 4)-smooth system. Assume that the following conditions are fulﬁlled. ≥ 1. At the origin the system has an equilibrium state of the saddle-focus type,

i.e., the root λ1,...,λn of the characteristic equation at O are such that

λ = ρ iω, ω =0, 1,2 ± 6 Reλi < ρ, i =3,...,n 1 σ = ρ −+ λ > 0. − − n In this case dim W u = 1 and dim W s = n 1. O O − 2. Assume that one of two separatrices of the saddle-focus comes back to O as t + , see Fig. 6.4. → ∞

41 3. As t + Γ tends to O tangentially to a two-dimensional hyper-plane → ∞ corresponding to the eigenvalues λ1 and λ2.

4. A certain value, called a separatrix value, which characterizes the global behaviour of Γ is non-zero.

The systems with the above properties form a subset of B1 of codimension one in the space of dynamical systems with C3 metrics. Shilnikov [71, 73] established that the behaviour of the trajectories in a neighbourhood of the separatrix Γ is rather complicated. In these works he fo- cused on the separation of the set of hyperbolic trajectories and its description via the language of symbolic dynamics. In Ovsyannikov and Shilnikov [53, 54] it was established that everywhere dense structural instability takes place on B1. In particular, the following theorem was proven:

Theorem 6.4 In B1 the system with

1. structurally unstable periodic trajectories and

2. structurally unstable Poincar´ehomoclinic curves are everywhere dense.

Further, in section 10, we present an example of an n, (n 4)-dimensional ≥ system possessing a wild attractor including a saddle-focus.

7 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 Morse-Smale systems to systems with complex dynamics. Such bifurcations interest us in the context of scenarios of the appearance of strange attractors, i.e., the routes to dynamical chaos.

42 The violation of the condition of the structural stability of Morse-Smale systems may occur in the following three cases:

1. The lost of the condition of hyperbolicity of equilibrium states or periodic trajectories.

2. The lost 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 is section 3).

Due 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 thirties. 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 50-60ths these cases were generalized to the high-dimensional case; moreover, new ones were added:

5 Period-doubling bifurcation (known, in fact, to Poincar´e);

6 Birth of an invariant torus from a structurally unstable periodic orbit. 15

15see [5] for more details

43 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 of 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. Consider a two-dimensional diffeomorphism with a structurally unstable heteroclinic trajectory behaving as shown in Fig. 7.1. Denote by λ1,2 and γ1,2 the characteristic roots at O such that λ < 1 and γ > 1. Assume that W s 1,2 | 1,2| | 1,2| O1 u and WO2 have a homoclinic quadratic tangency along a heteroclinic trajectory Γ. Let is define

ln λ θ = | 2|, − ln γ | 1| which we call a modulus. Palis [55] had established the following result: Two diffeomorphisms X and X˜ both belonging to the boundary B1 (see the previous section) are homeomorphic if their moduli are equal. Such diffeomorphisms form a bifurcation surface B1 of codimension one. Thus, the set of such systems may be subdivided into a continuum of classes of topological equivalence distinguished by their moduli. Since the notion of the modulus is important in the following discussion, let us give its complete 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 a locally non-constant contin- M M uous functional h is deﬁned such that for two systems X and X˜ from to be M equivalent it is necessary that h(X) = h(X˜). We will say that X has at least

44 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 Ω-equivalence to hold for systems, then the systems of the given type are Ω-rough. Next we need the notion of an Ω-modulus. This may be introduced in a similar manner as above with the only difference that Ω-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(x, µ) (7.1) having a non-rough equilibrium state of saddle-saddle type at the origin for µ = 0. This means that in a small neighbourhood of the origin O, the system can be written in the following form

x˙ = µ + x2, y˙ = y, z˙ = z.−

For µ < 0, it has two saddle equilibrium states O1 and O2, respectively; O1 s has its two-dimensional stable invariant manifold WO1 and a one-dimensional u s unstable invariant manifold WO1 , whereas for O2 , dim WO2 = 1 and u dim WO2 = 2 , see Fig. 7.2. When µ = 0, the point O is of saddle-saddle type s u which has stable manifold WO and unstable manifold WO homeomorphic to semi-planes (Fig. 7.3). s u Let us assume that at µ = 0, WO and WO intersect each other transversely along the trajectories Γ ,..., Γ , m 2. 1 m ≥ Theorem 7.1 (Shilnikov [78]) . As µ increases through 0, the equilibrium state O disappears and an unstable set Ω is born whose trajectories are in one-

45 to-one correspondence with the Bernoulli scheme on m symbols. As µ 0+, Ω → is tightened into a bouquet deﬁned by:

O Γ Γ . ∪ 1 ∪···∪ m

This examples demonstrates that Morse-Smale systems may be bounded from Smale systems by a bifurcation surface of codimension-1. The next example demonstrates the transition through a codimension-1 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]. r Consider a C -smooth one-parameter family of dynamical systems Xµ in n R . Suppose that the ﬂow has a periodic orbit L0 of the saddle-node type at

µ = 0. Choose a neighbourhood U0 of L0 which is a solid torus partitioned by the (n 1)-dimensional strongly stable manifold W ss into two regions: the − L0 node region U + where all trajectories tend to L as t + , and the saddle 0 → ∞ − u region U where the two-dimensional unstable manifold WL0 bounded by L0 u lies. Suppose that all of the trajectories of WL0 return to L0 from the node region U + as t + and do not lie in W ss, as shown Fig. 7.4. Moreover, since → ∞ u ¯ u any trajectory of W is bi-asymptotic to L0, WL0 is compact.

Observe that systems close to X0 and having a simple saddle-node periodic trajectories close L0 form a surface B of codimension-1 in the space of dynamical systems. We assume also that the family Xµ is transverse to B. Thus, when − µ < 0, the orbit L0 is split into two periodic orbits, namely: Lµ of the saddle + type and stable Lµ . When µ> 0 L0 disappears.

It is clear that Xµ is a Morse-Smale system in a small neighbourhood U of the set W u for all small µ< 0. The non-wandering set here consists of the two + − s + periodic orbits Lµ and Lµ . All trajectories of U W − tend to Lµ as t + . \ Lµ → ∞ At µ = 0 all trajectories on U tend to L0. The situation is more sophisticated when µ > 0. It was established in [Afraimovich and Shilnikov [15]] that if W u is a smooth submanifold of Rn,

46 then a two-dimensional stable invariant surface (a torus) may be generated from W s. In the case where W s 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 Poincar´ehomoclinic orbits. u Note that if X0 has a global cross-section, W is always homeomorphic to a torus. The Poincar´emap to which the problem under consideration is reduced, may be written in the form

x¯ = f(x,θ,µ), (7.2) θ¯ = mθ + g(θ)+ ω + h(x,θ,µ), mod 1, where f, g and h are periodic functions of θ. Moreover, f C1 0 and h C1 k k → k k → 0 as µ 0, m is an integer and ω is a parameter defined in the set [0, 1). → Diffeomorphism (7.2) is defined in a solid-torus Dn−2 S1, where Dn−2 is a disk × x < r, r > 0 Observe that (7.2) is a strong contraction along x. Therefore, k k mapping (7.2) is close to the degenerate map

x¯ =0, (7.3) θ¯ = mθ + g(θ)+ ω, mod 1. This implies that its dynamics is determined by the circle map

θ¯ = mθ + g(θ)+ ω, mod 1, (7.4) where 0 ω < 1. Note that in the case of the ﬂow in R3, the integer m in ≤ (7.2)-(7.4) may be 0, 1, 1. −

Theorem 7.2 (Shilnikov-Turaev) If m =0 and if

max g′(θ) < 1, k k then for sufficiently small µ > 0, the original flow has a periodic orbit both of which length and period tend to infinity as µ 0. →

This is that “blue sky catastrophy” we mentioned above. In the case where ¯ u m = 1, the closure WL0 is a two-dimensional torus. Moreover, it is smooth provided that (7.2) is a diﬀeomorphism. In the case where m = 1 W¯ u is a − L0

47 Klein bottle, also smooth if (7.2) is a diffeomorphism. In the case of the last ¯ u theorem WL0 is not a manifold. We must say that in the general case, if the diffeomorphism (7.4) has an interval [θ1,θ2] where m + g′(θ) < 1, | | 1 2 then the original system possesses a countable set of intervals [µn,µn] accumulating to zero where the system Xµ has stable periodic orbits. In the case of Rn (n 4) the constant m may be any integer. ≥ Theorem 7.3 (Shilnikov-Turaev) Let m 2 and let m + g′(θ) > 1. Then for | |≥ | all µ > 0 sufficiently small, the Poincarémap (7.2) has a hyperbolic attractor homeomorphic to the Smale-Williams solenoid, while 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 onto that of employing the bifurcations of an invariant torus. We are not developing here the theory of such bifurcations but restrict ourself by consideration of a modelling situation. Consider a one-parameter family of smooth dynamical systems

x˙ = X(x, µ) which possesses an invariant m-dimensional torus Tm with a quasi-periodic trajectory at µ = 0. Assume that the vector ﬁeld may be recast as

y˙ = C(µ)y, z˙ = µ + z2, (7.5) θ˙ = Ω(µ) in a neighbourhood of Tm. Here, z R1, y Rn−m−1, θ Tm and Ω(0) = ∈ ∈ ∈ (Ω1,..., Ωm). The matrix C(µ) is stable, i.g., its eigenvalues lie to the left of the imaginary axis in the complex plane. At µ = 0 the equation of the torus is y = 0, the equation of the unstable manifold W u is y = 0,z > 0, and that of the strongly unstable manifold W ss partitioning the neighbourhood of Tm into a node and a saddle region, is z = 0. We assume also that all of the trajectories

48 of the unstable manifold W u of the torus come back to it as t + . Moreover → ∞ they do not lie in W ss. On a cross-sections transverse to z = 0 the associated Poincar´emap may be written in the form

y¯ = f(y,θ,µ), (7.6) θ¯ = Aθ + g(θ)+ ω + h(x,θ,µ), mod 1, where A is an integer matrix, f, g, and h are periodic functions of θ. Moreover

f C1 0 and h C1 0 as µ 0, ω = (ω ,...,ω ) where 0 ω < 1. k k → k k → → 1 m ≤ k Denote G(θ)= Aθ + g(θ).

If G′(θ) < 1 for all θ (for example, if A = 0 and if g(θ) is small), then the | | shortened map is contracting for all µ > 0 as well as the original map (7.6). Since a contracting mapping has only one ﬁxed point, we arrive at the following statement

Proposition 7.1 If G′(θ) < 1 for all small θ, then the ﬂow X has a unique k k µ attracting periodic orbit for all µ> 0 suﬃciently small.

