ebook img

Defining Chaos PDF

0.66 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Defining Chaos

Defining Chaos Brian R. Hunt1 and Edward Ott1 University of Maryland, College Park MD 20742, USA (Dated: 10 April 2015) In this paper we propose, discuss and illustrate a computationally feasible definition of chaos which can be applied very generally to situations that are commonly encountered, including attractors, repellers and non- periodically forced systems. This definition is based on an entropy-like quantity, which we call “expansion entropy”, and we define chaos as occurring when this quantity is positive. We relate and compare expansion entropy to the well-known concept of topological entropy, to which it is equivalent under appropriate con- ditions. We also present example illustrations, discuss computational implementations, and point out issues arising from attempts at giving definitions of chaos that are not entropy-based. 5 Toward the end of the 19th century, Poincar´e was concerned with weather forecasting and accordingly 1 0 demonstrated the occurrence of extremely com- focusedon thechaosattribute oftemporally exponential 2 plicated orbits in the Newtonian dynamics of increaseofthesensitivityoforbitlocationstosmallinitial threegravitationallyattractingbodies. Thiscom- perturbations. As we will discuss, these two attributes r p plexityisnowcalledchaosandhasreceivedavast can be viewed as “two sides of the same coin”. A amount of attention since Poincar´e’s early dis- We think of a definition of chaos as being “good” if it covery. In spite of this abundant past and cur- conforms to common intuitive notions of chaos (such as 8 rent work, there is still no broadly applicable, complexorbitstructureandorbitsensitivity)and,atthe 2 convenient, generally accepted definition of the same time, has the following three desirable features: ] term chaos. In this paper, we advocate a par- D ticular entropy-based definition that appears to • Generality: The definition should work for almost C be very simple, while, at the same time, is read- alltheexamplesthattypicalreadersofthisjournal . ily accessible to numerical computation, and can are likely to judge as chaotic. n be very generally applied to a variety of often- i • Simplicity: The definition should be fairly concise l encountered situations, including attractors, re- n and not too technical. pellers, and non-periodically forced systems. We [ also review and compare various previous defini- • Computability: The definition should allow a prac- 3 tions of chaos. tical, straightforward computational implementa- v tionfordiscerningtheexistenceofchaosinamodel. 6 9 Considering the issue of generality, one would like a 8 7 definition of chaos to be applicable not only to attrac- I. INITIAL DISCUSSION 0 tors, but also to non-attracting sets, often called re- . pellers. With respect to chaotic repellers3–5, we note 1 While the word chaos is widely used in science and that they are central to the physically relevant topics of 0 5 mathematics, there are a variety of ways of defining fractalbasinboundaries6,chaotictransients,andchaotic 1 it. Thus, for this 25th anniversary issue of the journal scattering7, occurring, for example in fluid dynamics8,9, : CHAOS, we are motivated to review issues that arise celestialmechanics10,chemistry11,andatomicphysics12. v i whenattemptingtoformulateagenerallyapplicabledef- Furthermore, again considering the issue of general- X inition of chaos, and to advocate a particular entropy- ity, due to their common occurrence in applications, r based definition that seems to us to be especially apt. we desire that our definition of chaos be applicable to a Wealsorelateourproposeddefinitiontopreviousdefini- non-autonomous dynamical systems (i.e., systems that tions. are externally forced by time dependent inputs), includ- Intuitively, perhaps the two most prominent (not nec- ing external inputs that are temporally quasi-periodic13, essarily independent) attributes of what scientists com- stochastic14–16, or are themselves chaotic. Here physical monly think of as chaos are the presence of complex or- examples include quasi-periodic forcing of atmospheric bit structure and extreme sensitivity of orbits to small jets17, quasi-periodic forcing of stellar luminosity vari- perturbations. Indeed, in the paper by Li and Yorke1 ation by two superposed stellar modal oscillations18,19, where the term chaos was introduced in its now widely advective transport in fluids with temporally and spa- accepted nonlinear dynamics context, the term was mo- tiallyirregularflowfields20–24,andphasesynchronismof tivatedbythesimultaneouspresenceofunstableperiodic chaos by noisy or chaotic drives25. We emphasize that, orbitsofallperiods, aswellasanuncountableinfinityof whenconsideringexternallyforcedsystems,weareinter- non-periodic orbits. Thus, Li and Yorke’s introduction ested in identifying chaos in the response of the system ofthisterminologywasmotivatedbythechaosattribute to a particular realization of the forcing, not in charac- of complex orbit structure. On the other hand, Lorenz2 terizing whether the forcing is chaotic. 