A test for a conjecture on the nature of attractors for smooth dynamical systems
aa r X i v : . [ m a t h . D S ] M a r Conjecture on the Nature of Attractors
A Test for a Conjecture on the Nature of Attractors for Smooth DynamicalSystems
Georg A. Gottwald and Ian Melbourne School of Mathematics and Statistics, University of Sydney, Sydney 2006 NSW,Australia a)2)
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK b) (Dated: 1 October 2018) Dynamics arising persistently in smooth dynamical systems ranges from regular dynamics (periodic, quasiperi-odic) to strongly chaotic dynamics (Anosov, uniformly hyperbolic, nonuniformly hyperbolic modelled byYoung towers). The latter include many classical examples such as Lorenz and H´enon-like attractors andenjoy strong statistical properties.It is natural to conjecture (or at least hope) that most dynamical systems fall into these two extremesituations. We describe a numerical test for such a conjecture/hope and apply this to the logistic map wherethe conjecture holds by a theorem of Lyubich, and to the Lorenz-96 system in 40 dimensions where thereis no rigorous theory. The numerical outcome is almost identical for both (except for the amount of datarequired) and provides evidence for the validity of the conjecture.PACS numbers: 05.45.-a, 05.45.Pq, 05.45.Ac, 05.45.JnKeywords: regular and chaotic dynamics, nonuniform hyperbolicity, SRB measures, Gallavotti-Cohen chaotichypothesis, Palis conjecture
A longstanding open problem in the theory of dy-namical systems, that continues to be the sub-ject of much discussion by mathematicians andphysicists, is the question of what constitutes atypical dynamical system. An answer would notonly constitute an immense theoretical advancewithin the theory of smooth dynamical systems,but would have a profound practical impact onour understanding and analysis of physical phe-nomena in the real world. In this work we for-mulate a conjecture on the nature of typical dy-namical systems stating that they are either reg-ular or chaotic in a way that assures good statis-tical properties such as existence of Sinai-Ruelle-Bowen (SRB) measures, exponential decay of cor-relations, large deviation principles as well as cen-tral limit theorems. Since the current state ofthe theory does not allow for a rigorous theoret-ical treatment of the conjecture, we devise a nu-merical test which we use to find corroboratingevidence for the conjecture.
1. INTRODUCTION
A central, but currently intractable, question in thetheory of smooth deterministic dynamical systems is tounderstand the types of attractors for typical systems. Aclassification of attractors would range from very regu- a) Electronic mail: [email protected] b) Electronic mail: [email protected] lar dynamics to very chaotic dynamics, including peri-odic sinks at one extreme and uniformly hyperbolic (Ax-iom A) attractors at the other extreme. The uniformlyhyperbolic attractors of Smale generalise the Anosovdiffeomorphisms and flows. (Smale’s definition of uni-formly hyperbolic includes the periodic case, but we shallabuse terminology and reserve the words “uniformly hy-perbolic” for the nonperiodic case.)Throughout this paper we are interested in both dis-crete time dynamical systems (noninvertible maps anddiffeomorphisms) and continuous time systems (flows).Similar comments and results apply to both. However,our notation and definitions will be confined to the dis-crete case, where f : R n → R n is a smooth map withcompact attractor Λ ⊂ R n . Our focus is primarily ondissipative systems, but the material goes over to Hamil-tonian systems with the obvious modifications.An important property of uniformly hyperbolic attrac-tors is the existence of a physical measure , or SRB mea-sure after Sinai, Ruelle and Bowen, which has the prop-erty that time averages converge to the space average fora set of initial conditions of positive volume (ie. positiveLebesgue measure). This is in contrast to the ergodictheorem for ordinary ergodic measures where the con-vergence takes place for a set that has full measure withrespect to the ergodic measure which however is usually aset of zero volume (since the ergodic measure is supportedon the attractor Λ which is usually of zero volume).
