We claim that looking at probability distributions of finite time largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and Poincar\ue9 recurrences in the-quite-delicate case of dynamical systems with weak chaotic properties

Artuso R.

Manchein C.

We claim that looking at probability distributions of finite time largest Lyapunov exponents, and more precisely studying their large deviation properties, yields an extremely powerful technique to get quantitative estimates of polynomial decay rates of time correlations and Poincaré recurrences in the-quite-delicate case of dynamical systems with weak chaotic properties

ARTUSO, ROBERTO

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Instability statistics and mixing ratesRoberto Artuso1,2,* and Cesar Manchein1,3,†1Center for Nonlinear and Complex Systems and Dipartimento di Fisica e Matematica,Università degli Studi dell’Insubria, Via Valleggio 11, 22100 Como, Italy2I.N.F.N. Sezione di Milano, Via Celoria 16, 20133 Milano, Italy3Departamento de Física, Universidade Federal do Paraná, 81531-980 Curitiba, Paraná, BrazilReceived 3 June 2009; revised manuscript received 7 August 2009; published 21 September 2009We claim that looking at probability distributions of finite time largest Lyapunov exponents, and moreprecisely studying their large deviation properties, yields an extremely powerful technique to get quantitativeestimates of polynomial decay rates of time correlations and Poincaré recurrences in the-quite-delicate case ofdynamical systems with weak chaotic properties.DOI: 10.1103/PhysRevE.80.036210 PACS numbers: 05.45.AcI. INTRODUCTIONA general problem of dynamical systems theory concernsboth the evaluation of time or space averages, and an under-standing of how finite order estimates of such averages con-verge to their asymptotic limit. The second feature is tightlyconnected to a quantitative estimation of mixing rates, that ishow time correlations asymptotically decay. Though in ex-tremely simplified models simple Markov chains, latticemodels with a finite transition matrix the connection be-tween convergence of finite order estimates and mixing prop-erties may be established quite easily, the issue in more gen-eral contexts is much more complicated, and many facets ofthe problem still remain as open problems. In particularwhen weakly chaotic systems are considered it is expectedthat sticking to regular structures in the phase space severelydegrades mixing properties, and critical, polynomial decay oftemporal correlations can be observed. We remark that mixedsystems are thought to be generic 1 and that slow polyno-mial decay may influence deeply deterministic transportproperties 2, as the Kubo formula suggests. We here pro-pose to look at large deviation properties of finite time esti-mates as an efficient tool to get quantitative informationabout mixing properties of the system. We emphasize thatthe idea of a relationship between the tails of finite timedistributions and memory effects is not new: for instanceinspection of the tails of finite time Lyapunov exponents wasused to get informations about qualitative changes in thedynamics of coupled maps in 3, the detection of smallregular islands in 4,5, or motion of a particle in a randomtime-dependent potential 6 see also 7 for an example inhyperbolic dynamics. A rigorous analysis, inspiring thepresent work, was presented in 8,9: originally 8 a class ofone-dimensional maps f was taken into account the theoret-ical analysis was extended to higher dimensional systems in10: if we denote by  the Lyapunov exponent, and byPnn the distribution of finite time estimatesnx0 =1nln df nxdx x0 1then, by fixing a threshold ˜ such that 0˜ we mayestimate the fraction of initial conditions yielding an estimatebelow the thresholdM˜n = −˜dnPnn . 2The quantity M˜n asymptotically vanishes if the system isergodic as we always suppose: for weakly chaotic system itmay however decay polynomially,M˜n 1n, 3in such a case a bound is proven 8 for correlation decay,Cn 1n−1, 4where, as usual,Cn = dxxf nxwhere  and  are taken in a suitable class of smooth ob-servables 11 and  is the invariant measure of the system.Actually in 8 M˜n is defined in a slightly different waya full equivalence is established only via further assump-tions: our definition accomplishes a twofold purpose: on theone side it allows a comparison with large deviation esti-mates in 9, and on the other side it yields an easily com-putable quantity.This actually leads to the main point of our paper: weargue that scrutinizing the way in which M˜n decays pro-vides an extremely efficient way of studying quantitativelythe decay of correlations, a key issue in the analysis of fullyand weakly chaotic dynamical systems. As a matter of factdirect quantitative estimates of mixing rates are known to bequite delicate numerically 12, and many efforts have beendevoted to devise how to tackle the problem by alternative*roberto.artuso@uninsubria.it†cmanchein@gmail.comPHYSICAL REVIEW E 80, 036210 20091539-3755/2009/803/0362105 ©2009 The American Physical Society036210-1approaches, such as return time statistics 12–15, which canbe rigorously shown to yield the correct answer in 1d inter-mittency 16.The plan of the paper is as follows: first we suggest, inview of recent results 9, that estimate 4 is not optimal,and the decay properties of M˜n and Cn are the same;then we check such a conjecture for a class of one-dimensional intermittent maps, where polynomial decayrates of correlations are known exactly. We then consider anintermittent area-preserving map, which is a prototype ex-ample of intermittency in higher dimensions. We finally re-investigate, by this technique, the problem of generic corre-lation decay in maps with a mixed phase space: in all casesthe results are coherent with exact results, and corroborateproposed conjectures, in a clean, controlled