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Generic Properties of Dynamical Systems Christian Bonatti Generic Properties of Dynamical Systems Christian Bonatti To cite this version: Christian Bonatti. Generic Properties of Dynamical Systems. Encyclopedia of Mathematical Physics, Academic Press, 2006, pp. 494-502. 10.1016/B0-12-512666-2/00164-4. hal-01348445 HAL Id: hal-01348445 https://hal.archives-ouvertes.fr/hal-01348445 Submitted on 27 Jul 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Generic Properties of Dynamical Systems represented as a point x in a geometrical space Rn. C Bonatti, Université de Bourgogne, Dijon, France In many cases, the quantities describing the state are related, so that the phase space (space of all possible states) is a submanifold M Rn. The time evolution R Introduction of the system is represented by a curve xt, t 2 drawn on the phase space M, or by a sequence xn 2 The state of a concrete system (from physics, M, n 2 Z, if we consider discrete time (i.e., every chemistry, ecology, or other sciences) is described day at the same time, or every January 1st). using (finitely many, say n) observable quantities Believing in determinism, and if the system is (e.g., positions and velocities for mechanical isolated from external influences, the state x0 of the systems, population densities for echological system at the present time determines its evolution. systems, etc.). Hence, the state of a system may be For continuous-time systems, the infinitesimal 1 evolution is given by a differential equation or vector generic properties, generic diffeomorphisms verify field dx=dt = X(x); the vector X(x) represents velo- simultanuously all the properties Pi. city and direction of the evolution. For a discrete-time A property P is C r-robust if the set of diffeo- system, the evolution rule is a function F : M ! M;if morphisms verifying P is open in Diff r(M). A x is the state at time t,thenF(x) is the state at the property P is locally generic if there is an (nonempty) time t þ 1. The evolution of the system, starting at open set O on which it is generic, that is, there is the initial data x0, is described by the orbit of x0,that residual set R such that P is verified on R\O. is, the sequence {(xn)n2Z j xnþ1 = F(xn)} (discrete The properties of generic dynamical systems time) or the maximal solution xt of the differential depend mostly on the dimension of the manifold M equation ax=dt = X(x) (continuous time). and of the C r-topology considered, r 2 N [ {þ1} (an important problem is that Cr-generic diffeo- General problem Knowing the initial data and the morphisms are not C rþ1 ): infinitesimal evolution rule, what can we tell about the long-time evolution of the system? On very low dimensional spaces (diffeomorphisms of the circle and vector fields on compact surfaces) the The dynamics of a dynamical system (differential dynamics of generic systems (indeed in a open and equation or function) is the behavior of the orbits, dense subset of systems) is very simple (called Morse– when the time tends to infinity. The aim of Smale) and well understood; see the subsection ‘‘dynamical systems’’ is to produce a general ‘‘Genericpropertiesofthelow-dimensional procedure for describing the dynamics of any systems.’’ system. For example, Conley’s theory presented in Inhigherdimensions,for Cr-topo logy, r>1,one the next section organizes the global dymamics of a has generic and locally generic properties related general system using regions concentrating the orbit to the periodic orbits, like the Kupka–Smale accumulation and recurrence and splits these regions property(seethesubsection‘‘Kupka–Smaletheo- in elementary pieces: the chain recurrence classes. rem’’)andtheNewhousephenomenon(seethe We focus our study on C r-diffeomorphisms F (i.e., F subsection‘‘Local 2C-genericityofwildbehavior and FÀ1 are r times continuously derivable) on a forsurfacediffeomorphisms’’).However,westill compact smooth manifold M (most of the notions and do not know if the dynamics of Cr-generic results presented here also hold for vector fields). Even diffeomorphisms is well approached by their for very regular systems (F algebraic) of a low- periodic orbits, so that one is still far from a dimensional space ( dim (M) = 2), the dynamics may global understanding of Cr-generic dynamics. be chaotic and very unstable: one cannot hope for a For the C1-topology, perturbation lemmas show that precise description of all systems. Furthermore, neither the global dynamics is very well approximated by the initial data of a concrete system