Definition 1.1
An ergodic measure µ supported on Λ isan SRB measure if there is a set B of positive volumesuch that lim n →∞ n n − X j =0 v ( f j x ) = Z Λ v dµ, onjecture on the Nature of Attractors 2for every continuous observable v : R n → R and for all x ∈ B .Uniformly hyperbolic attractors have numerous strongstatistical properties. In particular, they have exponen-tial decay of correlations up to a finite cycle . Definition 1.2
An attractor Λ with ergodic measure µ has exponential decay of correlations if there exists a con-stant γ ∈ (0 ,
1) such that for all smooth v, w : R n → R there is a C > (cid:12)(cid:12)(cid:12)Z Λ v w ◦ f n dµ − Z Λ v dµ Z Λ w dµ (cid:12)(cid:12)(cid:12) ≤ Cγ n . More generally, the attractor has exponential decay ofcorrelations (up to a finite cycle) if there exists k ≥ , . . . , Λ k such that for all i =1 , . . . , k it is the case that f (Λ i ) = Λ i +1 (with k + 1 = 1)and f k : Λ i → Λ i has exponential decay of correlations.From now on, we omit the words “up to a finite cycle”and speak simply of exponential decay of correlations.The SRB measure µ on a uniformly hyperbolic attrac-tor enjoys this property and it suffices that v and w areLipschitz (or even H¨older, in which case the constant γ depends on the H¨older class). There are numerous otherstatistical properties such as central limit theorems thathold for uniformly hyperbolic attractors. These are de-scribed in a more general setting below. Remark 1.3
Decay of correlations for uniformly hyper-bolic (even Anosov) flows is rather less well understood.Only partial results exist ; however statistical limitlaws such as central limit theorems and invariance prin-ciples remain valid for uniformly hyperbolic flows .Smale conjectured (see for example Section 2.5 inPalis ) that for typical dynamical systems (typical inthe sense of C r open and dense, r ≥
1) periodic sinks anduniformly hyperbolic attractors comprise the full rangeof possibilities. This conjecture turned out to be false,and moreover the notion of typicality turned out to beinadequate even in situations where the conjecture holds(see for example items (i)–(iii) below). Over the last 40–50 years, numerous examples have arisen that make itnecessary to enlarge the notions of being very regular orvery chaotic.(i) KAM tori with quasiperiodic dynamics are non-robust in a topological sense (they are destroyedby C r small perturbations,) but they are unavoid-able in a probabilistic sense (the set of parametersthat give rise to KAM tori has large measure). Fordissipative systems a similar phenomenon arises inNaimark-Sacker bifurcation from a periodic solu-tion.(ii) The logistic map (see Section 3 for more details) isa one-parameter family of one-dimensional maps. For each value of the parameter there is a uniqueattractor that attracts almost every trajectory. Foran open and dense set of parameters, the attrac-tor is a periodic sink. However, Jakobson showedthat the complementary set of parameters has posi-tive measure. More recently, Lyubich proved thatalmost every parameter in this complementary setsatisfies the so-called Collet-Eckmann condition and hence constitutes strongly chaotic (though notuniformly hyperbolic) dynamics.(iii) H´enon-like attractors arise near quadratic ho-moclinic tangencies and are strongly chaotic .These are again unavoidable in a probabilisticsense.(iv) Geometric Lorenz attractors are topologically ro-bust but nonuniformly hyperbolic examples ofstrongly chaotic systems . Tucker showedthat these include the classical Lorenz attractor .The strongly chaotic attractors mentioned above –uniformly hyperbolic, Collet-Eckmann, H´enon-like, (ge-ometric) Lorenz – have the common property that theyare modelled by a Young tower with exponential tailsas introduced by Young . (For Lorenz attractors, it isthe Poincar´e map that is modelled by a Young tower.)Roughly stated, a dynamical system f : Λ → Λ is mod-elled by a Young tower with exponential tails if thereexists a set Y ⊂ Λ with return time function τ : Y → Z + (not necessarily the first return time) and return map F = f τ : Y → Y such that (i) F is uniformly hyper-bolic, and (ii) the likelihood of a large return time τ isexponentially small.Numerous strong statistical properties have beenproved for such attractors modelled by Young towers:existence of an SRB measure, exponential decay of corre-lations and central limit theorems , large deviation prin-ciples , Berry-Ess´een estimates and local limit theo-rems , invariance principles . There is also an en-larged class of attractors that possess polynomial decayof correlations; where this decay is summable the abovestatistical properties apply.In a sense that can be made precise, there is an equiva-lence between the existence of a Young tower and strongstatistical properties . This observation uses the work ofAlves et al. and Melbourne & Nicol .Since there are good reasons for hoping (if not believ-ing) that most attractors are either highly regular orenjoy strong statistical properties, and in the absenceof convincing counterexamples, one possibility is to de-fine strongly regular attractors to be the periodic andquasiperiodic ones, and strongly chaotic attractors to bethe ones modelled by a Young tower with exponentialtails. This leads naturally to the following deliberatelyimprecise conjecture. Conjecture 1.4
Typically (in a sense that we do notmake precise), the attractors for smooth dynamical sys-tems fall into one of the following two classes:onjecture on the Nature of Attractors 3(a) Regular dynamics: Λ is a periodic or quasiperiodicsink.(b) Chaotic dynamics: Λ is modelled by a Young towerwith exponential tails.(In the case of flows, this statement is at the level of thePoincar´e map.)A precise conjecture would require a precise definitionof “typically”, probably leading to the failure, though notnecessarily the relevance, of the conjecture.