2 An important point for consideration of non- II. EXPANSION ENTROPY periodically forced chaotic systems is that the notion of a compact invariant set, which is typically used in A. Definition definitions of chaos for autonomous systems (including Poincar´e maps of periodically forced systems), may not Our definition of expansion entropy, which we denote be appropriate or convenient for situations with non- H , is closely related to previous definitions of topolog- periodic forcing. Furthermore, in practice, it may be 0 ical entropy, to which it is equivalent under appropri- difficulttolocateordetectaninvariantsetthatisnotan ate conditions (see Sec. IV). Expansion entropy uses the attractor. Thus,ratherthandefiningchaosforaninvari- linearization of the dynamical system and a notion of antset, wewillinsteadconsideranotionofchaosforthe volume on its state space; thus, unlike topological en- dynamicswithinanygivenboundedpositive-volumesub- tropy, it is defined only for smooth dynamical systems. setS ofthestatespace. WecallsuchasetS arestraining On the other hand, expansion entropy does not require region. For autonomous systems, chaos for an invariant the identification of a compact invariant set. As we will set can be detected by taking S to be a neighborhood of discuss, the differences in the definitions may make the the desired invariant set. criterionH >0attractiveasageneraldefinitionofchaos 0 in smooth dynamical systems. We assume that the state space of the dynamical sys- tem is a finite-dimensional manifold M. (For example, In our opinion, the currently most satisfactory way of if the manifold M is n-dimensional Euclidean space, the defining chaos for autonomous systems is by the exis- state x at a given time is a vector [x ,x ,...,x ] where tence of positive topological entropy or metric entropy. 1 2 n each x is a real number. If some of the coordinates are We note, however, that the standard definitions of these i angle variables, they can be taken modulo 2π, result- entropies are quite difficult to straightforwardly imple- ing in manifolds such as a circle, cylinder or torus.) We ment in a numerical procedure. In addition, while gen- writeµ(S)forthevolume28 ofasubsetS ofM,andwrite eralizations to the original definitions of topological and dµ(x) for integration with respect to this volume; for n- metric entropy have been proposed, we view it as desir- dimensional Euclidean space, one can take dµ(x)=dnx. able to have a relatively simple definition that is appli- Given a dynamical system on M, we will use an integral cable very broadly. However, we do not address here the to define its expansion entropy on a closed subset S (the questionofidentifyingchaosinexperimentaldata,which restraining region) that has positive, finite volume. The presents additional challenges, especially in the cases of set S need not be invariant under the system. non-attracting sets and externally forced systems. We consider a deterministic dynamical system to be defined by an evolution operator, by which we mean a family f of maps f :M →M, with the interpretation t(cid:48),t Motivated by the considerations above, in Sec. II we that if x and x(cid:48) are the states of the system at times t introduce and discuss the definition of an alternate en- andt(cid:48),respectively,thenx(cid:48) =ft(cid:48),t(x). (Forexample,ft(cid:48),t tropy quantity that we call “expansion entropy”. The couldrepresentthesolutionfromtimettot(cid:48) ofasystem expansion entropy of an n-dimensional dynamical sys- of differential equations.) The family f must satisfy the tem on a restraining region S is the difference between identities ft,t(x) = x and ft(cid:48)(cid:48),t(x) = ft(cid:48)(cid:48),t(cid:48)(ft(cid:48),t(x)). The two asymptotic exponential rates: first, the maximum maps ft(cid:48),t are defined for t(cid:48),t being integer-valued (dis- over d ≤ n of the rate at which the system expands d- cretetime)orbeingrealnumbers(continuoustime),with dimensional volume within S; and second, the rate at the restriction that t(cid:48) ≥ t if the system is noninvertible. which n-dimensional volume leaves S (this rate is 0 for Weallowthesystemtobenonautonomous,includingthe an invariant set). We define chaos as the existence of case where ft(cid:48),t is a realization of a stochastic dynamical positive expansion entropy on a given restraining region. system.29 Ifthesystemisautonomous,thenft(cid:48),t depends Expansion entropy generalizes (to nonautonomous sys- onlyont(cid:48)−t,andinthiscasewewilloftenwriteft(cid:48),t =fT tems and noninvariant restraining regions) a quantity where T = t(cid:48) −t. Regardless, we assume that ft(cid:48),t is a that was formulated by Sacksteder and Shub26 in the differentiable function of x. caseofanautonomoussystemonacompactmanifold. In Recall that the singular values of a matrix A are the this restricted case, by the results of Kozlovski27, expan- squarerootsoftheeigenvaluesofA(cid:62)A. ThinkingofAas sion entropy is equal to topological entropy for infinitely a linear transformation, the image of the unit ball under differentiable maps. In Sec. III we present examples of A is an ellipsoid, and the singular values of A are the the application of our definition of expansion entropy to semiaxes of this ellipsoid. Let G(A) be the product of various systems, and also provide illustrative numerical the singular values of A that are greater than 1; if none evaluationsofexpansionentropyforsomeoftheseexam- of the singular values are greater than 1, let G(A) = 1. ples. Section IV discusses topological entropy and pre- Then G(A) is roughly the number of (cid:15)-balls needed to vious work on computation of this quantity. Section V cover the image of an (cid:15)-ball under A. discusses issues that arise in previous non-entropy-based IfthematrixAisn×n,considerad-dimensionalplane definitions of chaos. P in n-dimensional Euclidean space, where d ≤ n. Let d 3 W be a d-dimensional ball in P , let A(W) denote the It can be shown that the exponential growth rate of d image of W under A, and let µ denote d-dimensional G(Df (x)) as t(cid:48) → ∞ is the sum of the positive Lya- d t(cid:48),t volume. Then G(A) is the maximum over orientations punov exponents of the trajectory starting at x at time of P and