nor the infinitesi- periodicorbits(seethesection‘‘C1-genericsystems: mal-evolution rule are known exactly: fragile proper- globaldynamicsandperiodicorbits’’).Onethen ties describe the evolution of the theoretical model, and dividesgenericsystemsin‘‘tame’’systems,witha not of the real system. For these reasons, we are mostly global dynamics analoguous to hyperbolic dynamics, interested in properties that are persistent, in some and ‘‘wild’’ systems, which present infinitely many sense, by small perturbations of the dynamical system. dynamically independent regions. The notion of The notion of small perturbations of the system dominatedsplitting(seethesection‘‘Hyperbolic requires a topology on the space Diffr(M)ofC r- propertiesof 1C-genericdiffeormorphisms’’)seems diffeomorphisms: two diffeomorphisms are close for to play an important role in this division. the C r-topology if all their partial derivatives of order r are close at each point of M. Endowed with this topology, Diff r(M) is a complete metric space. The open and dense subsets of Diffr(M)providethe Results on General Systems natural topological notion of ‘‘almost all’’ F. Genericity Notions of Recurrence is a weaker notion: by Baire’s theorem, if OTi, i 2 N,are Some regions of M are considered as the heart of the dense and open subsets, the intersection i2N Oi is a dense subset. A subset is called residual if it contains dynamics: such a countable intersection of dense open subsets. A Per(F) denotes the set of periodic points x 2 M of property P is generic if it is verified on a residual F, that is, Fn(x) = x for some n > 0. subset. By a practical abuse of language, one says: A point x is recurrent if its orbit comes back ‘‘C r-generic diffeomorphisms verify P’’ arbitrarily close to x, infinitely many times. Rec(F) denotes the set of recurrent points. A countable intersection of residual sets is a residual The limit set Lim(F) is the union of all the set. Hence, if {Pi}, i 2 N, is a countable family of accumulation points of all the orbits of F. 2 A point x is ‘‘wandering’’ if it admits a neighbor- with an associated Lyapunov function. Conley hood Ux M disjoint from all its iterates (1978) proved that n \ F (Ux), n > 0. The nonwandering set (F) is the set of the nonwandering points. RðFÞ¼ ðAi [ RiÞ R(F) is the set of chain recurrent points, that is, i2N points x 2 M which look like periodic points if we This induces a natural partition of R(F)inequiva- allow small mistakes at each iteration: for any lence classes: x y if x 2 Ai , y 2 Ai. Conley proved ">0, there is a sequence x = x0, x1, ..., xk = x that x y iff, for any ">0, there are "-pseudo orbits where d(f (xi), xiþ1) <" (such a sequence is an from x to y and vice versa. The equivalence classes "-pseudo-orbit). for are called chain recurrence classes. Now, considering an average of the Lyapunov A periodic point is recurrent, a recurrent point is a functions one gets the following result: there is a limit point, a limit point is nonwandering, and a i continuous function ’: M ! R with the following nonwandering point is chain recurrent: properties: PerðFÞRecðFÞLimðFÞ ðFÞRðFÞ ’(F(x)) ’(x) for every x 2 M, (i.e., ’ is a All these sets are invariant under F,and (F) and Lyapunov function); R(F) are compact subsets of M. There are diffeo- ’(F(x)) = ’(x) , x 2R(F); morphisms F for which the closures of these sets are for x, y 2R(F), ’(x) = ’(y) , x y;and distinct: the image ’(R(F) is a compact subset of R with empty interior. A rotation x 7! x þ with irrational angle 2 RnQ on the circle S1 = R=Z has no periodic This result is called the ‘‘fundamental theorem of points but every point is recurrent. dynamical systems’’ by several authors (see The map x 7! x þ (1=4)(1 þ cos (2x)) induces Robinson (1999)). on the circle S1 a diffeomorphism F having a Any orbit is ’-decreasing from a chain recurrence unique fixed point at x = 1=2; one verifies that class to another chain reccurence class (the global (F) = {1=2} and R(F) is the whole circle S1. dynamics of F looks like the dynamics of the gradient flow of a function , the chain recurrence An invariant compact set K M is transitive if there classes supplying the singularities of ). However, is x 2 K whose forward orbit is dense in K. Generic this description of the dynamics may be very rough: points x 2 K have their forward and backward if F preserves the volume, Poincare´’s recurrence orbits dense in K: in this sense, transitive sets are theorem implies that (F) = R(F) = M; the whole M dynamically indecomposable.
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