A test for Conjecture 1.4
Although it is hard to see how to test directly for Con-jecture 1.4, there are certain implications that can betested numerically. Suppose that Λ is an attractor fora map or diffeomorphism f : R n → R n and that µ isan ergodic invariant measure on Λ. Let v : R n → R bea smooth observable. Recall that the power spectrum S : [0 , π ] → [0 , ∞ ) is given by S ω = lim n →∞ n S ω ( n ) , S ω ( n ) = Z Λ (cid:12)(cid:12)(cid:12) n − X j =0 e ijω v ◦ f j (cid:12)(cid:12)(cid:12) dµ. Since S π − ω = S ω we restrict from now on to the interval[0 , π ].The following dichotomy was established by Melbourne& Gottwald . Theorem 1.5
Let Λ be a periodic or quasiperiodic sink,or an attractor modelled by a Young tower with exponen-tial tails for a smooth map f : R n → R n . Suppose that µ is the SRB measure on Λ . Let v : R n → R be a C ∞ observable. Typically,(a) In the periodic/quasiperiodic case, S ω = 0 almosteverywhere. Moreover, K ω = lim n →∞ log S ω ( n ) / log n = 0 for all but finitely many ω ∈ [0 , π ] .(b) In the Young tower case, there is a constant s > such that S ω ≥ s for all but finitely many valuesof ω . In particular, K ω = lim n →∞ log S ω ( n ) / log n = 1 for all but finitely many ω ∈ [0 , π ] . Remark 1.6
Often in the physics literature, regular andchaotic dynamics is distinguished in terms of the powerspectrum . Broadband power spectrum (where thereexists an interval or at least a set of positive measure onwhich S ω is positive) is seen as being the signature ofchaotic dynamics. The dichotomy in Theorem 1.5 is significantly stronger.In case (a), we are requiring that S ω ( n ) grows slower thanany polynomial rate. In contrast, the requirement that S ω = 0 is compatible with values of K ω anywhere in [0 , S ω ( n ) grows like n/ log n , then S ω = 0but K ω = 1.In case (b), the power spectrum is positive andbounded away from zero for all but finitely many points,which is rather more than claiming broadband spectrum.Based on Theorem 1.5, our conjecture can be testedas follows. Consider a parameterized family of smoothdynamical systems, with parameter a ∈ R . For each fixedvalue of a , compute the family of limits S ω and checkwhether they are almost all zero or almost all one. Sucha test can be carried out numerically by taking valuesof a and ω that are reasonably dense and estimating thegrowth rate of S ω ( n ). Remark 1.7
A slightly weaker version of the conjecturewould be to include Young towers with polynomial tails(rather than only those with exponential tails). Our testdoes not distinguish between these situations. However,we do not know of any persistent examples in smooth dy-namics where an attractor is modelled by a Young towerwith subexponential tails, but not by a Young tower withexponential tails.There are similarities and differences between the testproposed above and the 0–1 test for chaos . The 0–1test is optimised to work with limited amounts of data.In particular, taking the median value of K ω for 100 ran-domly chosen values of ω greatly accelerates the conver-gence of the test. The test described in this paper isa much more stringent examination of the dichotomy inConjecture 1.4 but requires much more data. Even forthe logistic map, the refined test in this paper requiresenormous amounts of data that would be impractical inthe 0–1 test for chaos.Our conjecture is related to the Palis conjecture andto the Gallavotti-Cohen chaotic hypothesis. Palis conjecture.