over d of µ (A(W))/µ (W). Thus, G(A) is t. Thus, the limit in Eq. (3) is, in a sense, an average of d d d thelargestpossiblegrowthratioofd-dimensionalvolumes this sum of positive Lyapunov exponents over trajecto- under A. BelowwewillapplyGtothederivativematrix riesthatremaininS forall(forward)time. Thecriterion Df , in which case it represents a local volume growth H > 0, which we propose for defining chaos, requires t(cid:48),t 0 ratio for the (typically nonlinear) map f . that this exponential volume growth rate strictly exceed t(cid:48),t Let S be the set of x such that f (x) ∈ S for all the exponential rate 1/τ at which trajectories leave S. t(cid:48),t t(cid:48)(cid:48),t + t(cid:48)(cid:48) between t and t(cid:48) (that is, the trajectory of x from t to Note that if S is forward invariant, e.g., an absorbing t(cid:48) under f never leaves S). Let neighborhood of an attractor, then 1/τ =0. + Some points of interest for this entropy definition are 1 (cid:90) that E (f,S)= G(Df (x))dµ(x). (1) t(cid:48),t µ(S) t(cid:48),t St(cid:48),t (i) it applies to non-autonomous systems, Definition of expansion entropy: We define the expansion entropy H0 to be (ii) it assigns an entropy value H0 to every restraining region S in the manifold M, and lnE (f,S) H (f,S)= lim t(cid:48),t . (2) 0 t(cid:48)→∞ t(cid:48)−t (iii) it directly suggests a computational technique for numerically estimating H (see Sec. IIC). 0 WeconsiderH andotherlimitingquantitiesbelowtobe 0 well-definedonlyifthelimitinvolvedexists.30Weremark Finally, notice that that if the system f is nonautonomous and the restrain- ing region S is not invariant, H0(f,S) could potentially H0(f,S(cid:48))≤H0(f,S) if S(cid:48) ⊂S. (5) depend on the starting time t in addition to f and S. Also, it can be shown that the value of H0 is invariant This property follows from the fact that St(cid:48)(cid:48),t ⊂ St(cid:48),t under differentiable changes of coordinates that are non- if S(cid:48) ⊂ S, and consequently E (f,S(cid:48)) ≤ E (f,S). t(cid:48),t t(cid:48),t singular on S. Thus,ifthereischaosaccordingtothedefinitionH >0 0 To help interpret the definition of H0(f,S), we now with respect to a restraining region S(cid:48), then there is also replace1/µ(S)with[1/µ(St(cid:48),t)][µ(St(cid:48),t)/µ(S)]inEq.(1). chaoswithrespecttoeveryrestrainingregionS thatcon- The definition (2) can then be expressed as tains S(cid:48). In particular, as illustrated by the example in Sec.IIIB,thisimpliesthattheexpansionentropywillde- H (f,S)= lim lnE˜t(cid:48),t(f,S) − 1 , (3) tect chaos (H0 > 0) within a restraining region S when 0 t(cid:48)→∞ t(cid:48)−t τ+ S also contains a nonchaotic attractor, even when the chaos exists only on a repeller. where 1 ln[µ(S)/µ(S )] = lim t(cid:48),t (4) B. Expansion Entropy of the Inverse System τ+ t(cid:48)→∞ t(cid:48)−t and In this section, we show that the expansion entropy of an autonomous, invertible system is the same as the ex- 1 (cid:90) E˜ (f,S)= G(Df (x))dµ(x). pansion entropy of the inverse system. (Note that this is t(cid:48),t µ(St(cid:48),t) St(cid:48),t t(cid:48),t also true for the topological entropy; see Sec. IV.) This equality results from the following identity for all invert- Thus,wecanviewH (f,S)asthedifferenceoftwoexpo- ible systems (not necessarily autonomous): 0 nentialrates,givenbythelimitsinEqs.(3)and(4),with thefollowinginterpretations. ImaginethatN initialcon- E (f,S)=E (f,S). t,t(cid:48) t(cid:48),t ditionsareuniformlysprinkledthroughoutthevolumeof S attimet,andthatN isverylarge(N →∞). Thesec- To verify this identity, notice that f is the inverse of t,t(cid:48) ond term 1/τ in Eq. (3) is then the exponential decay f . Below we use the notation x(cid:48) =f (x), and conse- + t(cid:48),t t(cid:48),t rate, as t(cid:48) increases, of the number of trajectories from quently x = f (x(cid:48)). Then Df (x(cid:48)) and Df (x) are t,t(cid:48) t,t(cid:48) t(cid:48),t these initial conditions that remain in S for all times be- inverses, and hence the singular values of Df (x(cid:48)) are t,t(cid:48) tween t and t(cid:48). The quantity E˜ (f,S) is the average the inverses of the singular values of Df (x). Since t(cid:48),t t(cid:48),t over these remaining trajectories of the maximum local the product of the singular values of a square ma- d-dimensional volume growth ratio along the trajectory. trix is the absolute value of its determinant, if A is Thus, the first term in Eq. (3) is the exponential growth invertible then |detA| = G(A)/G(A−1). In particu- rate of this average. lar, |detDf (x)| = G(Df (x))/G(Df (x(cid:48))). Also, t(cid:48),t t(cid:48),t t,t(cid:48) 4 f (S )= S . Writing E (f,S) as an integral over theselogarithms. Notonlydoesthisallowustoestimate t(cid:48),t t(cid:48),t t,t(cid:48) t,t(cid:48) x(cid:48) and then making the change of variables x(cid:48) =f (x), thesamplingerror,italsoproducesamorereliablemean t(cid:48),t estimate than computing lnEˆ (f,S) for a single sample 1 (cid:90) T E (f,S)= G(Df (x(cid:48)))dµ(x(cid:48)) of 100N points. Example illustrations of this computa- t,t(cid:48) µ(S) t,t(cid:48) tional approach are given in Secs. IIIB, IIIC, and IIIE. St,t(cid:48) 1 (cid:90) Specializing to the case of an autonomous invertible = G(Df (x(cid:48)))|detDf (x)|dµ(x) µ(S) t,t(cid:48) t(cid:48),t system f, since Sec. IIB shows that the expansion en- 1 (cid:90)St(cid:48),t tropy of f and f−1 are the same, one could do a nu- = G(Df (x))dµ(x)=E (f,S). merical computation of H using either f or f−1. The µ(S) t(cid:48),t t(cid:48),t 0 St(cid:48),t questionthenarisesastowhichofthesetwoalternatives ispreferablefromthepointofviewofcomputationalcost Iff isautonomousandinvertible,wewritef =f t(cid:48),t t(cid:48)−t and accuracy. In the