An introduction to chaos dynamic systems

Acta Applicandae Mathematicae 13 (1988), 221-226. 221 Book Review R. L. Devaney: An Introduction to Chaotic Dynamical Systems, Ben- jamin/Cummings, 1986. The dynamical systems which are discussed in this volume are differentiable maps ~: M--~ M. The dynamical aspects come from the interpretation: the points of M are supposed to represent the possible states of some 'system' - the time is discrete and parametrized by N or Z - a state x e M at time n leads to a state ~(x) at time n + 1. This interpretation is important to understand the type of questions which are asked about these dynamical systems. In the present context, M is finite-dimensional, often a vector space but sometimes a closed manifold (mainly a sphere or a torus). The map ~ is assumed to be differentiable; sometimes it is assumed to be a diffeomorphism (i.e., to have a differentiable inverse) but not always (if not, we speak of an endomorphism). This volume introduces the reader to recent work in this field. The author aimed at, and in my opinion succeeded in, writing a text which requires a minimum of mathematical sophistication, e.g., by avoiding 'manifolds' when not essential and by preferring exercises about special examples to general theorems. This makes the book, I think, very useful as a textbook, even for undergraduate courses. Before entering into a further discussion, I give a brief description of the content of the three parts in which this book is divided. The first part is a discussion of 1-dimensional dynamical systems, i.e., dynami- cal systems with dim(M) = 1. It was only relatively recent that it was realized that such systems have interesting properties. The author uses this very low-dimen- sional context to introduce a number of notions which are also of importance in higher dimensions, like hyperbolicity, symbolic dynamics, t01~ological conjugacy, chaos, structural stability, bifurcation, Morse-Smale diffeomorphisms, and homoclinic points. Then it also contains a discussion of concepts and results which are typically one-dimensional in the sense that they depend on the ordering of M = R: Sarkovskii's theorem, Schwarzian derivative, kneading theory, and the period doubling sequence (although period doubling also occurs in higher dimensions). In the second part, there is a discussion of higher-dimensional systems. The main part deals with systems which have some form of hyperbolicity. It includes the surprisingly complicated attractors which can occur in 2-dimensional diffeomorphisms (the Plykin attractors). After a discussion of the invariant 222 BOOK REVIEW manifold theorem, this part concludes with a section on the H6non map mainly consisting of a long list of exercises by which one is introduced to certain aspects of this intriguing example through the application of many of the previous results. In the third and final part, one considers complex dynamical systems: M is the complex plane C or the Riemann sphere ~-3 and ~0: M---~ M is holomorphic, often polynomial. The main concepts differ from those in the previous part: the attractors here are usually quite simple and the emphasis is more on the boundaries of the domains of attraction (which often coincide with the Julia set). Also the methods are quite different: they depend strongly on the classical theory of complex functions of one variable. One of the main examples uniting these three parts is the 1-parameter family of quadratic endomorphisms ~p,(x) =/x" x" (1 - x). In the first part /z and x are real; in the second part we meet the H6non map, which is just a 'thickening' of this endomorphism to a diffeomorphism in R 2 using the principle: if ~:R--->R is an endomorphism and b#0 then ~pb:R2--->R2, defined by q~b(X, y) = (q~(X) + y, b" x), is a diffeomorphism which 'reduces to ~p' for b = 0; in the third part we again meet the same example but now x and/z are in the complex plane. I think that this presentation with emphasis on examples and exercises is very suitable for the geometric theory of dynamical systems which, in the present state, is in the first place a collection of examples. In the rest of this review I shall concentrate on the discussion of a few notions which are central to the subject of this volume but which are also somewhat controversial in the sense that people disagree about 'the right definition' or about their relevance. The author of the present volume seems to avoid these discussions. In what follows, I comment on some of these controversial points, mainly in order to provide some information on the different points of view. (As references for other good introductions to dynamical systems, but written from a different point of view, I want to mention [1] which treats dynamical systems from the point of view of the natural sciences and [4] which gives a pictorial introduction to dynamical systems.) Chaos. We first quote the author's definition (p. 50). DEFINITION 8.5. Let V be a set. F: V~ V is said to be chaotic on V if 1. f has sensitive dependence on initial conditions. 2. f is topologically transitive. 3. periodic points are dense in V. To summarize, a chaotic map possesses three ingredients; unpredictability, indecomposability, and an element of regularity. A chaotic system is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down or decomposed into two subsystems (two invariant open subsets) which do not interact under f because of topological transitivity. And, in the midst of this random behavior, we nevertheless have an element of regularity, namely the periodic points which are dense. BOOK REVIEW 223 The sensitive dependence on initial conditions means that there is a 8 > 0 such that for all x e V and all neighbourhoods U of x in V there are y ~ U and n > 0 such that the distance between Fn(x) and Fn(y) is at least 3. Conditions 2 and 3 in the above definition are satisfied in all the examples which are considered, but in my opinion can be omitted in a formalization of the notion 'chaos': is the disjoint union of two chaotic systems not chaotic? is a system not chaotic because it does not have 'an element of regularity'? The