As already mentioned, Smale’s con-jecture regarding the ubiquity of periodic sinks anduniformly hyperbolic attractors turned out to be false.Hence it became necessary to formulate a weakened state-ment. Over the years, Palis gave a number of conjec-tures in this direction; we refer to the original workby Palis for statements of these conjectures (rathermore precise than ours!) and progress towards their ver-ification.The emphasis in the Palis conjectures is global, focus-ing on (i) the finitude of attractors possessing SRB mea-sures, with the property that the union of their basinsaccounts for a set of initial conditions of full measure,and (ii) the stability of these attractors under perturba-tions. Our conjecture is more local since we have saidnothing about the finitude of attractors, nor their stabil-ity under perturbations. However, for typical attractorsonjecture on the Nature of Attractors 4taken on their own, we make stronger statements abouttheir statistical properties.
Gallavotti-Cohen chaotic hypothesis.
The chaotichypothesis proposes that chaotic systems should beconsidered as Anosov systems for practical purposes.Since the property of part (b) of Theorem 1.5 is cer-tainly valid for Anosov systems, our numerical test canbe viewed as a test also of the chaotic hypothesis.The remainder of the paper is organised as follows. InSection 2, we describe how to test parametrized fami-lies of dynamical systems for their conformity to Con-jecture 1.4. In Section 3, we carry out this test for thelogistic map. This provides a benchmark for our testsince the conjecture is known to be valid by Lyubich .In Section 4, we carry out the test for the 40-dimensionalLorenz-96 system, which is regarded as highly impor-tant in meteorological studies, and which is far beyondthe current understanding of rigorous dynamical systemstheory. Nevertheless, the numerical results for Lorenz-96are similar to those for the logistic map except for theamount of data required for convergence. We concludewith a brief summary in Section 5.
2. THE NUMERICAL TEST
Consider a smooth family of maps f a : R n → R n where a ∈ R is a parameter. For convenience, we assume thatall trajectories are bounded.Suppose that a ∈ R and that Λ ⊂ R n is an attractorfor f a . For values of ω chosen randomly from [0 , π ] wecompute K ω as defined in Theorem 1.5. Then accordingto Conjecture 1.4 and Theorem 1.5 we anticipate that K ω takes the constant value 0 or 1 independent of ω (forall but finitely many ω ).To carry out this procedure numerically, we note thatcomputing S ω ( n ) directly is unfeasible since Λ (and µ )are not given. However, by the ergodic theorem, for µ -almost every x ∈ Λ, S ω ( n ) = lim J →∞ J J − X j =0 | p ω ( j + n ) − p ω ( j ) | , (2.1)where p ω ( n ) = n − X ℓ =0 e iℓω v ( f ℓa x ) . (2.2)Moreover, assuming the conjecture, typically µ can betaken to be an SRB measure and x can be chosen froma set of positive Lebesgue measure.Equally, assuming the conjecture, typically x ∈ R n lies in the basin of an attractor Λ (depending on x and a ) and S ω ( n ) can be computed using (2.1) and (2.2).From this K ω = lim n →∞ log S ω ( n ) / log n can be com-puted. Again, if Conjecture 1.4 is valid, then for typical f a and x it should be the case that K ω takes the con-stant value either 0 or 1 independent of ω (for all butfinitely many ω ).There are some finite computation issues that need tobe addressed. The most crucial one is that the definitionof K ω involves a double limit, first as J → ∞ and thenas n → ∞ . We will ignore this issue to begin with andreturn to it at the end of the section.To implement the test, we take as initial condition x = f x where x is chosen at random and fixedthroughout. (Neglecting this transient of 1000 iterates isnot strictly necessary but speeds up the calculations.) Afinite but reasonably dense set of parameters a is speci-fied. For each value of a , we compute K ω for 100 (say)randomly chosen values of ω ∈ [0 , π ]. Given the finitenessof the data, it is necessary to specify small open inter-vals I and I containing 0 and 1 respectively, such that K ω ∈ I r is viewed as an r for r = 0 ,