remainder of this section, we argue and f−1 = f . Then T −T thatitiscomputationallypreferabletocalculateH from 0 f if f is volume contracting in S, while calculation from H (f−1,S)= lim ln[E (f−1,S)]/T 0 T→∞ T,0 f−1 is preferable if f is volume expanding in S. In order = lim ln[E (f,S)]/T to see this, we generalize Eq. (4) to define both forward −T,0 T→∞ and backward exponential decay rates = lim ln[E (f,S)]/T 0,−T T→∞ = lim ln[E (f,S)]/T =H (f,S) 1 1 T,0 0 T→∞ τ± =Tl→im∞T ln[µ(S)/µ(S±T,0)] (7) Herethefirstequalityisbydefinition,thesecondequality is a change of notation, the third equality follows from whereS is,asinSec.IIA,thesetofinitialconditions the time-reversal identity for E derived above, and the ±T,0 attime0whosetrajectoriesunderf remaininS between fourth equality uses the fact that f is autonomous. times 0 and ±T, respectively. That is, in terms of the previouslystatednumericalprocedureforcalculatingthe expansion entropy, 1/τ is the exponential temporal de- C. Discussion of Numerical Evaluation of Expansion + cayrateofthenumberofinitialconditionssprinkleduni- Entropy formly throughout S at time 0 that lead to orbits that never leave S up to time T, while 1/τ is the analogous With respect to point (iii) in Sec. IIA, we can imag- − quantity taking the initially sprinkled points backward ine a computation of H proceeding as follows. First, 0 from time 0 to time −T. Since, to estimate the inte- randomly sprinkle a large number of initial conditions gral in Eq. (1), we need to compute the average expan- {x ,x ,...,x } uniformly in S. Then evolve each tra- 1 2 N sionratesonlyfromthoseinitialconditionsthathavenot jectory f (x ) and the corresponding tangent map T,0 i left S, statistics at any given T are improved when the Df (x ) forward in time, continuing to evolve only as T,0 i number of such orbits is largest. Further, the estimate long as the trajectory remains in S. At a discrete se- of the limit T → +∞ dictates that we make T large. quence of times T, compute These two considerations indicate that the forward (re- spectively, backward)calculationofH willbecomputa- N 0 EˆT(f,S)=N−1(cid:88)(cid:48) G(DfT,0(xi)), (6) tionallymoreefficientifτ+ >τ− (respectively,τ− >τ+). Subtractingthedefinition(7)of1/τ fromthedefinition i=1 + of 1/τ , and using the fact that S = f (S ) for − −T,0 T,0 T,0 where the prime on the summation symbol signifies that an autonomous invertible system, we obtain only those i values for which f (x ) remains in S up T,0 i to time t are included in the sum. From our definition oEf E(fin,SE)q..P(l1o)t,tiwnge lsneeEˆth(aft,SEˆ)Tv(efr,sSu)s Tis,afonressutffiimcaietnetolyf (1/τ+)−(1/τ−)=Tl→im∞T1 ln{µ(fT,0(ST,0))/µ(ST,0)}. T,0 T largeN andT,weexpecttofindanapproximatelylinear relationship. Accordingly, we can estimate H as the The right hand side is positive (respectively, negative) 0 slopeofastraightlinefittedtosuchdata(seealsoJacobs when the map is volume expanding (respectively, con- et al.31 for a similar approach in two dimensions). tracting) in S. Thus, τ > τ if the map is volume ex- − + Asinothersuchprocedures,judgmentandexperimen- panding,whileτ >τ ifthemapisvolumecontracting. + − tation are called for in determining reliable choices of N In particular, if S is a neighborhood of an attractor, it is and the range of T over which to do the fit, and such best to employ a forward time calculation. We note that choices will be constrained by computer resources. In the common examples ofthe H´enon map and the Lorenz practice,wefinditusefultochooseanumber,say100,of system are uniformly volume contracting at all points in differentsamplesofsizeN, computelnEˆ (f,S)foreach state space (implying that τ+ >τ−), while Hamiltonian T sample, and take the mean and standard deviations of systems are volume preserving (implying that τ+ =τ−). 5 D. Generalization to q-order expansion entropy 1.5 In past work on fractal dimension, the box-counting 1 dimension has been generalized to a spectrum of di- mensions often denoted D , where the box-counting di- q mension corresponds to q = 0, and the index q can 0.5 be any nonnegative number32–34. In addition, a spec- ) trum of entropy-like quantities, again depending on an x ( index q ≥ 0, has been introduced by Grassberger and f 0 Procaccia34,35, where q =0 corresponds to the topologi- calentropy,andq =1correspondstothemetricentropy. Thus, motivated by these past works, it is natural to in- -0.5 troduce an analogous spectrum of q-order expansion en- tropies,H ,andtoconsiderwhethertheyareusefulwith q respect to the issue of defining chaos. -1 InAppendixA,weintroduceanddiscussanaturalway -1 -0.5 0 0.5 1 1.5 of specifying Hq. In particular, the form defining Hq is x specified so that it gives Eqs. (1) and (2) when q = 0, gives an expansion entropy analogue of the entropy of FIG.1. AmapillustratingacasewhereS =[−1,1.5]contains Grassberger and Procaccia34,35, and also gives a corre- achaoticrepeller(inS(cid:48) =[0,1])andanattractingfixedpoint spondence for q =1 with previous results for the metric (x=−1/2). entropyofrepellers4,36 andwithPesin’sformula37 forthe metric entropy for attractors. However, as we will argue in Appendix A, q = 0 is special in regard to defining Once again, H (f,S)=0. 0 chaos. In particular, Appendix A will consider H for For fixed points (or periodic orbits) of higher- q anexampleinwhichS containsanattractingfixedpoint dimensional systems, similar arguments can be made, and a chaotic repeller (see Sec. IIIB). For this example, though they are more complicated in the case when the it is shown that H >0, while H for q >0 can be zero. fixed point has both stable and unstable directions. The 0 q Thus,H successfullydetectsthechaoswithinS,butH essenceofthesecalculationsisthatanygrowthinthein- 0 q for q >0 may not. tegrandGofEq.