This result is analogous to the result of theorem 7.2. It yields us a new example of the blue sky catastrophe. Observe that the restriction of the Poincar´emap on the invariant torus is close to the shortened map

θ¯ = Aθ + g(θ)+ ω, mod 1, (7.7)

This implies, in particular, that if (7.7) is an Anosov map for all ω (for example when the eigenvalues of the matrix A do not lie on the unit circle of the complex plane, and g(θ) is small), then the restriction of the Poincar´emap is also an Anosov map for all µ> 0. Hence, we arrive at the following statement

Proposition 7.2 If the shortened map is an Anosov map for all small ω, then for all µ> 0 sufficiently small, the original flow possesses a hyperbolic attractor which is topologically conjugate to the suspension over the Anosov diffeomorphism.

49 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 of the length if any tangent vector field grows exponentially under the action of the differential of the map. An example is the algebraic map

θ¯ = Aθ, mod 1, such that the spectrum of the integer matrix A lies strictly outside the unit circle, and any neighboring map is also expanding. If (G′(θ))−1 < 1, then the k k shortened map

θ¯ = ω + G(θ)= ω + Aθ + g(θ), mod 1, (7.8) is expanding for all µ> 0. Shub [81] had established that expanding maps are structurally stable. The study of expanding maps and their connection to smooth diﬀeomorphisms was continued by Williams [Williams [83]]. Using the result of his work we come to the following result which is analogous to our theorem 7.2, namely

Proposition 7.3 If (G′(θ))−1 < 1, then for all small µ > 0, the Poincar´e k k map possesses a hyperbolic attractor locally homeomorphic to a direct product of Rm+1 and a Cantor set.

An endomorphism of a torus is called an Anosov covering if there exists a continuous decomposition of the tangent space into the direct sum of stable and unstable submanifolds just like 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 (7.7) is an Anosov covering if, we assume, det A > 1 | | and if g(θ) is sufficiently small. Thus, the following result is similar to the previous proposition

Proposition 7.4 If the shortened map (7.7) is an Anosov covering for all ω, then for all small µ > 0 the original Poincar´emap possesses a hyperbolic attractor locally homeomorphic to a direct product of Rm+1 and a Cantor set.

50 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 Ω-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 9.

8 Strange attractors of the Lorenz type

In 1963 Lorenz [44] suggested the model:

x˙ = σ(x y), y˙ = rx− y− xz, (8.1) z˙ = bz−+ xy− − in which he discovered numerically a vividly chaotic behaviour of the trajectories when σ = 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. For a number of years this work of Lorenz did not attract special attentions. The breakthrough occurred in a mid seventies when this model becomes the center of attention of mathematicians on one side and on the other side of researches from other ﬁelds such as non-linear optics, magneto-hydrodynamics etc. Notice that from the viewpoint of mathematics the Lorenz model, as well as its extension

x˙ = y, y˙ = x(1 z) λy x3, (8.2) z˙ = αz−+ x2−, − − where λ > 0, α > 0 amd B > 0 may 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.

51 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 may be obtained through a finite number of bifurcations. In particular, in the Lorenz model (due 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 done in [6, 7]. The fact that we are considering only three-dimensional systems is not important, in principle, because the general case where only one characteristic value is positive for the saddle while the others have negative real parts, and the value least with the modulus is real, the result is completely similar to the three-dimensional case. 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. Suppose that in the domain U B each system X has an equilibrium state ⊂ O of the saddle type. In this case the inequalities λ1 < λ2 < 0 < λ3 hold for the roots λi = λi(X), i =1, 2, 3 of the characteristic equation at O, and the saddle value σ(X) = λ2 + λ3 > 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 u u trajectories Γ1,2 = Γ1,2(X) originating from it, by W = W (X). It is known that both W s and W u depend smoothly on X on each compact subset. Here it is assumed that in a certain local map V = (x , x , x ) , containing O, X can { 1 2 3 } be written in the form

x˙ i = λixi + Pi(x1, x2, x3), i =1, 2, 3. (8.3)

Suppose that the following conditions are satisﬁed for the system X U (see 0 ⊂ Fig. 8.1):

1. Γ (X ) W s(X ), i =1, 2 , i.e., Γ (X ) is doubly asymptotic to O. i 0 ⊂ 0 i 0

2. Γ1(X0) and Γ2(X0) approach to O tangentially to each other.

52 Further considerations will require some concepts and facts presented in

[Shilnikov [71, 73]]. The condition λ1 < λ2 implies that the non-leading manifold ss s WO of WO, consisting of O and the two trajectories tangential to the axis x1 s s s at the point O, divides WO into two open domains: W+ and W−. Without loss of generality we may assume that Γ (X ) W s (X ), and hence Γ is tangent i 0 ⊂ + 0 i to the positive semiaxis x2. Let v1 and v2 be sufficiently small neighborhoods of the separatrix “butterfly” Γ=¯ Γ O Γ . Let stand for the connection 1 ∪ ∪ 2 Mi s component of the intersection of W+(X0) with vi, which contains Γi(X0). In the general case is a two-dimensional C0-smooth manifold homeomorphic Mi ether to a cylinder or to a Möbius band. The general condition lies in the fact that certain values A1(X0) and A2(X0), called the separatrix values, should not be equal to zero.

It follows from the above assumptions that X0 belongs to the bifurcation set 2 2 1 B1 of codimension two, and B1 is the intersection of two bifurcation surfaces B1 1 1 and B2 each of codimension one, where Bi corresponds to the separatrix loop Γ¯ = O Γ . In such a situation it is natural to consider a two-parameter family i ∪ i of dynamical systems X(µ), µ = (µ ,µ ), µ <µ , X(0) = X , such that X(µ) 1 2 | | 0 0 2 intersects with B1 only along X0 and only for µ = 0. It is also convenient to 2 assume that the family X(µ) is transverse to B1 . By transversality we mean s that for the system X(µ) the loop Γ1(X(µ)) “deviates” from W+(X(µ)) by a s value of the order of µ1, and the loop Γ2(X(µ)) “deviates” from W+(X(µ)) by a value of the order of µ2. It is known from [Shilnikov [73]] that the above assumptions imply that in the transition to a system close to X0 the separatrix loop can generate only one periodic orbit which is of the saddle type. Let us assume, for certainty, that the loop Γ (X ) O generates a periodic orbit L for µ > 0 and Γ (X ) O 1 0 ∪ 1 1 2 0 ∪ generates the periodic orbit L2 for µ2 > 0. The corresponding domain in U, which is the intersection of the stability regions for L1 and L2, i.e., i.e., the domain in which the periodic orbits L1 and

L2 are structurally stable, will be denoted by U0. A stable manifold of Li for the system X U will be denoted by W s and the unstable one by W u. If ⊂ 0 i i u u the separatrix value Ai(X0) > 0, Wi is a cylinder; if Ai(X0) < 0, Wi is a

53 M¨obius band. Let us note that, in the case where is an orientable manifold, M s Wi will also be a cylinder if Ai(X0) > 0. Otherwise it will be a M¨obius band. However, in the forthcoming analysis the signs of the separatrix values will play an important role. Therefore, it is natural to distinguish the following three main cases

Case A (orientable) A1(X0) > 0, A2(X0) > 0, Case B (semiorientable) A1(X0) > 0, A2(X0) < 0, Case C (nonorientable) A1(X0) < 0, A2(X0) < 0.

In each of the above three cases the domain U0 also contains two bifurcation 1 1 surfaces B3 and B4 :

1. In Case A, B1 corresponds to the inclusion Γ W s and B1 corresponds 3 1 ⊂ 2 4 to the inclusion Γ W s; 2 ⊂ 1 2. In Case B, B1 corresponds to the inclusion Γ W s and B1 corresponds 3 1 ⊂ 1 4 to the inclusion Γ W s; 2 ⊂ 1

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 u “along” Γ1(X0) and Γ2(X0), and if both Wi are M¨obius bands, i =1, 2, u the unstable manifold W3 of the periodic orbit L3 is a cylinder. In this case the inclusions Γ W s and Γ W s correspond to the surfaces B1 1 ⊂ 2 2 ⊂ 3 3 1 and B4 , respectively.

1 1 Suppose that B3 and B4 intersect transversally over the bifurcational set 2 1 B2 , see Fig. 8.2. In a two-parameter family X(µ) this means that the curves B3 1 1 and B4 intersect at some point µ = (µ11,µ12). In a small neighbourhood of the 1 1/α origin of the parameter plane equations of B3 will be of the form µ1 = a1µ2 (1+ 1/α 1/α2 ... ), form µ1 = b1µ2 (1+... ) in cases A and B, and form µ1 = c1µ2 (1+... ) 1 1/α in case C. Equations of B4 will be of the form µ2 = a2µ1 (1 + ... ) in case A, 1/α2 1/α2 and of the form µ2 = b2µ1 (1 + ... ) form µ2 = c2µ1 (1 + ... ) in cases B and C. Here, α = λ /λ , a ,...,c are positive, and the ellipsis stand for − 2 3 1 2 terms which tend to zero when the argument tends to zero. Let us denote a 1 1 domain lying between B3 and B4 by U1. It can be shown (this can be deduced from the results of Shilnikov [78]) that there exists in U a suﬃciently small

54 ǫ-neighbourhood Uǫ of the system X0 such that a one-dimensional limiting set, homeomorphic to the suspension over the Bernoulli subshift on two symbols, will exists for each X U U . This statement may be proved if one considers the ∈ ǫ ∩ 1 mapping T of a certain transversal to Γ1(X0) and Γ2(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.8.3) with the following properties:

1. The Euclidean coordinates (x, y) can be introduced on D such that

D = (x, y): x 1, y < 2 . n | |≤ | | o 2. The equation y = 0 describes a connection component S of the intersection W s D such that no ω-semitrajectory that begins on S possesses any point O ∩ of intersection with D for t> 0.

3. The mapping T (X): D D and T (X): D D are deﬁned along 1 1 7→ 1 2 7→ the trajectories of the system X, where

D1 = (x, y): x 1, 0

D2 = (x, y): x 1, 1 y < 1 , n | |≤ − ≥ o and Ti(X) is written in the form x¯ = f (x, y), i (8.4) y¯ = gi(x, y), where f ,g Cr, i =1, 2. i i ∈

4. fi and gi admit continuous extensions on S, and

∗∗ ∗∗ lim fi(x, y)= xi , lim gi(x, y)= yi , i =1, 2. y→0 y→0

5. T1D1 Pi1 = (x, y):1/2 x 1, y < 2 , ∈ n ≤ ≤ | | o

T2D2 Pi2 = (x, y): 1 x 1/2, y < 2 . ∈ n − ≤ ≤− | | o Let T (X) T (X) D , (f,g)= (f ,g ) on D , i =1, 2. ≡ i | i ≡ i i i

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

(a) (f ) < 1, k x k  (b) 1 (g )−1 f > 2 (g )−1 (g ) (g )−1 f  −k y k·k xk k y k·k x k·k y · yk  p  (8.5)  (c) (g )−1 < 1  k y k  −1 −1  (d) (gy) fy gx < (1 fx )(1 (gy) ).  k · k·k k −k k −k k   Hereafter, = sup . k·k (x,y)∈D\S | · |

s It follows from the analysis of the behaviour of trajectories near WO that in a small neighbourhood of S the following representation is valid:

f = x∗∗ + ϕ (x, y) yα, g = y∗∗ + ψ (x, y) yα, 1 1 1 1 1 1 (8.6) f = x∗∗ + ϕ (x, y)( y)α, g = y∗∗ + ψ (x, y)( y)α, 2 2 2 − 2 2 2 − where ϕ ,...,ψ are smooth with respect to x, y for y = 0, and T (x) satisfies 1 2 6 i estimates (8.2) for sufficiently small y. Moreover, the limit of ϕ1 will be denoted by A1(X) and that of ψ2 by A2(X). The functionals A1(X) and A2(X) will be also called the separatrix values in analogy with A1(X0) and A2(X0) which were introduced above. Let us note that for a system lying in a small neighbourhood 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 X U. It is convenient to assume, for simplicity, that A (X) ∈ 1,2 do not vanish. It should be also noted that the point Pi with the coordinates ∗∗ ∗∗ (xi ,yi ) is the first point of intersection of Γi(X) with D. Let us consider the constant q =

−1 −1 2 −1 −1 1+ fx (gy) + 1 (gy) (fx) 4 (gy) gx (gy) fy k kk k −k k −k 1 k− k kk kk p 2 (gy) k k (8.7) Conditions (8.7) implies that q > 1 and this (together with conditions (8.5)) implies that all periodic points are of the saddle type.