1. Define M = { ω : K ω ∈ I } ,M = { ω : K ω ∈ I } ,M u = { ω : K ω I ∪ I } , with M + M + M u = 100. (Here u stands for undecided.)As K ω is computed with greater and greater precision,a consequence of the conjecture is that either M → M → M u → { M , M } → . The numerical test that we propose can now be statedmore precisely. We make three choices of intervals(i) I = ( − . , . I = (0 . , . I = ( − . , . I = (0 . , . I = ( − . , . I = (0 . , . A equally spaced valuesof the parameter a and 100 values of ω ∈ [0 , π ] chosenat random. (The value of A will depend on the lengthof the range of interesting parameters for the dynamicalsystem.) Then we analyse the convergence to zero of thefollowing four quantities as the limit J → ∞ and n → ∞ is approached: Q u = P a M u ,Q ′ u = { a : M u > } , and Q min = P a min { M , M } ,Q ′ min = { a : min { M , M } > } . The quantity Q u ∈ { , , . . . , A } denotes the totalnumber of values of ω and parameter values a for whichthe value of K is undecided (i.e. it lies outside I ∪ I ),whereas the quantity Q ′ u ∈ { , , . . . , A } denotes theonjecture on the Nature of Attractors 5number of parameter values for which K ω is undecidedfor more than 10% of the choices of ω . Similarly for Q min and Q ′ min with the number of undecideds replacedfor each a by the minimum of the number of 0s and thenumber of 1s.Our choices for the intervals I and I around 0 and 1are somewhat arbitrary; if the conjecture is true thaneventually the four quantities Q u , Q ′ u , Q min , Q ′ min willreach zero, regardless. The amount of data to achieve thisdepends on the choice of intervals, but this dependenceis not relevant for Conjecture 1.4. On the other hand,our numerical experiments indicate that K ω very quicklylies between 0 and 1 (within a small error), and we havechosen the sharper lower limit − . I and upper limit1 . I with this in mind.We note that the convergence to zero need not bemonotone. For example, suppose that a parameter value a yields a chaotic attractor of class (b), so that eventually M = 0 and M = 100. If the convergence is sufficientlyslow, then it is possible that M = 95 and M = 5 (say)for N too small. For moderate values of N , the situationmight improve to M = 15, M = 85. In this case, theparameter a contributes adversely to Q min ′ for N moder-ate but not for N small. An example of this is shown inFigure 2 where the number of outliers for Q ′ min increasesfrom zero to one as N increases within the range of ourexperiment. The double limit
As promised, we discuss the issues regarding the doublelimit in the formula K ω = lim n →∞ lim J →∞ log (cid:16) J J − X j =0 | p ω ( j + n ) − p ω ( j ) | (cid:17) / log n. Under certain conditions, it should be possible to provethat for any τ ∈ (0 , K ω = lim J →∞ log (cid:16) J J − X j =0 | p ω ( j + J τ ) − p ω ( j ) | (cid:17) / log J τ . (2.3)However, there is no way to tell how large J needs tobe for a given τ to be effective, rendering formula (2.3)unsuitable for a numerical test. We follow the simplerroute of replacing J τ by δJ where δ is a small constant(depending on the family of dynamical systems). By in-spection for a few randomly chosen values of a and ω , wecheck that δ is sufficiently small for the range of n usedin the numerical test. In Sections 3 and 4, we verify that δ = 0 .
01 suffices for the logistic map and the Lorenz-96system, respectively.Suppose that N denotes the number of iterates avail-able for the numerics, so we have computed f ja x for j = 0 , . . . , N −
1. Then S ω ( n ) can be computed for n = 0 , , . . . , δN and we can use log S ω ( δN ) / log δN asan estimate for K ω . Speeding up the test
We have already mentioned that taking a short tran-sient (say 1000 iterates) speeds up the convergence in thetest. There are further devices for speeding up the testthat we observed while developing the 0–1 test for chaos.First, it is useful to define the modified mean squaredisplacement D ω ( n ) = S ω ( n ) − ( R Λ v dµ ) − cos nω − cos ω . Note that R Λ v dµ = lim n →∞ n P n − j =0 v ( f j x ) can becomputed using the ergodic theorem, and that K ω =lim n →∞ log D ω ( n ) / log n . For Young towers with expo-nential tails that are mixing, it was proved that the con-vergence as n → ∞ is now uniform in ω . Even inthe nonmixing case, numerics show this to be a use-ful modification. To avoid taking logarithms of nega-tive numbers, we set D ω ( n ) = D ω ( n ) + C where C =max k =1 ,...,δN | D ω ( k ) | . (Since C is a constant once theamount of data is specified, the growth rate in n is un-changed.) Then we replace S ω ( n ) by D ω ( n ) in all theformulas.Second, to make more efficient use of the data, we com-pute D ω ( n ) for all n ≤ δN and perform linear regressionon log D ω ( n ) plotted against log n (cf. Gottwald & Mel-bourne ).