(1)ast(cid:48)−tincreasesisbalanced(upto a time-independent multiplicative constant) by a reduc- tion in the volume of S . The conclusion remains that t(cid:48),t III. ILLUSTRATIVE EXAMPLES H =0, i.e., that isolated period orbits are not chaotic. 0 A. Attracting and repelling fixed points B. Example: A one-dimensional map with a chaotic repeller and an attracting fixed point Consider a one-dimensional differentiable map f with afixedpointx ;forsimplicity,weassumethatDf(x )(cid:54)= 0 0 ±1. LettherestrainingregionSbeanintervalcontaining Consider the one-dimensional map f shown in Fig. 1 x on which |Df(x)| =(cid:54) 1; that is, f is either uniformly and the two restraining regions S and S(cid:48) ∈ S, where we 0 expanding or uniformly contracting on S. In either case, takeS =[−1,1.5]andS(cid:48) =[0,1]. Themapislinearwith we show below that the expansion entropy H (f,S) is derivative 3 on [0,1/3] and linear with derivative −2 on 0 zero, i.e., that fixed points are not chaotic according to [1/2,1],mappingeachoftheseintervalsto[0,1]. Forthis our definition. example, the fixed point x = −1/2 attracts almost all Inthecaseofanattractingfixedpoint,|Df (x)|<1, initialconditionswithrespecttoLebesguemeasureinS. t(cid:48),t and hence G(Df (x)) = 1, for all x ∈ S and t(cid:48) > t. On the other hand, S(cid:48) contains an invariant Cantor t(cid:48),t Also, S = S for t(cid:48) >t. Then from Eq. (1) and (2) we set that is commonly called a chaotic repeller. We now t(cid:48),t have E (f,S)=µ(S) for t(cid:48) >t and H (f,S)=0. show that according to our definition, f exhibits chaos t(cid:48),t 0 In the expanding case, |Dft(cid:48),t(x)| > 1 for trajectories by having positive expansion entropy H0 in both S and that remain in S from time t to t(cid:48). Also, S is a subin- S(cid:48). The invariant Cantor set consists of all initial con- t(cid:48),t terval of S whose endpoints map to the endpoints of S ditions in [0,1] whose trajectories never land in the in- under f . Thus, terval (1/3,1/2), i.e., those whose base 6 expansion does t(cid:48),t not contain the digit 2. The set S(cid:48) of initial condi- 1 (cid:90) tions whose trajectories remain in [0T,,01] from time 0 to E (f,S)= |Df (x)|dx t(cid:48),t µ(S) t(cid:48),t time T consists of 2T intervals corresponding to all pos- St(cid:48),t (cid:12) (cid:12) sible strings of length T of the letters L and R, where 1 (cid:12)(cid:90) (cid:12) = (cid:12) Df (x)dx(cid:12)=1. L denotes an iteration in the interval [0,1/3] and R de- µ(S)(cid:12)(cid:12) St(cid:48),t t(cid:48),t (cid:12)(cid:12) notes an iteration in the interval [1/2,1]. A string with 6 k L’s and T −k R’s corresponds to an interval of length 3−k2k−T, on which DfT = 3k(−2)T−k. Then the inte- N =1000 gralofG(DfT)=|DfT|oneachsuchintervalis1. Thus, 30 E (f,S(cid:48)) = 2T, and hence H (f,S(cid:48)) = ln2. In accor- T,0 0 dancewithproperty(5),sinceScontainsS(cid:48),wealsohave H0(f,S)=ln2. )S020 In Appendix A, in addition to defining the quantity ; f ( Hq discussed in Sec. IID, we also evaluate Hq for the ^ET map in Fig. 1. We find for the smaller restraining re- 10 n gion S(cid:48) that H (f,S(cid:48)) > 0 for all q ≥ 0, but for the l q larger restraining region S there is a critical value q <1 c such that H (f,S) = 0 for q ≥ q . For q = 1, we 0 q c have H (f,S(cid:48))=(2/5)ln(5/2)+(3/5)ln(5/3)>0, while 0 10 20 30 40 50 60 1 T H (f,S)=0. OurinterpretationisthatH (f,S)isdom- 1 1 inated by the dynamics of Lebesgue almost every initial N =100000 conditionwhosetrajectoryapproachesthefixedpointat- 50 tractor, while H (f,S) is dominated by the chaotic sad- 0 dle in S(cid:48). We therefore conclude that H (and similarly 1 H for q > 0) is not an appropriate tool for detecting 40 q non-attracting chaos in a restraining region. Next we use this example to illustrate the numeri- ) S030 cal computation of H0. Our procedure, as explained in ;f Sec. IIC, is to choose a sample size N and range of T ( T values and to do the following for each T. Using Eq. (6), ^E 20 n we compute an estimate Eˆ of E for each of 100 dif- l T T,0 ferentsamplesofN pointseachintherestrainingregion, andcomputethemeanandstandarddeviationofthe100 10 samples. The results for N =1000 and N =100,000 are showninFig.2forS(cid:48) andinFig.3forS. Theestimated 0 valueofH istheslopeofthesolidcurveinanappropri- 0 0 20 40 60 ate scaling interval. The scaling interval for a given N T can be judged by consistency of the results with a larger valueofN,inadditiontosmallnessoftheerrorbarsand FIG. 2. Computation of lnEˆ versus T for the map of T straightnessofthecurve. Noticethatthesomewhatarbi- Fig. 1 with restraining region S(cid:48). For each T, we computed trarilychosenrestrainingregionS yieldsnearlyaslonga lnEˆ (f,S(cid:48)) for 100 different samples, with N = 1000 ran- T scalingintervalastherestrainingregionS(cid:48) thatischosen domly chosen initial conditions in each sample (top figure) with knowledge of the invariant Cantor set. and N =100,000 randomly chosen initial conditions in each samples (bottom figure). The solid curve shows the mean of the100samples,andtheerrorbarsshowtheirstandarddevi- ation. AswediscussedinSec.IIC,theslopeofthesolidcurve C. Example: A random one-dimensional map should,inthelimitoflargeN andT,approximateH (f,S(cid:48)). 