56 In terms of mappings, the conditions for the existence of periodic orbits L1 and L can be formulated quite simply: it is required that P D , P D in 2 1 ∈ 2 2 ∈ 1 Case A; P D , P D in Case B; P D , P D in Case C. 1 ∈ 2 2 ∈ 2 1 ∈ 1 2 ∈ 2 The point of intersection of Li with D will be denoted by Mi, and its coor- ∗ ∗ dinates by xi and yi . As already mentioned, the periodic orbit L3 will exist in ∗ ∗ Case C together with L1 and L2. It intersects D at two points M3(x3,y3 ) and ∗ ∗ M4(x4,y4 ), 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 s for the intersections of the connection components of the stable manifolds Wi of the periodic orbits Li, i = 1, 2 with D containing ﬁxed points Mi, can be represented in the form y = y (x), x 1. In Case C the equation for the i | | ≤ connection component D W s, containing point M , can also be written in the ∩ 3 i form y = yi(x), i =3, 4.

Let us deﬁne the functionals R1 and R2 as follows: in Case A R = [y∗∗ y (x∗∗)], R = y∗∗ y (x∗∗); 1 − 1 − 2 2 2 2 − 1 2 in Case B R = [u∗∗ y (x∗∗), R = y∗∗ y (x∗∗); 1 − 1 − 1 2 2 2 − 1 2 in Case C R = [y∗∗ y (x∗∗)], R = [y∗∗ y (x∗∗)]; 1 − 1 − 4 1 2 − 2 − 3 2 ∗∗ ∗∗ where (ui , vi ) are the coordinates of T Pi in case B (see Fig.8.4) It can be easily seen now that R = 0 for the system X B1 and R = 0 if X B1 and 1 ∈ 3 2 ∈ 4 that the domain U consists of systems X for which R1 > 0 and R2 > 0. The condition R1 > 0 means that:

1. in Case A, L1 and L2 have a heteroclinic trajectory;

2. in Case B, L1 has a structurally stable homoclinic curve;

3. In Case C, L3 also has a structurally stable homoclinic curve.

If R2 > 0, it is only L3, which has a homoclinic curve in Case 3. In Case A and B, L1 and L2 will have a heteroclinic curve for R2 > 0. + The domain X Uǫ R1 < 0, R2 > 0 will be denoted by U2 and the n ∈ | o + domain X Uǫ R1 < 0, R2 < 0 by U1 . n ∈ | o 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

57 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 ω-limiting and, in the general case, where it is locally maximal, is an attractor. Let Σ denote the closure of the set of points of all the trajectories of the mapping T (X), which are contained entirely in D. Due to above considerations , Σ is non-trivial and contains a countable set of periodic orbits in U U + U + ∪ 1 ∩ 2 for Case C and in U U + for Case B. For Case A in U + U + and for Case B in 1 ∪ 1 1 ∪ 2 + U2 , the above cited situation is possible, and also a trivial one where Σ consists of M1 M2 and heteroclinic points. The corresponding bifurcational surface that divide these sets of systems will satisfy the following equations:

In Case A: R3 = 0 and R4 =0, where

R = v∗∗ + y (u∗∗), R = v∗∗ y (u∗∗), 3 − 1 1 1 4 2 − 2 2 ∗∗ ∗∗ where [ui , vi ] are coordinates of points T (X)Pi, i =1, 2;

In Case B: R3 =0, where

R = v∗∗ + y (u∗∗) 3 − 3 1 3 ∗∗ ∗∗ 2 and u3 , v3 are coordinates of the point T (X)P1. + + It is obvious that in Case A the surface R4 = 0 lies in U2 , and R3 on U1 (the same is true for Case B).

Σ is described most simply in the domain U1. Here the following theorem holds.

Theorem 8.1 If X U , T (X) Σ is topologically conjugated with the Bernoulli ∈ 1 | scheme (σ, Ω2) with two symbols.

58 IF X U˜ U , Σ will be a one-dimensional set. We shall not concentrate ∈ 2\ 2 on this case. Let us only note that Σ is unstable and its closure is similar, in many respects, to the structure of Σ for X U . We conﬁne ourselves to the ∈ 2 following theorem.

Theorem 8.2 The system X U , has a two-dimensional limiting set Ω, which ∈ 2 satisﬁes the following conditions:

1. Ω is structurally unstable.

2. [Γ Γ O] Ω. 1 ∪ 2 ∩ ⊂ 3. Structurally stable periodic orbits are everywhere dense in Ω.

4. Under perturbations of X periodic orbits in Ω disappear as a result of

matching to the saddle separatrix loops Γ1 and Γ2.

Note that in this case the basic periodic orbits will not belong to Ω. In terms of mappings, the properties of Ω can be formulated in more detail. Let us ﬁrst single out a domain D˜ on D as follows: we assume that in Case A

D˜ = (x, y) D1 D2 y2(x)

D˜ = (x, y) D1 D2 y12(x)

Theorem 8.3 Let X U . Then: ∈ 2 I. Σ˜ 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.

59 II. D˜ is foliated by a continuous stable foliation H+ into leaves, satisfying the Lipschitz conditions, along which a point is attracted to Σ˜; inverse images of the discontinuity line S : y = 0 (with respect to the mapping T k, k =1, 2,...) are everywhere dense in D˜.

III. There exits a sequence of T (X)-invariant null-dimensional sets ∆ , k k ∈ Z , such that T (X) ∆ is topologically conjugated with a ﬁnite topological + | k Markov chain with a nonzero entropy, the condition ∆ ∆ being k ∈ k+1 satisﬁed, and ∆ Σ˜ as k . k → → ∞ IV. The non-wandering set Σ Σ˜ is a closure of saddle periodic points of 1 ∈ T (X) and either Σ = Σ˜ or Σ =Σ+ Σ−, where: 1 1 ∪

1. Σ− is null-dimensional and is an image of the space Ω− of a certain TMC (G−, Ω−, σ) under the homeomorphism β : Σ− Σ− which conjugates 7→ σ Ω− and T (X) Σ−; | | l(X) Σ− = Σ− ,l(X) < , m ∞ m[=1 where T (X)Σ− =Σ− , Σ− Σ− = m m m1 ∩ m2 ∅ for m = m and T (X) Σ− is transitive; 1 6 2 | m 2. Σ+ is compact, one-dimensional and

3. if Σ+ Σ− = , Σ+ is an attracting set in a certain neighbourhood; ∩ ∅ 4. if Σ+ Σ− = , then Σ+ Σ− = Σ+ Σ− for a certain m, and this ∩ 6 ∅ ∩ m ∩ m intersection consists of periodic points of no more than two periodic orbits, and

− + (a) if Σm is ﬁnite, Σ is ω-limiting for all the trajectories in a certain neighbourhood;

− + (b) if Σm is inﬁnite, Σ is not locally maximal, but is ω-limiting for all the trajectories in D˜, excluding those asymptotic to Σ− Σ+. \

60 ′ Let U2 deﬁne a set of systems from U2, for which the separatrices Γ1 and Γ2 ′′ tend to O or to periodic orbits, and let U2 stand for a set of such systems from U ′ , for which Γ and Γ tend to O as y . 2 1 2 → ∞ Theorem 8.4

′′ I. U2 is dense in U2.

II. For X U ′ : ∈ 2 (a) the mapping T (X) admits a ﬁnite Markov partition of Σ˜ whose boundary belongs to a ﬁnite set of leaves from H+(X);

(b) for a TMC (G, Ω, σ), constructed according to this partition, there exists a continuous mapping β : Ω Σ˜, which is one-to-one on − 7→ the residual set of Ω and is such that the diagram

Ω β−1(Σ˜ S) σ Ω \ ∩ −→ β β˜ ↓ ↓ Σ˜ S T Σ˜ \ −→ is commutative;

(c) Σ˜ is locally homeomorphic to the direct product of a Cantor set by a segment at each point, which does not belong to the Markov partition boundary.

III. Σ+ is an attractor for X U ′′. ∈ 2 IV. T (X) does not admit a finite Markov partition of Σ˜ for X U U ′ . 16 ∈ 2\ 2 We will say that the rational case takes place if X U ′ . It follows from the ∈ 2 2 definition of the rational case that a bifurcation set BX of codimension 2, which 1 1 is an intersection of two bifurcational surfaces B1X and B1X of codimension 1, is connected with each system X U ′ . Let us call the surface B1 a rational ∈ 2 iX 1 one, if Γi tends either to O or to a periodic orbit; if Γi tends to O then BiX will be called a bifurcational surface of the first type, and if Γi tends to the periodic orbit, a bifurcational surface of the second type. Surfaces of the first type are

16See [64] on Markov partitions.

61 of interest in that the bifurcation of periodic orbits is connected only with them — a periodic orbit may disappear only via matching either to a separatrix loop or to a contour. Transition through surfaces of the second type are responsible, in general, for local re-arrangements of an attractor. However, there may be exceptional re-arrangements, namely those related to a global internal crisis of Σ 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 condition under which the existence of the Lorenz attractor are guaranteed. Consider a finite-number parameter family of vector field defined by the system of differential equations

x˙ = X(x, µ), (8.8) where x Rn+1, µ Rm, and X(x, µ) is a Cr-smooth functions of x and µ. ∈ ∈ Assume that following two conditions hold

A. System (8.8) has a equilibrium state O(0, 0) of the saddle type. The eigenvalues of the Jacobian at O(0, 0) satisfy

Reλ < Reλ < λ < 0 < λ . n ··· 2 1 0

B. The separatrices Γ1 and Γ2 of the saddle O(0, 0) returns to the origin as t + . → ∞ Then, for µ > 0 in the parameter space there exists an open set V , whose boundary contains the origin, such that in V system (8.8) possesses the Lorenz attractor in the following three cases [75]: Case 1.