3. LOGISTIC MAP
The logistic map, or quadratic family, f a ( x ) = ax (1 − x ), 0 ≤ a ≤
4, is a convenient example to begin with sinceit is very well understood. For each value of a , there isa unique attractor Λ ⊂ [0 , a ∈ P ⊂ [0 , (see also Benedicks& Carleson ) proved that Leb( P ) <
4. By Lyubich for almost every a ∈ [0 , \ P , the Collet-Eckmann con-dition holds, and this implies the existence of a Youngtower with exponential tails (see Theorem 7 in Young ).Hence Conjecture 1.4 is valid for this family.We confine our numerics to the parameter range 3 . ≤ a ≤ , . ⊂ P and corresponds entirely to peri-odic sinks of low period (at most period 4).The first step is to determine a suitable value of δ . Toachieve this, we chose various values of a ∈ [3 . ,
4] and ω ∈ [0 , π ] at random and plotted log D ω ( n ) against log n for various ranges of n = 1 , . . . , N . Theorem 1.5 impliesthat the graph should be linear, but in practice given N iterates of the dynamical system, the graph is linear onlyup to a certain point. A typical example is shown inFigure 1. (The graphs for different choices of N need notcoincide for a given n since the averaging in (2.1) is over adifferent range. The strange (and inconsistent) behaviourfor large n confirms that there is insufficient averagingonce n is too large relative to N .) It is evident fromonjecture on the Nature of Attractors 6the graphs that δ = 0 . δ = 0 . a and ω confirm that δ = 0 .
01 isa safe choice for the entire range of values of a , ω and N in our numerical test. From now on we fix this value of δ for the logistic map. log n l og D ω ( n ) FIG. 1. Graph of log D ω ( n ) against log n for the logistic mapwith a = 3 . ω = 1 . n = 1 , . . . , N with N = 50 , N = 100 , N = 200 , N = 400 , δ = 0 .
01, blue circles for δ = 0 . δ = 0 .
1. The four graphs are spacedapart vertically so that they can be seen separately, with N =50 ,
000 at the bottom, up to N = 400 ,
000 at the top.
Our numerical results for the logistic map are shownin Figure 2. The results are consistent with the theory,based on Lyubich , which dictates that the four quan-tities Q u , Q min , Q ′ u , Q ′ min converge to zero as N → ∞ .However, it is also clear that there are a handful of casesthat are converging very slowly, with little appreciableimprovement from N = 100 ,
000 to N = 500 , a ≈ .
57 leads tovery slow convergence in any numerical method for dis-tinguishing regular and chaotic dynamics. Nevertheless,by we know that the conjecture is true for this exam-ple, so the difficulty is not with the conjecture itself, butwith the numerical verification of the conjecture. Under-standing these limitations to this (or any) numerical testis instructive when applying it to examples where thereis no proof of convergence.To this end, it is useful to first contrast these resultswith the 0–1 test for chaos which uses the median valueof K ( ω ), and hence converges very quickly for most val-ues of the parameter a , see Figure 3. The problematicparameters are indeed the ones near the onset of chaos a ≈ .
57 and also near the first periodic window a ≈ . a ≈ .
83 does not cause a problem.(Of course, periodic windows are dense but at this levelof resolution, where a is increased in increments of 0 . M u is shown as a func-tion of the parameter a for a range of iterates N . It isshown that by the time we reach N = 500 ,
000 iterates,the nonconvergence of the four quantities in Figure 2is due almost entirely to two values of the parameter, a = 3 .
58 and a = 3 .
4. LORENZ-96 MODEL
In this section, we consider the Lorenz-96 model dx i dt = x i − ( x i +1 − x i − ) − x i + a with i = 1 , · · · , m , (4.1)where x = x m . This system of ordinary differentialequations was first introduced by Lorenz as an idealisedmodel for midlatitude atmospheric dynamics . Weconsider the case m = 40 with the parameter a varyingin the interval [3 ,
7] in increments of 0 .