0 Thedashedlinehasslopeln2,whichisthevalueweobtained Consider the one-dimensional random map analytically for H0(f,S(cid:48)). θ =[θ +α +Ksinθ ] mod 2π (8) t+1 t t t where(cid:104)···(cid:105) denotesanaverageoverη. Ifθ isuniformly η 0 where K > 0 and α0,α1,α2,... are independent ran- distributed, then θ0,θ1,θ2,... are independent and uni- domvariablesthatareuniformlydistributedinthecircle formly distributed, so that [0,2π). We take the restraining region S to be the entire circle. T−1 (cid:89) Notice that (cid:104)|dθ /dθ |(cid:105) = (cid:104)|1+Kcosθ |(cid:105) T 0 θ0,θ1,...,θT−1 t θt t=0 (cid:12)(cid:12)(cid:12)(cid:12)ddθθT(cid:12)(cid:12)(cid:12)(cid:12)=T(cid:89)−1|1+Kcosθt|, =(cid:104)|1+Kcosθ|(cid:105)Tθ 0 t=0 Thissuggeststhatforatypicalrealizationoftherandom and that inputs, H0 ≈λ, where E =(cid:104)max(|dθ /dθ |,1)(cid:105) , λ=ln(cid:104)|1+Kcosθ|(cid:105) . T,0 T 0 θ0 θ 7 10 N =1000 30 8 )S20 )S 6 ;f ;f ( ( T T ^E ^E n10 n 4 l l 2 0 0 10 20 30 40 50 60 T 0 N =100000 0 10 20 30 40 50 50 T FIG.4. ComputationoflnEˆ versusT fortherandomcircle T 40 map (8) with K =1.5. For each T, we computed lnEˆ (f,S) T for100differentsamples,withN =1,000,000randomlycho- sen initial conditions in each sample. Each initial condition ) S 30 used a different realization of the random sequence of maps. ; f The solid curve shows the mean of the 100 samples, and the ( ^ET error bars show their standard deviation. The dashed line n 20 has slope ln(cid:104)|1+1.5cosθ|(cid:105)θ; this estimate of the expansion l entropy appears to be slightly larger than the slope of the computed data. 10 D. Example: Shear map on the 2-torus 0 0 20 40 60 This example illustrates a case where orbits are dense T and,asinchaos,typicalnearbyorbitsseparatefromeach other with increasing time. However, the rate of sepa- FIG.3. ComputationoflnEˆ versusT forthemapofFig.1 T ration is linear, rather than exponential in time. The withrestrainingregionS. ThisistheanalogueofFig.2with S(cid:48) replaced by S. example is the following map of the 2-torus: φ =[φ +θ ] mod 2π, θ =[θ +ω] mod 2π, (9) t+1 t t t+1 t where ω/(2π) is irrational38,39. As shown in Fig. 5, the For 0<K ≤1, image under this map of a curve C looping around the θ,φ-torus once in the θ direction is a curve C(cid:48) that loops λ=ln(cid:104)(1+Kcosθ)(cid:105) =ln1=0, θ onceintheφdirection,aswellasonceintheθ direction, with the number of φ loops increasing by one on each while for K >1, subsequent iterate. Orbits are dense and nearby initial conditionswithdifferentvaluesofθseparatelinearlywith λ=ln(cid:104)|1+Kcosθ|−(1+Kcosθ)+1(cid:105) >ln1=0. θ t. ToevaluateH forthismapfromthedefinitionEq.(2), 0 with the restraining region S being the entire torus, we For 0 < K ≤ 1, each map is a diffeomorphism, so note that dθt/dθ0 >0, and (cid:18)1 1(cid:19) (cid:18)1 t(cid:48)−t(cid:19) Df = and Df = t+1,t 0 1 t(cid:48),t 0 1 E =(cid:104)max(dθ /dθ ,1)(cid:105) <(cid:104)dθ /dθ +1(cid:105) =2. T,0 T 0 θ T 0 θ0 for all φ and θ. For t(cid:48) −t (cid:29) 1, the singular values of Thus, ET,0 is not exponentially increasing, so indeed Dft(cid:48),t are approximately t(cid:48)−t and (t(cid:48)−t)−1. Thus, for H = 0 for 0 < K ≤ 1. Numerical experiments agree large t(cid:48)−t, 0 withtheargumentabovethatH >0forK >1, though 0 E (f,S)≈(2π)2(t(cid:48)−t), establishing that the transition to chaos (according to t(cid:48),t our definition) occurs exactly at K = 1 would require a and H = 0 (since (t(cid:48) − t)−1log(t(cid:48) − t) → 0 as t(cid:48) → 0 more definitive study. In Fig. 4, we show the computed ∞). Hence this example is not chaotic according to our lnEˆ (see Sec. IIC and IIIB) versus T for K =1.5. definition. T 8 15 )10 C S ? ;f ( T ^E n 5 l C 0 0 0 5 10 15 20 25 3 T FIG. 5. Image C(cid:48) of a circle C given by φ=constant under FIG. 7. Computation of lnEˆ versus T for the H´enon map the shear map (9). T with a = 4.2 and b = 0.3. For each T, we computed lnEˆ (f,S) for 100 different samples, with N =100,000 ran- T domly chosen initial conditions in each sample. The solid curve shows the mean of the 100 samples, and the error bars show their standard deviation. The dashed line has slope (b) (c) H =ln2. 0 with b=0.3, the results of Devaney and Nitecki40 imply that for a ≥ 3.4, the map has a topological horseshoe, andfora≤5.1,thenonwanderingsetiscontainedinthe square−3≤x,y ≤3,whichwetaketobetherestraining f(S) region S. Fig. 7 shows the results of a numerical compu- tationfora=4.2oflnEˆ (seeSec.IICandIIIB)versus T T, which agrees well with the value H =ln2. (a) (d) 0 S S IV. TOPOLOGICAL ENTROPY FIG.6. Ahorseshoemapf thatislinearfortrajectoriesthat remain in the restraining region S. In this section, we define topological entropy, and dis- cuss its relation to (and equivalence with, in appropriate circumstances) both expansion entropy and the related E. Example: Horseshoe and H´enon map notion of volume growth. Theoriginaldefinitionoftopologicalentropy,byAdler, Figure 6 shows the action of a horseshoe map in the Konheim, and McAndrew41, was for a continuous map plane on a unit square S, which we also take to be the f on a compact topological space X. If X is a met- restraining region. The step (a) → (b) represents a uni- ric space, an equivalent definition of topological entropy form horizontal compression and vertical stretching of due to Dinaburg42 and Bowen43 is as follows. (Equiva- the square. Let ρ > 2 be the ratio by which the ver- lence to the original definition was proved by Bowen44.) tical length of the square is stretched, and assume that For (cid:15) > 0, two points x and y in X are called (T,(cid:15))- bending deformations in the step (b) → (c) take place separated if the distance between their kth iterates sat- only in the shaded region. The fraction of the orig- isfies d(fk(x),fk(y)) > (cid:15) for some 0 ≤ k < T. A finite inal square that remains in the square after one iter- setofpointsP ⊂X issaidto(T,(cid:15))-spanX ifthereisno ate is 2/ρ (see Fig. 6(d)), and after t(cid:48) −t iterates, the pointinX thatis(T,(cid:15))-separatedfromeverypointinP. fraction is (2/ρ)t(cid:48)−t. Also, G(Df ) = ρt(cid:48)−t, yielding Let n(T,(cid:15)) be the minimum number of points needed to t(cid:48),t E (f,S)=ρt(cid:48)−t(2/ρ)t(cid:48)−t =2t(cid:48)−t,andH =ln2. Thus, (T,(cid:15))-span X, and let N(T,(cid:15)) be the maximum number t(cid:48),t 0 by our definition the horseshoe map is chaotic in S. of points in X that can be pairwise (T,(cid:15))-separated. Let For the H´enon map lnn(T,(cid:15)) h (f,(cid:15))=limsup xt+1 =a+byt−x2t, yt+1 =xt n T→∞ T 9 and surfaceareas, etc. Expansionentropy, byestimatingvol- ume growth locally, allows an analogous computation to lnN(T,(cid:15)) hN(f,(cid:15))=limsup T . (10) be done without having to compute and measure mul- T→∞ tidimensional surfaces. It is analogous to the approach Itisnothardtoshowthatn(T,(cid:15))≤N(T,(cid:15))≤n(T,(cid:15)/2). used by Jacobs et al.31 for two-dimensional maps. This implies the analogous relation between h and h , Anotherapproachtocomputingentropyisbysymbolic n N which implies that they have the same limit as (cid:15) → 0. dynamics: partition the state space into numbered sub- Define the topological entropy h of f on X by sets, and estimate the exponential growth rate (as time increases) of the number of different sequences of sub- h(f,X)= limh (f,(cid:15))= limh (f,(cid:15)). (11) n N sets that can be visited by a finite-time trajectory. This (cid:15)→0 (cid:15)→0 approach can yield good estimates with well-chosen par- The notions of expansion entropy H and topological 0 titions, but inadequate partitions may lead to underesti- entropy h are both well-defined in the case when f is a mation, and in some cases symbolic dynamics indicates smooth, autonomous system on a compact manifold M positiveentropywhenthetopologicalentropyisactually and the restraining region S is all of M. In this case, zero.53 Sacksteder and Shub26 defined a quantity they called h 1 We conclude this section with a brief discussion of that is equivalent to expansion entropy. Subsequently, the connection between the definitions of expansion en- Przytycki45 proved that h is an upper bound on h if 1 tropy and topological entropy in the case of a smooth, f is a C1+γ diffeomorphism for γ > 0; this proof was autonomous system on a compact manifold M, with re- extendedtononinvertiblemapsbyNewhouse46. Though straining region S = M. In Appendix B, we argue that there are examples47 for which the two quantities differ, for sufficiently small (cid:15) > 0, the quantity E (f,S) of Kozlovski27 proved that h = h for C∞ maps. Thus, T,0 1 Eq. (1) approximates N˜(T,(cid:15))/N(0,(cid:15)), where N˜(T,(cid:15)) is H (f,M) = h(f,M) for a sufficiently smooth map f on 0 themaximumnumberoftrajectoriesthatareadistance(cid:15) a compact manifold M. apartateithertime0orattimeT. NotethatN˜(T,(cid:15))isa From our point of view, these results leave open con- lower bound on N(T,(cid:15)), because the latter distinguishes sideration of important issues regarding nonautonomous between trajectories that are (cid:15) apart at some time be- systems and the role of restraining regions. For exam- tween 0 and T; however, at least for hyperbolic systems ple, suppose now that J is a compact invariant set of the difference between N˜ and N should be inconsequen- an autonomous system f on a (not necessarily compact) tial. Eqs. (10) and (11) first take a limit with respect to manifoldM. IfJ hasvolumezero,H (f,J)isundefined, 0 T and then (cid:15). Normalizing by N(0,(cid:15)) does not change butwecandefineH foraneighborhoodS ofJ thatcon- 0 the limit: tains no other invariant sets. In this case, we conjecture thatH0(f,S)=h(f,J)iff isC∞. Moregenerally,when lnN(T,(cid:15))−lnN(0,(cid:15)) the restraining region S contains multiple invariant sets, h(f,S)= limlimsup we conjecture (consistent with Eq. (5)) that H (f,S) is (cid:15)→0 T→∞ T 0 ln(N(T,(cid:15))/N(0,(cid:15))) the maximum topological entropy of f on an invariant = limlimsup . (12) subset of S. (cid:15)→0 T→∞ T Our notion of expansion entropy is related to the We have argued above that notion of volume growth defined by Yomdin48 and Newhouse46. Yomdin defines the exponential rate v (f) d N˜(T,(cid:15)) of d-dimensional volume growth of a smooth map f on a E (f,S)= lim . compactmanifoldM,andprovesthatv (f)≤h(f,M)if T,0 (cid:15)→0 N(0,(cid:15)) d f isC∞. Newhousedefinesthevolumegrowthratemore Thus,byEq.(2),thedefinitionofH differsfromthedef- generally for a neighborhood U of a compact invariant 0 set J ⊂M, and proves that h(f,J) is bounded above by inition of h primarily because it uses N˜(T,(cid:15)) ≤ N(T,(cid:15)), themaximumoverdofthed-dimensionalvolumegrowth and because the limits with respect to T an (cid:15) are taken rate on U. See also Gromov49 for a discussion of these in the reverse order. results. Based on these results, Newhouse and Pignataro50 proposed and implemented algorithms for computing V. DEFINITIONS OF CHAOS THAT DO NOT INVOLVE entropy of two-dimensional diffeomorphisms (including ENTROPY: SENSITIVE DEPENDENCE, LYAPUNOV Poincar´e sections of three-dimensional differential equa- EXPONENTS AND CHAOTIC ATTRACTORS tions) by computing the exponential growth rate of the length of an iterated curve. Other algorithms51,52 com- A concept often associated with chaos is that of sensi- pute the growth rate of the number of disconnected arcs tivedependence,theideathattheorbitsfromtwonearby resulting from the iteration of an initial line segment initial conditions can move far apart in the future. In within a neighborhood of a two-dimensional chaotic sad- the mathematical literature the most common definition dle or repeller. Of course, these methods could be ex- of “sensitive dependence” is as follows. (This definition tended to higher dimensions by considering growth of is also sometimes called “weak sensitive dependence.”) 