A Γ1 and Γ return to the origin tangentially to each other along the dominant

direction corresponding to the eigenvalue λ1;

62 B 1 λ1 Reλi <γ< 1, νi > 1, γ = , νi = ; 2 −λ0 − λ0

A The separatrix values A1 and A2 (see above) are equal to zero.

In the general case, the dimension of the parameter space is four since we may choose µ1,2, to control the behaviour of the separatrices Γ1,2 and µ3,4 = A3,4. In the case of the Lorenz symmetry, we need two parameters only. Case 2.

A Γ and Γ belong to the non-leading manifold W ss W s and enter the 1 ∈ saddle along the eigen-direction corresponding to the real eigenvector λ2

B 1 λ1 Reλi <γ< 1, νi > 1, γ = , νi = ; 2 −λ0 − λ0 In the general case, the dimension of the phase space is equal to four. Here,

µ3,4 control the distance between the separatrices. Case 3.

A Γ / W ss; 1,2 ∈ B γ = 1;

C A = 0, and A < 2. 1,2 6 | 1,2| In this case m = 3, µ = γ 1 3 − In the case where the system is symmetric, all of these bifurcations are of codimension 2. In A. Shilnikov [68, 69] it was shown that both subclasses (A) and (C) are realized in the Shimizu-Marioka 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

63 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. Since there is a foliation (see above) we may reduce the Poincar´emap to the form

x¯ = F (x, y), (8.9) y¯ = G(y), where the right-hand side is, in general, continuous, apart from the discontinuity line y = 0, and G is piece-wise monotonic. Therefore, it is natural to reduce (8.9) to a one-dimensional map

y¯ = 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 G′ > 1. The | | latter is possible only when the stable ﬁliation is smooth. 17. To conclude we remark that the topological dimension of strange attractors of the Lorenz type is equal to two. As for the fractal dimension, it does not exceed three. In section 9 we will discuss the way of constructing a strange attractor of a higher topological dimension.

9 Quasiattractors. Dynamical phenomena in the Newhouse regions

Hyperbolic attractors, despite their “attraction”, have never been observed in non-linear dynamics. Perhaps, because of the scenarios of their appearance are not simple from the practical point of view. It is quite 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

17For smoothness of the stable foliation in mapping of the Lorenz type, see Robinson [60] and Shashkov and Shilnikov [63]

64 that strange attractors of the Lorenz type occur in concrete applications. This is not the case of 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 imbedded 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 phenomenon under consideration, such as well-developed turbulence. The ex- ception 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]. The quasi-attractors 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 Poincaréhomoclinic trajectories. Let us consider first the two-dimensional case. let O be a fixed point of the saddle type with eigenvalues λ < 1 and γ < 1. Assume that the stable | | | | and unstable manifolds of O have a quadratic tangency along the structurally unstable homoclinic curve Γ.

Theorem 9.1 (Gonchenko, Turaev and Shilnikov [34]) If λγ is a trajectory of | | the third type 18, then in the set B1 of such diﬀeomorphisms the diﬀeomorphisms with a countable set of stable periodic points are everywhere dense.

Since 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. 18see section 6

65 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 since it concerns three-dimensional systems with a negative divergence, for example, in Lorenz model with large Rayleigh numbers, in systems with spiral chaos, in Chua’s circuit, as well as some in diffeomorphisms whose Jacobians are less then one, for example, the Hénon map etc. Let us pause on the Henon map [42] for a while. This is a more studied map. So, for example, it is known from [25] that when b is small, the | | values of the parameter a for which the stable periodic points exist, form an open, everywhere dense set. At the same time, numerical experiments exhibit a rather complex behavior of the trajectories in the Hénon map. It should be also mentioned that Benedics and Carleson [16] had proved the existence of a Cantor set of the values a of a positive measure for values of which the Hénon map has a strange attractor. In the Benedict-Carleson attractor, all of the periodic points are structurally stable whereas the attractor itself is not. Small perturbations of the parameter a may destroy such an attractor. This situation resembles the situation which takes place for the orientable C2-smooth circle diffeomorphism: quasi-periodic trajectories exist for irrational values of the Poincarérotation number but are structurally unstable. 2 In the case of a one-parameter family of C -smooth diffeomorphisms Xµ with properties formulated in theorem 9.1 when µ = 0, the associated map T k+m : σ0 Π (see section 8) after rescalling takes the form close to the k 7→ 0 Henon map, see Fig. 6.3a. The case where λγ > 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 ln λ θ = | |. (9.1) − ln γ | | We can go further in the case of systems with homoclinic tangencies of the third type: such systems have a countable set of Ω-moduli (Gonchenko,

66 Shilnikov [31]. Below we consider also a high-dimensional case. Note at once that the results obtained for this case cannot always be generalized. Of a special interest here are questions on the co-existence 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, the 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, which concerns the strong results, may be found in the series of papers of Gonchenko, Shilnikov and Turaev [33, 34, 35]. Let a family of vector ﬁelds X be Cr-smooth ((r 3)) with respect to all of 0 ≥ its arguments. We assume that the system X satisﬁes the properties A) E) 0 − below.

A) X0 has a saddle periodic orbit L0 with multipliers λi, γj such that

λ λ < 1 < γ le γ γ , | m| ≤ · · · ≤ | 1| | 1| | 2| ≤ · · · ≤ | n| Denote λ = λ , γ = γ . Suppose also that either | 1| | 1| A1) λ is real and λ> λ , 1 | 2| or

A2) λ = λ¯ = λeiϕ, (ϕ =0, π) and λ> λ ; 1 2 6 | 3| or

A3) γ is real and γ < γ , 1 | 2| or

A4) γ =γ ¯ = γeiψ, (ψ =0, π) and γ > γ ; 1 2 6 | 3| B) the saddle value σ = λγ = 1; 1 | | 6 C) the stable W s and the unstable W u manifolds of L have a quadratic tangency along the homoclinic curve Γ . In particular, dim(W¯ s W¯ u )=2 0 M ∩ M ¯ s ¯ u s u where WM and WM are the subspaces tangential to W and W , respectively, at the point M Γ . ∈ 0

67 The study of dynamics of such systems is usually reduced to that of the

Poincare map on some cross-section S to periodic orbits L0. This map is constructed as a superposition of two maps T0(µ) and T1(µ) such that T0 is a map along the orbits close to the periodic orbit Lµ and T1 is a global map along the orbits in a neighbourhood of Γ . The point O = L S is a saddle ﬁxed 0 ∩ 0 point; its invariant manifolds we also denote by W s and W u. Let M + W s ∈ and M − W u be two points of intersection Γ with S, then we assume that ∈ 0 − − + the map T1 is deﬁned in a neighbourhood of a point M , T1(M )= M . We denote the nonleading stable and unstable manifolds of O by W ss and by W uu, respectively, and the subspaces tangent to the manifolds W s, W u, W ss, W uu at the point O by W¯ s, W¯ u, W¯ ss and W¯ uu. Similarly, we let W s+, W u+, W¯ s+ and W¯ u+ denote the leading directions. We can now show that W ss and W uu, as associated leaves, may be uniquely embedded into invariant Cr−1- smooth folia- ss uu s u tions F and F on Wloc and Wloc, respectively, and that there exist invariant C1-smooth manifolds H and H tangential to W¯ u W¯ s+ and W¯ s W¯ u+, u s ⊕ ⊕ H W u , H W s , [Hirsh, Pugh and Shub [43]]. u ⊂ loc s ⊂ loc We assume also that

D) M + / W ss, M − / W uu ∈ ∈ ss + −1 uu E) 1. T1(Hu) is transverse to F at M and 2. T1 (Hs) is transverse to F at M −.

r 1 G) the curve Xµ in the space of C -smooth vector ﬁelds is transverse to H .

It is rather convenient to distinguish between the leading and non-leading coordinates. Let x, y be the leading coordinates, and let u, v be the non-leading coordinates. Then, the map T0 may be written in the form

(¯x, u¯) = A(x, u) + (f (x,u,y,v,µ),f (x,u,y,v)) 1 2 (9.2) (¯y, v¯) = B(y, v) + (g1(x,u,y,v,µ),g2(x,u,y,v)) where, ∂f f (x, y, 0, 0)=0, f (0, 0,y,v)= i (0, 0,y,v)=0, 1 i ∂x ∂g g (0, 0,y,v)=0, g (x, u, 0, 0) = i (x, u, o, 0)=0, 1 i ∂y

68 r−1 and fi and g - C are smooth functions. + + + Let Π0 and Π1 be suﬃciently small neighbourhoods of the points M (x ,u , 0, 0) and M −(0, 0,y−, v−):

+ + Π0 = (x0,u0,y0, v0)/ (x0 x ,u0 u ) ǫ0, (y0, v0) ǫ0 , { k − − k≤ k k≤ } (9.3) Π = (x ,u ,y , v )/ (x ,u ) ǫ , (y y−, v v−) ǫ . 1 { 1 1 1 1 k 1 1 k≤ 0 k 1 − 1 − k≤ 0} The map T ′ : Π Π along the trajectories of the system in a neighbourhood 0 7→ 1 of L is deﬁned on σ Π , where σ consists of a countable set of non-crossing µ 0 ⊂ 0 0 0 ∞ n 0 ′ 0 ”strips” σ and ¯ T σ = T σ Π , where k¯ stands for a number n n=k 0 n ⊂ 1 (maximal) of theS iterations under which the map of the strips is deﬁned; k¯ 1 n 0 depends on the size of the chosen neighbourhood. Denote σn = T0 σn.