definition as given above is a way to formalize the notion of 'chaotic dynamical system'. Other authors, e.g., see [1] - especially practitioners of the experimental sciences - take as a basic notion 'chaotic orbit' or even 'chaotic time series'. Especially, this last notion is basic: time series appear as the outcome of experiments and often are the only information which we have about the dynamics of a system. So it is often by the time series that we have to judge whether a dynamical system is chaotic or not. For time series, the notion of chaos is associated with continuous power spectra and autocorrelations which decrease to zero. If one tries to interpret these notions for dynamical systems one ends up in ergodic theory (both power spectra and autocorrelations are obtained by averaging). So the restriction which the author imposed on himself, namely not using measure-theoretic notions, makes it hard to relate his notion of chaotic dynamics to corresponding notions used elsewhere. Fractal. This notion is only described intuitively in the present volume (p. 37). The Cantor Middle-Thirds set is an example of a fractal. Intuitively, a fractal is a set which is self-similar under magnification. In the Cantor Middle-Thirds set, suppose we look only at those points which lie in the left-hand interval [0, 31]. Under a microscope which magnifies this interval by a factor of three, the "piece" of the Cantor set in [0, I] looks exactly like the original set. More precisely, the linear map L(x)= 3x maps the portion of the Cantor set in [0, 31] homeomorphically onto the entire set. See Exercise 10. This process does not stop at the first level: one may magnify any piece of the Cantor set at the nth stage of the construction by a factor of 3 n and obtain the original set. See Exercise 11. Still these fractal sets appear everywhere in dynamical systems and the possibility of investigating these sets and their scaling properties is one of the main reasons for studying complex analytic dynamical systems. The common definition of a fractal set is a set whose Hausdorff dimension is bigger than its topological dimension, see [3]. This is a completely rigorous definition (once the Hausdorff dimension and the topological dimension are defined), but it misses the point of self-similarity under magnification. Also, consulting other authors, I could not find a formalization of this last notion, and I think this is because no one succeeded in finding such a formalization. In order to see why, let us just try a formal definition and then see the consequences, t~onsider the following tentative definition. We say that a compact subset X c R n is self-similar under magnification if 224 BOOK REVIEW there is a smooth map q~: U---~R ", U a neighbourhood of X, such that - = x ; - q~ [X is a local homeomorphism; - for some constant C > 1 and all tangent vectors v of R n in points of X, IId (v)ll t> C . Iio11. Indeed, since q~"{X is also a local homeomorphism and since it is expanding (locally at least by a factor C"), we see that a small neighbourhood of x in X, magnified by the linear map (dq~")x, is equal, up to a deformation due to the nonlinear part of ~0", to a bigger neighbourhood of q~(x). On the other hand, one can argue that for a linear map L: R"--*R" the sets K and L(K) are only similar if L is conformal (i.e., preserves angles); in that case one would require the derivative dq~ in the above definition to be everywhere conformal. In any case, there are two situations where this last point does not matter: in the one-dimensional case (where any linear map is conformal) and in the case where q~ is a complex analytic endomorphism in one complex variable (because such mappings are conformal). In the one-dimensional case, the nontrivial hyperbolic invariant sets discussed in §I, 5 are such self-similar fractals. In the case of complex analytic endomorphisms, the Julia set is often self-similar in the above sense (namely if the derivative of ~ or of ~0 ~ is expanding everywhere on the Julia set). Here we see another problem: if a complex endomorphism q~ has a critical point in its Julia set, then (J(q~), q~) certainly does not satisfy the above definition. Still, if one looks at computer-generated pictures of such sets, they look self- similar, e.g., see Map 25 in [4] - how does one catch this in a formal definition? This is not all. The hyperbolic invariant sets of diffeomorphisms, as discussed in the second part of the book under review, are also often fractal in the sense that the Hausdorff dimension is bigger than the topological dimension (this is the case both for the horseshoe and for the Plykin attractors). Also, they seem to have self similarity but the above definition does not apply. If A is a compact hyperbolic invariant set for a diffeomorphism tp on R" (and if the periodic orbits are dense in A) then ~ is not expanding on A but A can locally be decomposed as a product Afo~ × A~ such that q~ is expanding in the A~oc direction and q~-i in the A~oc direction (see [5] Chapter III for more details). So here the expanding map in the first definition is replaced by a much more complicated map. So there is some intuitive notion of self-similarity which is applicable to many invariant sets of dynamical systems. But a closer examination shows that there may be different types of self similarity and that this may be related with the difficulty of formalizing this notion. Genericity. The author defines (p. 16) large sets (in the topological sense) as open and dense subsets after showing that this may be misleading, i.e., that there are open and dense subsets of R with arbitrarily small Lebesgue measure. This is not BOOK REVIEW 225 very convincing, since the intuitive notion of 'almost all' seems to correspond to 'full measure'. One might think (or hope) that in practice (in dynamical systems) one does not meet such pathologies as open and dense sets with small measure. A nice counter example to this hope is the devil's staircase which is discussed in §I, 14 (this is the graph of the function p(to) which is defined as the rotation number of the map