10 Definition: A continuous map, f : M → M, on the can never exhibit chaos. Yet quasi-periodically forced compactmetricspaceM hassensitivedependenceifthere systems are of practical interest and can have attractors exists a ρ > 0 such that for each δ > 0 (no matter how withapositiveLyapunovexponent,asituationgenerally small) and each x ∈ M, there is a y ∈ M that is within thought of as chaotic. Another point to make in connec- the distance δ of x and for which at some later time t, tion with the example (13) is that it presents a problem |ft(x)−ft(y)|>ρ. for the Devaney definition of chaos even when G is in- Thatis,nomatterhowclosetogethertheinitialcondi- dependent of θ: in that case z = G(z ), on its own, t+1 t tionsxandy are,ifwewaitlongenough,theorbitsfrom might indeed satisfy the conditions for Devaney-chaos; these initial conditions will separate by more than some however, by considering the state to be x = (z,θ) with fixed value ρ. Notice that this definition of sensitive de- θ quasi-periodic, the Devaney chaos condition (ii) is not t pendence does not say anything about the rate at which satisfied, even though there is no change in the chaotic these orbits diverge from each other: this rate might, for dynamics of z. example, be exponential (e.g., as for situations with a AccordingtoBanksetal.,Devaney-chaosonlyrequires positive Lyapunov exponent), or linear (e.g., as for the satisfaction of the two conditions that there be a dense example in Sec. IIID). orbit and a dense set of periodic orbits. Robinson38, on Another often used concept assigns sensitive depen- the other hand, notes that of the three conditions origi- dence to the dynamics on a compact invariant set (the nallyspecifiedbyDevaney,therequirementofadenseset space M is now not necessarily compact) as follows. ofperiodicorbitsdoesnotseemas“centraltotheideaof Definition: A continuous map f has sensitive depen- chaos”astheothertwoconditions(sensitivedependence dence on a compact invariant set J of a metric space M and a dense orbit). Thus he (and also, independently, if for every δ >0 (no matter how small) and every point Wiggins56) proposes the following definition. x ∈ J, there is a point y ∈ J within a distance δ of x Definition of Robinson-chaos: The same as such that, at some later time t, |ft(x)−ft(y)| > ρ for Devaney-chaos except that condition (ii) is deleted. some fixed value ρ>0. This definition, by not requiring periodic orbits, has The following is a definition of chaos, based on that the benefit of potentially allowing more consistent treat- given by Devaney54. ment of forced systems, like (13). However, there is Definition of Devaney-chaos: A continuous map still, in our opinion, a drawback. This occurs, e.g., with f :M →M, with M a compact metric space, is chaotic reference to the shear map example (9) of Sec. IIID, if it satisfies the following three conditions. which was considered by Robinson38 (see also Hunt and Ott39). As discussed in Sec. IIID, orbits are dense and (i) f has sensitive dependence. nearby points typically separate linearly with t. Thus, thisexampleisRobinsonchaotic. However,thetwoLya- (ii) f has periodic orbits that are dense in M. punov exponents are zero. While this example satisfies (iii) f has an orbit that is dense in M, i.e., there exists theRobinson-chaosdefinition, due totheslow, linear-in- aninitialconditionx∗suchthatforeachy ∈M and time, separation of orbits, such dynamics has previously each δ >0 (no matter how small), at some time t, been classified as nonchaotic (see literature on so-called theorbitfromx∗ willbewithinthedistanceδ from strange nonchaotic attractors13). Indeed, this linear-in- y: |ft(x∗)−y|<δ. time separation of nearby orbits presents comparatively littlepredictiondifficultyascomparedtotheexponential This definition can be converted to define Devaney divergence emphasized by Lorenz. chaosforacompactinvariantset,J =f(J),byreplacing Li and Yorke1 define the notion of a “scrambled set”, M in conditions (ii) and (iii) by J, and condition (i) by and the presence of a scrambled set can be taken as an- “f has sensitive dependence on J”. other definition of chaos. While this works well in the It was pointed out by Banks et al.55 that conditions original context of one-dimensional maps considered by (ii) and (iii) of Devaney’s definition, imply his condition Li and Yorke, as we will see, it is not as appropriate for (i). Thus condition (i) for Devaney-chaos can be omit- higher dimensional systems. ted. A serious drawback of Devaney’s definition of chaos Definition of a scrambled set: For f : M → M is that it excludes significant cases that are sometimes with M a compact metric space, an uncountably infinite considered and that most would regard as chaotic. For subset J of M is scrambled if, for every pair x,y ∈ J example, consider a map with quasi-periodic forcing, with x(cid:54)=y, z =G(z ,θ ), θ =[θ +ω] mod 2π, (13) t+1 t t t+1 t limsup|ft(x)−ft(y)|>0, liminf|ft(x)−ft(y)|=0. t→∞ t→∞ where ω/(2π) is an irrational number. Regarding this as a dynamical system with a state x = (z,θ), we see Thus, by the second Li-Yorke condition, the orbits that, because of the quasi-periodic behavior of θ, there from x and y come arbitrarily close to each other an are no periodic orbits of this system. Hence, the system infinite number of times, while by the first Li-Yorke con- (13) fails condition (ii) for Devaney chaos. Thus accord- dition, thedistancebetweentheorbitsfromxandy also ingtothedefinitionofDevaneychaos, asystemlike(13) exceeds a fixed positive amount an infinite number of

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.