Introduce the coordinates (xi,ui,yi, vi) in neighbourhoods Πi. The map n 0 T0 σn can be then written in a cross-form

n x1 = A1 x0 + ψ1(x0,u0,y1, v1) n  u1 = A2 u0 + ψ2(x0,u0,y1, v1)  −n (9.4)  y0 = B1 y1 + ψ3(x0,u0,y1, v1)  −n v0 = B2 v1 + ψ4(x0,u0,y1, v1)  ˆn −n ˆ where ψ r−2 λ , i = 1, 2, ϕ r−2 γˆ , j = 3, 4, λ > λ > k ikC ≤ k j kC ≤ max λ , λ/γ , γ < γˆ < min γ ,γ2 , A < λ˜−1, B−1 < γ˜−1, and {| i| | |} {| j| } k 2k 2

n n in the case A1) A1 = λ ; cos(nϕ) sin(nϕ) In the case A2) An = λnE(nϕ), where E(nϕ)= − ; 1 sin(nϕ) cos(nϕ) In the case A3) B−n γ−n; cos(nψ) sin(nψ) In the case A4) N −n = λnE( nψ), where E(nψ)= − . 1 − sin(nψ) cos(nψ) Taking into account the conditions C) - F) the map T : S S along 1 7→ orbits of the system Xµ in a neighbourhood Γ0 recast as :

+ x − x0 x = F (x1,u1,y1 y , v0), − + u − −  u0 u = F (x1,u1,y1 y , v0),  − − − (9.5)  y0 = G(x1,u1,y1 y , v0), v v− = H(x ,u ,y − y−, v ), 1 − 1 1 1 − 0  x where F x, F u, G, H Cr are smooth functions, ∂F (0, 0, 0, 0, 0) = b = 0, ∂y1 6 G(x ,u ,y y−, v )= D(x ,u ,y y−, v ) (y y−ψ)2 +C(x ,u , v¯ ), (9.6) 1 1 1 − 0 1 1 1 − 0 · 1 − 1 1 0

69 D(0, 0, 0, 0) = d, ψ(0, 0, 0) = 0, C(0, 0, 0) = 0, ∂C (0, 0, 0) = c = 0 and y y− = ∂x 6 1 − ∂G − ψ(x1,u1, v0) is a solution of the equation (x1,u1,y1 y , v0) = 0. ∂y1 − 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 σ < 1, then γ is real and positive and d < 0, and if σ > 1, then λ is real and positive and dc > 0. The second class consists of systems with real and positive γ and λ such that d> 0 and c< 0. All the rest go into the third class. We consider a small neighbourhood U of the contour L Γ. 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, Γ . 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 0 010 0 and { } {··· ··· ··· ···} 0 020 0 are identiﬁed. {··· ··· ··· ···} For systems of the third class, N also has nontrivial hyperbolic subsets, bu they do not in general exhaust all of N. Moreover, under the motion along the 1 surface B3 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 Ω-equivalence. The multipliers λ and γ such that λ = λ and γ = γ are said to be leading 1 j | i| | j | and the remainder are non-leading. Let ps denote the number of the leading λi and let pu denote the number of leading γj . We will then say that X has type (p ,p ). We also set θ = l ln λ/ ln γ. s u − Theorem 9.3 Let X ,X B1. Then for Ω-equivalence 19 of X and X it 1 2 ∈ 3 1 2 is necessary that θ1 = θ2 in the case of systems of type (1,1); that θ1 = θ2 and

ϕ1 = ϕ2 for systems of type (2,1); that θ1 = θ2 and ψ1 = ψ2 for systems of type (1,2); and for systems of type (2,2): ϕ1 = ϕ2 and ψ1 = ψ2, and also θ1 = θ , except, perhaps, for the case ϕ = ϕ = ψ = ψ 2π/3,π/2, 2π/5,π/3 . 2 1 2 1 2 ∈{ } The quantities ψ, ϕ and θ are the fundamental moduli in the sense that 1 they are deﬁned everywhere on B3 . If ψ/2π, ϕ2π, or θ is irrational, then there 19We talk about Ω-equivalence via a homeomorphism that is homotopic to the identity in U.

70 also exist other moduli, for example, analogous to the quantity τ introduced in [Gonchenko and Shilnikov [28, 29, 30]]. Moreover, we have

1 ∗ Theorem 9.4 In B3 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.

For systems from B∗ the quantities θ computed for the corresponding saddle periodic orbits are moduli of Ω-equivalence (theorem 9.3). Thus, we have a theorem which generalizes a result from [33, 34] to the higher-dimensional case.

1 Theorem 9.5 In B3 the systems with a countable number of moduli of Ω- equivalence are dense.

We note that if the Ω-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 lead to inﬁnitely degenerate bifurcations. Thus, we have

1 Theorem 9.6 In B3 the systems with a countable number of structurally unstable homoclinic curves of arbitrary orders of tangency and with a countable number of structurally unstable periodic orbits, both with multipliers ρ =1 and with ρ = 1, of arbitrary orders of degeneracy are dense. − Let X be a system from B1 (not necessarily of the third class). We consider r 1 a C -smooth ﬁnite-parameter family Xµ, transversally intersecting B at X for µ = 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. From Gonchenko, Shilnikov and Turaev [35, 37] we obtain

Theorem 9.7 There exists a sequence of open sets ∆i accumulating to µ = 0 such that X for µ ∆ lie in the Newhouse region, and the values of µ for µ ∈ i which the periodic orbit Lµ has a structurally unstable homoclinic curve of the third class are dense in ∆i.

From the two last theorems we get

71 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 Ω-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 1 C (ps + pu + 1)-dimensional C -smooth manifold W , depending continuously on the system. Any orbit L N has (m p + 1)-dimensional strong stable and ∈ − 2 (n p + 1)-dimensional strong unstable invariant Cr-smooth manifolds W ss − u L uu ss uu and WL , respectively, such that the trajectories belonging to WL (resp. WL ) tend exponentially to L as t + (resp. t ) with exponent not less → ∞ → −∞ in modulus than λˆ /T (resp. γ/Tˆ ), where T is the period of the orbit L, 0 < | | λˆ < λ, and γ>γˆ . The intersection W C S is deﬁned by an equation of the ∩ form (u, v) = fC(x, y), where fC (0, 0) = 0 and ∂fC(0, 0)/∂(x, y)=0. The ss u manifolds WL and W uL in the intersection with S are collections of leaves 1 1 transversal to Bu and Bs , respectively, and having the form (x, y, v) = fss(u) and (x,y,u)= fuu(v)

The dynamics on W C essentially depends on the modulus of the product of

ps pu the leading multipliers, i.e., onσ ˜2 = λ γ .

Theorem 9.10 The trajectories of the intersection to W C of any system close to X have for σ˜ not less than (p [p /θ]) 1 negative Lyapunov exponents, 2 s − u ≤ and for σ˜ > 1 that have not less than (p [p /θ]) 1 positive Lyapunov 2 u − s ≤ exponents.

Here [ ] denotes the integer closest to but strictly less than . · · · Forσ ˜2 < 1 we distinguish the three classes:

+ (1 ) (ps,pu)=(1, 1) or (ps,pu)=(2, 1) for λγ < 1.

+ (2 ) (ps,pu) = (2, 1) λγ < 1, and (ps,pu) = (1, 2) or (ps,pu) = (2, 2) for λγ2 < 1.

72 + 2 (3 ) (ps,pu)=(2, 2) for λγ > 1.

− − − Forσ ˜2 we distinguish the three classes (1 ), (2 ) and (3 ), which are obtained respectively from (1+) (3+) by replacing t by t. It follows from − − theorem 9.10 that in case (1−) (resp. (1+)) the restriction to W C of any system close to X cannot have periodic orbits with more than l stable (resp. unstable) multipliers. From this and theorem 9.9 we come to

Theorem 9.11 If σ˜2 > 1 or L has nonleading unstable multipliers, then all trajectories of both X and any system suﬃciently close to it are unstable.

We have noticed already in section 7 that the normal form of a system with a triple degenerate periodic orbit with the multipliers ( 1, 1, 1) may have − − the Lorenz attractor for certain parameter values. As we know now from the theorems above such periodic orbits of systems of dimensions n 5 exists in the ≥ Newhouse regions. Moreover, it should be noted that in the Newhouse regions along with stable periodic orbits, there co-exist a countable number of strange attractors. This implies that a quasiattractor may have a very complex hierarchy of embedded limiting sets, in other words, accurate numerical experiments may reveal the existence the stability windows of stable periodic orbits as well as those of strange attractors.

10 Example of wild strange attractor

In this section, following the paper of Shilnikov and Turaev [79], we will distinguish a class of dynamical systems with strange attractors of a new type. The peculiarity of such an attractor is that it may contain a wild hyperbolic set. We remark that such an attractor is to be understood as an almost stable, chain-transitive closed set. Let X be a smooth (Cr, r 4) ﬂow in Rn (n 4) having an equi- ≥ ≥ librium state O of a saddle-focus type with characteristic exponents γ, λ − ±

73 iω, α ,..., α where γ > 0, 0 <λ< Reα , ω = 0. Suppose − 1 − n−3 j 6 γ > 2λ (10.1)

This condition was introduced in [53] where it was shown, in particular, that it is necessary in order that no stable periodic orbit could appear when one of the separatrices of 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 R1, y R2, z Rn−3) such that ∈ ∈ ∈ the equilibrium state is in the origin, the one-dimensional unstable manifold of O is tangent to the x-axis and the (n 1)-dimensional stable manifold is tangent − to x =0 . We also suppose that the coordinates y correspond to the leading { } 1,2 exponents λ iω and the coordinates z correspond to the non-leading exponents ± α. Suppose that the flow possesses a cross-section, say, the surface Π : y = {k k 1, z 1, x 1 . The stable manifold W s is tangent to x = 0 at O, k k≤ | |≤ } { } therefore it is locally given by an equation of the form x = hs(y,z) where hs is a smooth function hs(0, 0)=0, (hs)′(0, 0) = 0. We assume that it can be written in such form at least when ( y 1, z 1) and that hs < 1 here. Thus, k k≤ k k≤ | | s s the surface Π is a cross-section for Wloc and the intersection of Wloc with Π has the form Π : x = h (ϕ, z) where ϕ is the angular coordinate: y = y cos ϕ, 0 0 1 k k y = y sin ϕ, and h is a smooth function 1 < h < 1. One can make 2 k k 0 − 0 h 0 by a coordinate transformation and we assume that it is done. 0 ≡ We suppose that all the orbits starting on Π Π return to Π, thereby defining \ 0 the Poincarémap: T : Π Π T : Π Π, where Π = Π x > 0 + + → − − → + ∩{ } Π = Π x < 0 . It is evident that if P is a point on Π with coordinates − ∩{ } (x, ϕ, z), then 1 1 lim T−(P )= P−, lim T+(P )= P+, x→−0 x→+0 where P 1 and P 1 are the first intersection points of the one-dimensional sep- − + aratrices of O with Π. We may therefore define the maps T+ and T− so that

1 1 T−(Π0)= P−,T+(Π0)= P+ (10.2)

Evidently, the region ﬁlled by the orbits starting on Π (plus the point O D and its separatrices) is an absorbing domain for the system X in the sense that

74 the orbits starting in ∂ enter and stay there for all positive values of time D D t. By construction, the region is the cylinder y 1, z 1, x 1 D {k k≤ k k≤ | |≤ } with two glued handles surrounding the separatrices (Fig.9.1). We suppose that the (semi)flow is pseudohyperbolic in . It is convenient for D us to give this notion a sense more strong than it is usually done [43]. Namely, we propose the following Definition. A semiflow is called pseudohyperbolic 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 sum of two subspaces N1 and N2 (continuously depending on

the point) such that the maximal Lyapunov exponent in N1 is strictly less

than the minimal Lyapunov exponent in N2: at each point M, for any vectors u N (M) and v N (M) ∈ 1 ∈ 2 1 u 1 v lim sup ln k tk < lim inf ln k tk t→+∞ t u t→+∞ t v k k k k where ut and vt 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:

V const eσtV t ≥ · 0

with some σ > 0; here, V0 is the volume of any region in N2 and Vt 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 N1 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.