1. Throughout,we integrate the system using a time step of 0 . ,
000 time steps, N data points after each 1000 time steps. As an observablewe take φ = x , so φ ( n ) = x ( t ) with t = 0 . n .In Figure 5 we show that again a value of δ = 0 . δ = 0 .
01 increases with the totalnumber of iterates N .Our numerical results for the Lorenz-96 model areshown in Figure 6. The results are consistent with Con-jecture 1.4 that the four quantities Q u , Q min , Q ′ u , Q ′ min converge to zero as N → ∞ . Indeed, the results in Fig-ure 6 are comparable to those in Figure 2 for which therewas a rigorous convergence proof. The main noticeabledifference is the speed of convergence since we have used6 times the amount of data, but that is not surprisinggiven the increase from one dimension to 40 dimensionsand the passage from discrete to continuous time.Again, it is also clear that there are only a few parame-ter values for which the convergence is slow. As with thelogistic map, this can be compared with the correspond-ing results for the more quickly convergent 0–1 test forchaos, see Figure 7, as well as the number of outliers inFigure 8.
5. SUMMARY
We have formulated a conjecture on the nature ofattractors of typical dynamical systems. Our Conjec-ture 1.4 states that typical dynamical systems are eitherregular, i.e. periodic or quasi-periodic, or strongly chaoticin the sense that they enjoy good statistical propertiesonjecture on the Nature of Attractors 7 N Q u range: 0 . . . . . . N Q m i n range: 0 . . . . . . N Q ′ u range: 0 . . . . . . N Q ′ m i n range: 0 . . . . . . FIG. 2. Graphs of Q u , Q min , Q ′ u , Q ′ min against the number of iterates N for the logistic map. In each case the results areshown for the three choices (i) I = ( − . , . I = (0 . , . I = ( − . , . I = (0 . , . I = ( − . , . I = (0 . , . a K FIG. 3. The median value of K ( ω ) plotted against the pa-rameter a for the logistic map. Blue crosses: N = 10 , N = 100 ,
000 iterates. Green diamonds: N = 500 ,
000 iterates. such as existence of SRB measures, exponential decay ofcorrelations, large deviation principles and central limitlaws.Certain implications of Conjecture 1.4 can be testednumerically and we have devised a numerical test ac-cordingly. The logistic map (for which a rigorous theoryexists ) was used as a benchmark for discussing vari- a M u N = 10000 N = 100000 N = 500000 FIG. 4. Graph of the percentage M u of undecided values of ω plotted against the parameter a for the logistic map, usingthe range (ii) I = ( − . , . I = (0 . , .
1) throughout.Blue crosses: N = 10 ,
000 iterates. Red circles: N = 100 , N = 500 ,
000 iterates. ous practical issues regarding the implementation of thetest. We then proceeded to the 40-dimensional Lorenz-96 system (for which no rigorous theory is available) andshowed convincing evidence that the conjecture is truealso in such more complex situations.In our numerical experiments, we have opted to fix aonjecture on the Nature of Attractors 8 log n l og D ω ( n ) FIG. 5. Graph of log D ω ( n ) against log n for the Lorenz-96system with a = 6 . ω = 0 . n = 1 , . . . , N with N = 50 , N = 100 , N = 200 , N = 400 , δ = 0 .
01, blue circles for δ = 0 . δ = 0 .
1. The four graphs are spacedapart vertically so that they can be seen separately, with N =50 ,
000 at the bottom, up to N = 400 ,
000 at the top. randomly chosen initial condition x and then varied theparameter a . The initial condition could also be varied,thereby possibly enlarging the class of examples used fortesting the conjecture. However, for the logistic map thiswould not add anything new since it is known that thereis a unique attractor for each value of a . For the Lorenz-96 system, there is no such uniqueness result, but since weare using the logistic map as a benchmark, it makes senseto keep the two implementations of the test as similaras possible. (As explained in our discussion of the Palisconjecture in the introduction, it is not our aim to exploreissues such as the number of coexisting attractors for afixed parameter a . Rather, we are exploring the natureof all attractors for typical initial conditions, for typicaldynamical systems.)The main focus in this paper has been on discrete-timedissipative systems, but the conjecture applies equallyto continuous time systems and to Hamiltonian systems.For the latter, where the notion of attractor does notmake sense, the conjecture would explore instead the na-ture of the typical asymptotic dynamics ( ω -limit sets) fortypical initial conditions. ACKNOWLEDGMENTS
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