x~- -~x+to+Es in27rxmod 1 for some fixed ~>0) , the set of points to c [0, 1] where p is locally constant is open and dense but the set of points where p is not locally constant has positive measure. This can be interpreted as: from the topological point of view there is always resonance, but from the measure theoretical point of view both resonance and nonresonance have positive prob- ability. These results have discredited the notion of genericity. It is by the way amusing to see how Devaney avoids the word generic: one does not find it in the index; on page 16 he defines a 'big set" as being open and dense without saying that the points of the big set are called generic points; in other places he describes nongeneric cases just as 'atypical' (p. 24) or generic cases as 'typical' (p. 82). He treats as main bifurcations the saddle-node, the period doubling, and the Hopf bifurcation without explaining that this is a quite canonical choice, since these are the only bifurcations of fixed points which occur in generic one-parameter families of diffeomorphisms. I think that although this notion of genericity conflicts in some cases with our intuition (or with measure theoretic notions), it is still useful as a means to distinguish the main cases (generic cases) from the more exceptional cases. Also, it should be noted that the genericity results proved by transversality, like the Kupka-Smale theorem and its generalizations, do not lead to contradictions with measure theoretic notions. Structural stability. The author defends this notion as extremely important in applications in the following way (p. 53). The notion of structural stability is extremely important in applications. Suppose our dynamical system is the solution of a differential equation or otherwise comes from a real world physical system. Ordinarily, the system itself will be only a model of real world phenomena: certain assumptions will have been made, and certain approximations and experimental errors will be present. Hence the dynamical system itself, albeit a completely accurate solution of the physical model, will nevertheless be only an approximation to reality since the model itself suffers this flaw. Now, if the dynamical system in question is not structurally stable, then the small errors and approximations made in the model have a chance of dramatically changing the structure of the real solution to the system. That is, our 'solution' could be radically wrong or unstable. If, on the other hand, the dynamical system in question is structurally stable, then the small errors introduced by approximations and experimental errors may not matter at all: the solution to the model system may be equivalent or topologically conjugate to the actual solution. This does not mean that the only interesting physical systems are the structurally stable ones. Indeed, most dynamical systems that arise in classical mechanics are not structurally stable. There are also simple examples of systems such as the Lorenz system from meteorology that are 'far' from being structurally stable. These systems cannot even be approximated in a sense to be made precise below by stable systems. Nevertheless, the concept of structural stability is an important one in applications of the theory of dynamical systems. 226 BOOK REVIEW This is, or was around 1960, the usual point of view. One may add in view of later results the following points. The notion of structural stability is based on topological conjugacy as equivalence relation. Since it became known that there are diffeomorphisms which cannot be C r approximated by C r structurally stable diffeomorphisms, it was felt that topological conjugacy might be too strong (i.e., giving too small equivalence classes) as equivalence relation. One can say the same about O-stability and l)-conjugacy. For this reason the interest has shifted from structural stability to persistence: although a system may not be structurally stable, it may have certain properties which persist if we perturb the system slightly, like the existence of transverse homoclinic or heteroclinic orbits. Another point is the relation between perturbations of a ditteomorphism q~ and errors due to the fact that t¢ is an imprecise mathematical model for the evolution of a concrete system. This is more complicated than suggested in the above quotation. The diffeomorphism q~: M--* M can be imprecise in a number of ways: the 'real model', or at least a model which gives better predictions, might be (a) a map if: M---~ M which is close to q~ (this is probably what the author had in mind); (b) a map ~: ~7/___~/~/together with a projection 7r: ~7/___~ M such that TrY(x) and q~Tr(x) are close for 'all' x c 1~/; (c) a map which assigns to each x e M a probability distribution on M which is essentially concentrated in a small neighbourhood of q~(x). And there are probably more possibilities. Restricting to nonprobabilistic models, I think type (b) is very common - for example, chemical reactions: points of M describe the state not taking into account the spacial variations of the concen- trations of the different chemicals, points of M describing the state including spatial variations. In the situation (b), one has to compare projections of ~b-orbits with q~-orbits. This has nothing to do with structural stability, bur rather with the shadowing of pseudo-orbits as defined by Bowen [2]. Finally, let me again point out that the above discussions are not meant as criticism of the mathematical exposition which I really like. They are mainly intended to discuss the interpretation of the mathematical concepts and results involved and to provoke a further discussion on these points. References 1. Berg6, P., Pomeau, Y. and Vidal, Ch.: L'ordre dans le chaos, Hermann, Paris, 1985. 2. Bowen, R.: On Axiom A diffeomorphisms, Amer. Math. Soc., 1978. 3. Mandelbrot, B.: The Fractal Geometry o[Nature, Freeman, San Francisco, 1982. 4. Peitgen, H. and Richter, P.: The Beauty o[Fractals, Springer, New York, 1986. 5. Shub, M.: Stabilit6 global des syst6mes dynamiques, Ast~risque 56 (1978). Groningen The Netherlands FLORIS TAKENS