75 Note that the property of pseudohyperbolicity 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

N1 and N2 depend continuously on the system. Hereat, the property of volume expansion in N2 is also stable with respect to small perturbations. Our definition is quite broad; it embraces, in particular, hyperbolic flows for which one may assume (N ,N ) = (N s,N u N )or(N ,N ) = (N s N ,N u) 1 2 ⊕ 0 1 2 ⊕ 0 where N s and N u are, respectively, the stable and unstable invariant subspaces and N0 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 N1 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 Poincarémap is expanding in a direction transverse to the contracting foliation.

In the present paper we assume that N1 has codimension three: dimN1 = n 3 and dimN = 3 and that the linearized flow (at t 0) is exponentially − 2 ≥ contracting on N1. Condition A means here that if for vectors of N2 there is a contraction, it has to be weaker than those on N1. To stress the last statement, we will call N1 the strong stable subspace and N2 the center subspace and will denote them as N ss and N c respectively. We also assume that the coordinates (x,y,z) in Rn are such that at each point of the space N ss has a non-zero projection onto the coordinate space D z, and N c has a non-zero projection onto the coordinate space (x, y). Note that our pseudohyperbolicity conditions are satisfied at the point O from the very beginning: the space N ss coincides here with the coordinate space z, and N c coincides with the space (x, y); it is condition (10.1) which guarantees the expansion of volumes in the invariant subspace (x, y). The pseudohyperbolicity of the linearized flow is automatically inherited by the orbits in a small neighborhood of O. Actually, we require that this property would extend into the non-small neighborhood of O. D According to [43], the exponential contraction in N ss implies the existence of an invariant contracting foliation ss with Cr-smooth leaves which are tangent N

76 to N ss. As in [7], one can show that the foliation is absolutely continuous. After a factorization along the leaves, the region becomes a branched manifold (since D is bounded and the quotient-semiflow expands volumes it follows evidently D that the orbits of the quotient-semiflow must be glued on some surfaces in order to be bounded; cf.[40]). The property of pseudohyperbolicity is naturally inherited by the Poincaré map T (T ,T ) on the cross-section Π: here, we have: ≡ + −

A∗ There exists a foliation with smooth leaves of the form (x, ϕ)= h(z) , |−1≤z≤1 where the derivative h′(z) is uniformly bounded, which possesses the following properties: the foliation is invariant in the sense that if l is a leaf, then T −1(l T (Π Π )) and T −1(l T (Π Π )) are also leaves of the + ∩ + + ∪ 0 − ∩ − − ∪ 0 foliation (if they are not empty sets); the foliation is absolutely continuous in the sense that the projection along the leaves from one two-dimensional transversal to another increases or decreases the area in a finite number of times and the coefficients of expansion or contraction of areas are bounded away from zero and infinity; the foliation is contracting in the sense that if two points belong to one leaf, then the distance between the iterations of the points with the map T tends to zero exponentially;

∗ B The quotient maps T˜+ and T˜− are area-expanding.

The fulfillment of conditions A∗, B∗ and relation (10.2) is sufficient for theorems 10.1-?? below to be valid. As in [7], one can find sufficient conditions for the fulfillment of A∗ and B∗.

Lemma 10.1 . Let us write the map T as

(¯x, ϕ¯)= g(x, ϕ, z), z¯ = f(x, ϕ, z), where f and g are functions smooth at x =0 and discontinuous at x =0: 6 1 1 lim (g,f) = (x−, ϕ−,z−) P−, lim (g,f) = (x+, ϕ+,z+) P+ . x→−0 ≡ x→+0 ≡ Let ∂g det =0 . (10.3) ∂(x, ϕ) 6

77 Denote −1 −1 ∂f ∂f ∂g ∂g ∂f ∂g A = ∂z ∂(x,ϕ) ∂(x,ϕ) ∂z , B = ∂(x,ϕ) ∂(x,ϕ) − −1 −1 ∂g ∂g ∂g C = ∂(x,ϕ) ∂z ,D = ∂(x,ϕ) . If lim C =0, lim A D = 0 (10.4) x→0 x→0 k kk k sup A D + sup B sup C < 1, (10.5) k kk k k k k k P ∈Π\Π0 p rP ∈Π\Π0 P ∈Π\Π0 then the map has a continuous invariant foliation with smooth leaves of the form (x, ϕ)= h(z) where the derivative h′(z) is uniformly bounded. If, |−1≤z≤1 additionally,

sup A + sup B sup C < 1, (10.6) P ∈Π\Π0 k k rP ∈Π\Π0 k k P ∈Π\Π0 k k then the foliation is contracting and if, moreover, for some β > 0

the functions A x −β,D x β, B, C are uniformly bounded and H¨older continuous,| | | | ∂ ln det D ∂ ln det D (10.7) and and D x β are uniformly bounded, ∂z ∂(x, ϕ) | | then the foliation is absolutely continuous. The additional condition

sup √det D + sup B sup C < 1 (10.8) P ∈Π\Π0 rP ∈Π\Π0 k k P ∈Π\Π0 k k guarantees that the quotient map T˜ expands areas.

It follows from [53, 54] that in the case where the equilibrium state is a saddle-focus, the Poincar´emap near Π = Π W s is written in the following 0 ∩ form under some appropriate choice of the coordinates.

(¯x, ϕ¯)= Q±(Y,Z), z¯ = R±(Y,Z). (10.9)

Here

Z =Ψ2(x, ϕ, z), (10.10)

78 where ρ = λ/γ < 1/2 (see (10.1)), Ω = ω/γ and, for some η>ρ,

∂p+|q|Ψ i = O( x η−p), 0 p+ q r 2 ; (10.11) k ∂xp∂(ϕ, z)q k | | ≤ | |≤ − the functions Q±, R± in (10.9) (“+” corresponds to x > 0 - the map T+,“-” corresponds to x< 0 - the map T−) are smooth functions in a neighborhood of (Y,Z) = 0 for which the Taylor expansion can be written down

Q± = (x±, ϕ±)+ a±Y + b±Z + ..., R± = z± + c±Y + d±Z + ... (10.12)

It is seen from (10.9)-(10.12) that if O is a saddle-focus satisfying (10.1), then if a = 0 and a = 0, the map T satisfies conditions (10.4) and (10.7) with + 6 − 6 β (ρ, η). Furthermore, analogues of conditions (10.3),(10.5),(10.6), (10.8) are ∈ fulfilled where the supremum should be taken not over x 1 but it is taken | |≤ over small x. The map (10.9),(10.10),(10.12) is easily continued onto the whole cross-section Π so that the conditions of the lemma were fulfilled completely. An example is given by the map

x¯ =0.9 x ρ cos(ln x +ϕ), | | | | ϕ¯ =3 x ρ sin(ln x +ϕ), (10.13) | | | | z¯ = (0.5+0.1z x η) sign x | | where 0.4= ρ < η As stated above, the expansion of volumes by the quotient-semiflow restricts the possible types of limit behavior of orbits. Thus, for instance, in there may D be no stable periodic orbits. Moreover, any orbit in has a positive maximal D Lyapunov exponent. Therefore, one must speak about a strange attractor in this case. Beforehand, we recall some definitions and simple facts from topological dynamics. Let XtP be the time-t shift of a point P by the flow X. For given ε>

0 and τ > 0 let us deﬁne as an (ε, τ)-orbit as a sequence of points P1, P2,...,Pk such that Pi+1 is at a distance less than ε from XtPi for some t > τ. A point Q will be called (ε, τ)-attainable from P if there exists an (ε, τ)-orbit connecting P and Q; and it will be called attainable from P if, for some τ > 0, it is (ε, τ)-attainable from P for any ε (this deﬁnition, obviously, does not depend

79 on the choice of τ > 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

XtP for any t. A compact invariant set C is called chain-transitive if for any points P C and Q CC and for any ε> 0 and τ > 0 the set C contains an ∈ ∈ (ε, τ)-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 neighborhood U there is a neighborhood V (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 neighborhood U(C) there exist ε0 > 0, τ > 0 and a neighborhood V (C) U such that the (ε0, τ)-orbits starting in V never leave U. It is ⊆ ∞ known, that a set C is orbitally stable if and only if C = U where U ∞ j { j}j=1 j\=1 is a system of embedded open invariant (with respect to the forward 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 ∂Uj enter inside Uj for a time interval not greater than some τj ; it is clear in this situation that (ε, τ)-orbits starting on ∂U lie always inside U if ε is sufficiently small and τ τ ). Since j j ≥ j 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 Π. Definition. We call the set of the points attainable from the equilibrium A state O the attractor of the system X. This definition is justified by the following theorem.

Theorem 10.1 The set is chain-transitive, completely stable and attainable A from any point of the absorbing domain . D The stability of follows immediately from the deﬁnition: it is known that A for any initial point (and for the point O, in particular) the set of attainable

80 points is completely stable (the system of absorbing domains is given by the sets of the points (ε , τ)-attainable from the initial point, with arbitrary ε +0 j j → and τ > 0). To prove the rest of the theorem note that the following lemma is valid.

Lemma 10.2 Points asymptotic to O as t + are dense in (in other → ∞ D words, the stable manifold of O is dense in ). D Indeed, take an arbitrary point on Π and let U be a neighborhood of Π. If U would not intersect with the pre-images of the surface Π = W s Π, then for 0 loc ∩ all i 0 the map T i is continuous and the areas of the projections of the ≥ |U i sets TU onto z = 0 along the leaves of the invariant foliation would increase exponentially (by virtue of property B∗), which contradicts the boundedness of

Π. Thus, the pre-images of Π0 (and these are the intersections of the stable manifold of O with the cross-section Π) are dense in Π, which implies that the stable manifold of O is dense in . D Lemma 10.2 implies immediately that O is attainable from any point of . D In particular, for any two points P and Q in , the point O is attainable from A P whereas Q is attainable from O according to the deﬁnition of . Hence, A Q is attainable from P , and to prove the chain-transitivity of it remains to A show that the (ε, τ)-orbits connecting P and Q can be chosen lying in . This, A however, follows from the complete stability of : for any δ > 0, an (ε, τ)-orbit A connecting P and Q does not leave the δ-neighborhood of if ε is suﬃciently A small, whence this (ε, τ)-orbit may be approximated by an (ε′, τ)-orbit which lies in where ε′ may be greater than ε but, clearly, ε′ 0 as ε 0, δ 0. A → → → Theorem 10.1 implies that is the least completely stable subset of : since A D 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 D contain the point O and, consequently, the set . Thus, is the intersection A A of all completely stable subsets of . D The next theorem shows that is a unique chain-transitive and completely A stable set in . D

81 Theorem 10.2 If an orbitally stable subset of contains points which do not D belong to , it contains points which are not chain-recurrent. A

Proof. Since the stable manifold of O is dense in , any orbitally stable subset D of contains the point O. Since any connected component of an orbitally stable D set is orbitally stable itself, we have that each of the components must contain O whence there is only one component; i.e., any orbitally stable subset of is D connected. Let C be such a set and suppose there is a point P C, P . ∈ 6∈ A Since is completely stable, there exists an absorbing domain U which contains A and does not contain P . The point O belongs to both and C: we have A A 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 ∂U. As we stated above, the (ε, τ)-orbits starting on ∂U always lie inside U, at a finite distance from ∂U, if ε is sufficiently small and τ is sufficiently large, which means that no point of ∂U 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 with the cross-section Π; we A will use this in the proof of theorem 10.4 Let q> 1 be the area expansion factor for the quotient map T˜: if V is a region in Π not intersecting Π0, then

S(T V ) > qS(V ), (10.14) 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. Besides, since S(Π) = 2S(Π ) and T Π Π, we have S(T Π )/S(Π ) < 2, and therefore + + ⊂ + +

Let us denote the separatrices of O by Γ+ and Γ−, and let P ± be the con- { i } secutive points of intersection of Γ± with Π; these sequences may be inﬁnite or ﬁnite, the latter if the corresponding separatrix forms a homoclinic loop (returns to O as t + ). → ∞

82 Theorem 10.3 The number N of connected components of the set Π is A ∩ ﬁnite and ln(q 1) 2 N < 2+ | − |. (10.15) ≤ ln q ± Each of the components contains at least one of the points Pi . Furthermore, for some integers N 1, N 1, N + N = N, satisfying the inequality + ≥ − ≥ + − q−N+ + q−N− > 1, (10.16) the set Π is represented in the form A ∩ Π= + + − − , (10.17) A ∩ A1 ∪···∪AN+ ∪ A1 ∪···∪AN− where + and − are the components containing the points P + and P − respec- Ai Ai i i tively. In this formula all the components ± are diﬀerent, Ai + Π = i

(see Fig. 9.2).

Let us take a one-parameter family Xµ of systems of the kind under consideration and assume that a homoclinic loop of the saddle-focus O exists at µ =0, i.e., one of the separatrices (say, Γ ) returns to O as t + . In other words, + → ∞ we assume that the family Xµ intersects, at µ = 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 µ varies, the loop splits and if M is the number of the last point of intersection of the separatrix Γ with the cross-section Π at µ = 0 (P + Π at µ = 0), then the distance + M ∈ 0 + between the point PM and Π0 changes with a “non-zero velocity” when µ varies. We choose the sign of µ so that P + Π when µ> 0 (respectively, P + Π M ∈ + M ∈ − when µ< 0).

83 Theorem 10.4 There exists a sequence of intervals ∆i (accumulated at µ = 0) such that when µ ∆ , the attractor contains a wild set (non-trivial ∈ i Aµ transitive closed hyperbolic invariant set whose unstable manifold has points of tangency with its stable manifold). Furthermore, for any µ∗ ∆ , for any ∈ i system Cr-close to a system X∗, its attractor also contains the wild set. µ A As far as the general case is concerned when the separatrix of the point O do not form the homoclinic loops, we note that as it follows from the density of the stable manifold in that the separatrix of the point O are non-wandering D trajectories. So we can suppose that they may be closed by small perturbations of the system. A similar problem has already arozen in the case of the Lorenz map but it was overcome (in Cr-topology) using certain speciﬁc properties of the map. By now we have a rather important lemma of Hayashi [41] which allows us to solve this problem. Hence the following statement is true

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 of µ are dense in the intervals ∆i for which an attractor of the system contain a periodic orbit of the saddle-saddle type along with its three-dimensional 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. Up to now we have not yet spoken about the statistical properties of the attractors. This is necessary since 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:

84 1. There exists a neighbourhood U, A U, such that if x U, then dist(x(t), A) ⊂ ∈ → 0 as t + . → ∞ 2. For any initial probability distribution P on A, its shift as t + to an 0 → ∞ invariant distribution P on A, independently of P0.

3. The probability distribution P is mixing, i.e., the autocorrelation function tends to zero as t . → ∞ 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 for Lorenz-like attractors as well [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, i.e., such an attractor contain the trajectories of the same topological type. This is not the case when we deal with the wild strange attractors since it may contain co-existing hyperbolic trajectories of various types. This is a challenge for the ergodic theory. Acknowledgments

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 Re- search 96-01001135 and by INTAS-93-0570.

85 References

[1] A.A.Andronov, E.A.Leontovich, I.E.Gordon and A.G.Maier [1971] The theory of bifurcations of dynamical systems on a plane, Izrael program of sci- entiﬁc translations, Izrael

[2] A.A.Andronov, E.A.Leontovich, I.E.Gordon and A.G.Maier [1973] The theory of dynamical systems on a plane, Izrael program of scientiﬁc translations, Izrael

[3] A.A.Andronov and L.Pontryagin [1937] Syst`emes Grossiers, Dokl. Acad. Nauk SSSR, 14, pp.247-251

[4] A.A.Andronov, E.A.Vitt and S.E.Khaikin [1966] Theory of oscillations, Pergamon Press: Oxford

[5] V.S. Afraimovich, V.I.Arnold, Yu.S.Il’yashenko and L.P.Shilnikov [1989], Bifurcation theory in Dynamical Systems V, Encyclopedia of Math. Sci- ences, 5, Springer-Verlag

[6] V.S.Afraimovich, V.V.Bykov and L.P.Shilnikov [1977], On the appearance and structure of Lorenz attractor, DAN SSSR, 234, pp.336-339

[7] V.S.Afraimovich, V.V.Bykov and L.P.Shilnikov [1983], On the structurally unstable attracting limit sets of Lorenz attractor type, Tran. Moscow Math. Soc., 2, pp.153-215

[8] V.S.Afraimovich and L.P.Shilnikov [1973], On critical sets of Morse-Smale systems. Trudy Moskov. Mat. Obsh. 28 pp.181-214; English transl. in Trans. Moscow Math. Soc. 28 [1973].

[9] V.S.Afraimovich and L.P.Shilnikov [1974], On some global bifurcations connected with disappearance of saddle-node ﬁxed point, Dokl. Acad. Nauk SSRR 219 pp.1981-1985

[10] V.S.Afraimovich and L.P.Shilnikov [1983], Strange attractors and quasi- attractors in Nonlinear Dynamics and Turbulence eds. by G.I.Barenblatt, G.Iooss and D.D.Joseph, Pitman, NY, pp.1-28

86 [11] D.Anosov [1967] Geodesic ﬂows on compact Riemannian manifolds of negative curvature, Trudy Math. Inst. im. V.A.Steklova, 90 [in Russian]

[12] D.V.Anosov, S.Kh.Aronson, I.U.Bronshtein and V.Z.Grines [1985], Smooth dynamical systems in Dynamical Systems, Encyclopedia of Math. Sciences, 1, Springer-Verlag

[13] D.V.Anosov and V.V.Solodov [1991], Hyperbolic sets in Dynamical systems 9, Encyclopedia of Math. Sciences, 66, Springer-Verlag

[14] S.Kh.Aronson and V.Z.Grines [1991], Cascades on surfaces in Dynamical systems with hyperbolic behaviour. Dynamical Systems 9, Encyclopedia of Math. Sciences, 66, Springer-Verlag

[15]

[16] M.Benedicks and L.Carleson [1991], The dynamics of the H´enon map, Ann. Math, 133(1), pp.73-169

[17] G.D.Birkhoﬀ [1935], Nouvelles recherche sur les syst`emes dynamiques, Memorie Pont Acad. Sci. Novi Lyncaei, 3(1), pp.85-216

[18] R.Bowen [1970], Markov partition for Axiom A diﬀeomorphisms, Amer. J. Math., 92(3), pp.724-747

[19] R.Bowen [1973], Symbolic dynamics for hyperbolic ﬂows, Amer. J. Math., 95(2), pp.429-460

[20] L.A.Bunimovich and Ya.G.Sinai [1980] Stochasticity of an attractor in the Lorenz model, in Nonlinear waves ed. A.V.Gaponov-Grekhov, Nauka, pp.212-226, [in Russian]

[21] V.V.Bykov and A.L.Shilnikov (1992) On boundaries of the region of existence of the Lorenz attractor, Selecta Math. Sovietica, v.11[4], pp.375-382

[22] O.L.Chua [1993], Global unfolding of Chua’s circuit, IEICE Trans. Fund. ECCS, 76-SA, pp.704-734

87 [23] W. de Melo and S.J. van Streen [1979] Diﬀeomorphisms on surfaced with fonite number of moduli, Ergog. Th. and Dynam. Systems, 7, pp.415-462

[24] A.Donjoy [1932] Sur les courbes défines par les équationa différentielles à la surface du tore, J. Math, 17(IV), pp.333-375

[25] R.Devaney and Z.Nitecki [1979], Shift automorphism in the H´enon mapping, Comm. Math. Phys., 67, pp.137-148.

[26] N.K.Gavrilov and L.P.Shilnikov [1972]. On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. Part 1, Math.USSR Sb., 17 (1972), pp.467-485 ; [1973] Part 2, Math.USSR Sb, 19, pp.139-156.

[27] S.V.Gonchenko. Moduli of Ω-conjugacy of two-dimensional diﬀeomor- phisms with a structurally unstable heteroclinic cycle. Matemat. sbornik, 187 (1996), No. 9, pp. 3-25, in Russian.

[28] S.V.Gonchenko and L.P.Shilnikov [1986], On dynamical systems with structurally unstable homoclinic curves, Soviet Math. Dokl., 33(1), pp.234-238.

[29] S.V.Gonchenko and L.P.Shilnikov [1987], Arithmetic properties of topological invariants of systems with a structurally unstable homoclinic trajectory, Ukrainian Math.J., 39(1), pp.21-28.

[30] S.V.Gonchenko and L.P.Shilnikov [1990], Invariants of Ω-conjugacy of diffeomorphisms with a structurally unstable homoclinic trajectory, Ukrainian Math.J., 42(2), pp.134-140.

[31] S.V.Gonchenko and L.P.Shilnikov [1993], On moduli of systems with a structurally unstable homoclinic Poincare curve. Russian Acad. Sci. Izv. Math. 41(3), 417-445.

[32] S.V.Gonchenko and L.P.Shilnikov. Geometrical properties of 2D- diﬀeomorphisms with a structurally unstable homoclinic orbit. Int. J. Bi- furcation and Chaos, 5 (1995), No.3, pp.21-28.

88 [33] S.V.Gonchenko, L.P.Shilnikov and D.V.Turaev [1992], On models with a structurally unstable homoclinic Poincar´ecurve, Sov. Math.Dokl., v.44[2], pp.422-426

[34] S.V.Gonchenko, L.P.Shilnikov and D.V.Turaev [1993], On models with non- rough Poincar´ehomoclinic curves, Physica D, v.62, pp.1-14

[35] S.V.Gonchenko, D.V.Turaev and L.P.Shilnikov [1993], Dynamical phenomena in multi-dimensional systems with a structurally unstable homoclinic Poincare curve. Russian Acad. Sci. Dokl. Math., 47(3), pp.410-415.

[36] S.V.Gonchenko, D.V.Turaev and L.P.Shilnikov [1993], On the existence of Newhouse domains in a neighborhood of systems with a structurally unstable Poincare homoclinic curve (the higher-dimensional case). Russian Acad. Sci. Dokl. Math. 47(2), pp.268-273.

[37] S.V.Gonchenko, L.P.Shilnikov and D.V.Turaev [1996] Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits. In- terdisc. J. CHAOS, 6(1), pp. 1-17.

[38] S.V.Gonchenko, L.P.Shilnikov and D.V.Turaev [1997], On the Newhose regions of two-dimensional diﬀeomorphisms close to those with a structurally unstable contour Trudy Steklov Mat. Inst., to be published

[39] J.Guckenheimer [1970], Endomorphisms of Riemann sphere, Global Anal- ysis, Proc. Symp. Pure Math., —Providence, R.I. Amer. Math. Soc., 14, pp.95-103.

[40] J.Guckenheimer and R.F.Williams [1979], Structural stability of Lorenz attractors, Publ. Math. IHES, 50, pp.59-72

[41] S.Hayashi, On the solution of C1-stability conjecture for ﬂow, Preprint

[42] , M.H´enon [1976], A two-dimensional mapping with a strange attractor, Comm. Math. Phys, 50, pp.69-77

[43] M. Hirsh, C.Pigh and M. Shub [1977], Invariant manifolds, Lecture Notes in Math. Springer-Verlag,

89 [44] E.N.Lorenz [1963] Deterministic non-periodic ﬂow, J. Atmos. Sci., 20, pp.130-141

[45] R.N.Madan (Editor) [1993] Chua’s circuit: A paradigm for chaos, World Sci. Series on Nonlinear Science, Ser. B, Vol.1 (Wold Sci, Singapore)

[46] R.Mane [1982] An orgodic closing lemma, Ann. Math, 116(2), pp.503-540

[47] V.K.Melnikov [1963], On stability of center periodically perturbated, Trudy Mosc. Mat. Ob., 12, pp.3-52 [in Russian]

[48] J.Milnor and R.Thurston [1977] On iterated maps of the interval I ans II. Unpublished notes, Prinseton University Press: Princeton

[49] S.Newhouse [1974], Diﬀeomorphisms with inﬁnitely many sinks. Topology, 13, pp. 9-18.

[50] S.Newhouse [1979], The abundance of wild hyperbolic sets and nonsmooth stable sets for diﬀeomorphisms.Publ. Math. I.H.E.S. 50, pp.101-151.

[51] S. Newhouse and J. Palis. Cycles and bifurcation theory. Asterisque. 31 (1976), 44-140.

[52] V.I.Oseledec [1968] A miltiplicative ergodic theorem: Lyapunov characteristic numbers, Trans. Moscow Math. Soc., 19, pp.197-231

[53] I.M.Ovsyannikov and L.P.Shilnikov [1986] On systems with a saddle-focus homoclinic curve, Mat. Sbornik, 58, pp.557-574; English translation in Math. USSR Sb. 58, pp.557-574 [1987].

[54] I.M.Ovsyannikov and L.P.Shilnikov [1991] Systems with a homoclinic curve of multidimensional saddle-focus type, and spiral chaos Mat. Sb. 182, pp.1043-1073; English translation in Math. USSR Sb. 73, pp. 415-443 [1992].

[55] J.Palis [1978], A diﬀerentiable invariant of topological conjugacies and moduli of stability, Asterisque, 51, pp. 335-346.

90 [56] M.M.Peixoto [1962] Structural stability on two-dimensional manifolds, Topology, 1, pp.101-120

[57] H.Poincaré[1899] Les Méthodes nouvelles de la Mècanique céleste, 3, Paris

[58] R.Plykin (1974) Sources and sinks of A-diﬀeomorphism of surfaces, Math. Sbornik, v.94, pp.243-264

[59] R.Plykin, E.A.Sataev and C.V.Shlyachkov [1991], Strange attractors in Dy- namical systems 9, Encyclopedia of Math. Sciences, 66, Springer-Verlag

[60] C.Robinson [1981], Diﬀerentiability of the stable foliation for the model Lorenz equations, Lect. Not. in Math., 898, p.302-315

[61] C.Robinson [1989], Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonlinearity, 2(4), pp.495-518

[62] M.Rychlik [1990], Lorenz attractor through Shilnikov type bifurcation, Er- god. Theory and Dyn. Syst., 10(4), 793-821

[63] M.V.Shashkov and L.P.Shilnikov L.P. [1994], On existence of a smooth invariant foliation for maps of the Lorenz type, Diﬀerential Equations, 30(4), pp.586-595 [in Russian]

[64] Ya.G.Sinai [1968], Markov partitions and U-diﬀeomorphisms, Fun. Anal., 2(1), pp.64-89

[65] Ya.G.Sinai [1980] Stochasticity of dynamical systems attractor in the Lorenz model, in Nonlinear waves ed. A.V.Gaponov-Grekhov, Nauka, pp.192-212

[66] S.Smale [1963] Diﬀeomorphisms with many periodic point. in Diﬀ. and Comb.Topology ed. by S.Cairus, Princeton Univ Press, Princeton, pp.63-86

[67] S.Smale [1967] Diﬀerentiable dynamical systems,Bulletin of Amer. Math. Soc., 73, pp.747-817

[68] A.L.Shilnikov [1986], Bifurcations and chaos in the Shimizu-Marioka system [In Russian], in Methods and qualitative theory of diﬀerential equations,

91 Gorky State University, pp.180-193 [English translation in Selecta Mathe- matica Sovietica 10 pp.105-117, [1991]]

[69] A.L.Shilnikov [1993], On bifurcations of the Lorenz attractor in the Shimizu-Marioka system, Physica D62 pp.338-346

[70] A.L.Shilnikov, L.P.Shilnikov and D.V.Turaev [1993], Normal forms and Lorenz attractors, Int. J. Bifurcations and Chaos, 1(4), pp.1123-1139

[71] L.P.Shilnikov [1965] A case of the existence of a countable set of periodic motions, Sov.Math.Dokl., 6, pp.163-166.

[72] L.P.Shilnikov [1969] On a new type of bifurcation of multi-dimensional dynamical systems, Soviet Math. Dokl., 10, pp.1368-1371

[73] L.P.Shilnikov [1970] A contribution to the problem of the structure of an extended neighbourhood of a rough state of saddle-focus type, Math.USSR Sb., 10, pp.91-102.

[74] L.P.Shilnikov [1980] Theory of bifurcations and the Lorenz model, Ap- pendix to the russian edition of The Hopf bifurcation and its applications eds. by J.E.Marsden and M.McCracken, Mir, Moscow

[75] L.P.Shilnikov [1981] The bifurcation theory and quasi-hyperbiloc attractors. Uspehi Mat. Nauk, 36, pp.240-241

[76] L.P.Shilnikov, [1994] Strange attractors and dynamical models, J. Circ., Syst. and Comp., 3(1) pp.1-10

[77] L.P.Shilnikov, [1994] Chua’s circuit: rigorous results and future problems, Int. J. of Bifurcation and Chaos, 4(3), pp.489-519

[78] L.P.Shilnikov [1967], On a Poincare-Birkhoﬀ problem. Math. USSR Sb., 3, pp.91-102.

[79] L.P.Shilnikov and D.V.Turaev [1996], An example of a wild strange attractor, Mat. Sbornik, to appear

92 [80] L.P.Shilnikov and D.V.Turaev [1995], On a blue sky catastrophy, Soviet Math. Dokl., 342, N.5, pp.596-599

[81] M.Shub [1978], Stabilit´eglobale des systems dynamiques, Asterisque, 56

[82] Ya.L.Umanskii [1990], Necessary and suﬃcient conditions of the topological equivalence of three-dimensional dynamical systems, Mat. Sbornik, 181(2), pp.212-239

[83] R.Williams [1970], Classiﬁcation of one-dimensional attractors, Proc. Symp. in Pure Math., pp.361-393

[84] R.Williams [1967], One-dimensional non-wandering sets, Topology, 6, pp.473-487

[85] R.Williams [1974], Expanding attractors, Publ. Math. IHES, 43, pp.169- 203

[86] R.Williams [1979], The structure of Lorenz attractors, Publ. Math. IHES, 50, pp.101-152

93 Figure Captions

Fig.2.1 An unclosed Poisson stable trajectory passing through the point x0 strikes

the shadowed cross-section inside the ǫ-neighbourhood of the point x0 inﬁnitely many times.

Fig.2.2 Examples of the heteroclinic contours.

Fig.3.1 The stable and the unstable manifolds of a saddle periodic orbit may be homeomorphic to either a cylinder (a) or to a M¨obis band (b).

Fig.3.2 A transverse homoclinic orbit.

Fig.4.1 A homoclinic curve Γ to the saddle periodic orbit L

Fig.4.2 The neighbourhood of L and Γ has the shape of a solid-torus with a handle.

0 Fig.4.3 The sketch illustrates the method of a construction of strips σk, k = k,¯ k¯ +1,... which lie in Π+ and the domains of the maps T kΠ+ Π−. 0 7→ The strips σk accumulate to the segment W s Π+ as k . 0 ∩ → ∞ Fig.4.4 Under action of the global part of the Poincar´emap an image of an initial strip intersects the initial strip forming a horseshoe.

Fig.4.5 The graph of TMC for Ω set of the trajectories lying entirely in a neighbourhood of O Γ. ∪ Fig.5.1 A geometry of DA-diﬀeomorphism before (a) and after the surgery (b).

Fig.5.2 A Witorius-Van Danzig solenoid.

Fig.6.1 A transverse (a) and a non-transverse (b) intersection of the unstable u manifold WO and the stable leaves of an invariant torus.

Fig.6.2 A homoclinic tangency

Fig.6.3 Three main types of homoclinic tangencies.

Fig.6.4 A homoclinic loop to a saddle-focus

Fig.7.1 A structurally unstable heteroclinic connection with a tangency

94 Fig.7.2 There are two saddles O1 and O2 for µ< 0.

Fig.7.3 A non-rough equilibrium state of the saddle-saddle type at the bifurcation moment µ = 0.

Fig.7.4 (a) 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 Poincar´emap on a transversal to the saddle-node periodic orbit.

Fig.8.1 Two one-dimensional separatrices Γ1 and Γ2 form a homoclinic butterﬂy.

Fig.8.2 A structure of the bifurcation set on the plane (µ1,µ2).

Fig.8.3 The images of the two halves of the return plane under the action of the Poincar´emap. Trajectories started within the plane next strike it

within the shaded areas. The points M1 and M2 are the ﬁrst points of the intersection the separatrices and the cross-section.

Fig.8.4 Three possible cases of the Poincar´e: (A) orientable, (B) semiorientable and (C) nonorientable.

Fig.10.1 A pseudo-projection in R3 of the neighbourhood of a homoclinic contour of a saddle-focus.

Fig.10.2 Figure illustrates schematically the connection components of the attractor A.