way.II. LARGE DEVIATIONS AND CORRELATION FUNCTIONThe way in which we defined M˜n 2 is in the form ofa large deviation result 17: a remarkable connection be-tween large deviations properties and correlation decay wasproven in 9 see also 18: if we consider a system forwhich correlation of smooth functions decay polynomiallythe smoothness is essential 11, see also 19 Cnn−then the following results hold, for Césaro finite time aver-ages of an observable :x	n−1	k=0n−1f kx	 − ¯ 		   C, 1n; 5 being again the invariant measure, while ¯ denotes thephase average of the observable. In one dimension, by takingx=ln	fx	 this suggests that bound 4 could be sharp-ened, in such a way that correlation decay and large devia-tions are characterized by the same power-law exponent: onthe other side in dimensions higher than one finite estimatesof Lyapunov exponents computed for instance by selectingthe largest eigenvalue of the product of jacobians along atrajectory lack composition structure of Césaro sums. Ourclaim is that, despite these provisos, the asymptotic decay ofM˜n quantitatively reproduces exactly the decay of corre-lations, and inspection of such a quantity provides an excel-lent tool to investigate the weak-chaos regime.Before applying such a method to a number of dynamicalsystems, we point out that the way in which we consider theexponent  is as the lowest possible value in a suitable classof zero mean smooth functions: it is well known that spe-cific choices of observables may yield faster decay see 20for a rigorous example in one-dimensional intermittent dy-namics.A. One-dimensional system: the Pikovsky mapOur first benchmark tool is represented by Pikovsky map21 Tz, which is implicitly defined byx = 12z1 + Tzxz, 0  x  1/2zTzx +12z1 − Tzxz, 1/2z x  1, 6while for negative values of x, the map is defined asTz−x=−Tzx 21. Such a map has remarkable features, asin x=1 two marginal fixed points of the Pomeau-Manneville type are present and z is the corresponding in-termittency parameter, while for any value of z the invariantmeasure is Lebesgue 21, as it can be seen by direct inspec-tion of the Perron-Frobenius operator. For this map the cor-relations decay polynomially, with an exponent that dependsupon the intermittency parameter z 22,Cn  n−1/z−1. 7This decay is very well reproduced by numerical simulationsfor M˜n, as we see from Fig. 1. We remark that the nu-merical simulations for the Pikovsky map were performed byusing 106 initial conditions uniformly chosen in the phasespace −1,1.When we want to check how well correlation decay isreproduced by large deviations, expressed by the decay ofM˜n, we have to be careful to take the large deviationparameter ˜ sufficiently far from the center of the distribu-tion: even if the general expectation is the asymptotic decaylaw will not depend upon the cutoff, finite time estimatesrequire that M picks up only contributions in the tail of thedistribution. As a matter of fact any choice of the cutoff ˜will determine a time scale ˜ such that only after such atransient the asymptotic decay is reproduced correctly. Thisis well illustrated in Fig. 2 where were chosen three values of˜ and we obtained the asymptotic decay for all cases, butwith different time scale ˜.In Fig. 3 we show the shape bimodal of normalizedfinite time distribution Pnn for map 6: as ergodicity pre-dicts, they tend to a Dirac  centered on  in the asymptoticlimit, but the tails are polynomially “fat” for finite times. Inthe inset inside Fig. 3 we can detect an interesting feature:10−410−310−210−1100101103 104 105Mλ(n)nz=1.5, ξ=2.00 ± 0.01z=2.0, ξ=1.00 ± 0.01z=4.0, ξ=0.33 ± 0.01~FIG. 1. Color online Numerical data symbols, and regressionfits lines of power-law decay of M˜n for Pikovsky map forthree different values of the intermittency exponent. The fit for thecase z=1.5 starts at n=3000. Theoretical values for , from Eq. 7,are =2, =1, and =1 /3.ROBERTO ARTUSO AND CESAR MANCHEIN PHYSICAL REVIEW E 80, 036210 2009036210-2by analyzing the distribution of Lyapunov exponents closerto zero in logarithmic scale it is possible see how the distri-bution obeys a power-law decay with time.As a final remark we point out that, though large deviationestimates such as Eq. 5 are symmetric with respect to theasymptotic phase average, from a physical point of view it isquite natural to focus the interest on the small instabilitybranch, as it is precisely anomalous proliferation of almoststable segments of trajectories that induces transition fromexponential to power-law decay of temporal correlations.B. Two-dimensional systems: a family of area-preservingmapsThe second dynamical system we consider is a family ofarea-preserving maps on the two torus − ,2, dependingon two parameters  and ,M,:yn+1 = yn + fxn mod 2xn+1 = xn + yn+1 mod 2, 8where fxn is defined byfxn = xn − 1 − sinxn. 9A map of this family was introduced in 23, and differentfeatures were analyzed in 24–27: we recall a few of therelevant properties. When 	0 the map is hyperbolic, whilefor =0 the fixed point at 0,0 becomes parabolic: in such acase dynamics close to the fixed point depends on the valueof , which plays the role of an intermittency parameter:correspondingly the decay of correlations is exponential inthe former case, while in the latter a power law is expected,with an exponent depending on . In 27 the following lawwas proposed:Cn  n−3+1/3−1; 10in the case =1 a rigorous bound is proven 26	Cn	n−2 while Eq. 10 predicts an exponent −3.Our numerical experiments encompass both regimes: inFig. 4a a hyperbolic parameter choice leads to an exponen-tial decay rate for M˜n, while in the intermittent case apower law is observed, with an exponent in agreement withEq. 10 see Fig. 4b. In this case the numerical resultswere obtained by using 106 and 108 initial conditions uni-formly distributed in the phase space, respectively.The case reported in Fig. 4b allows to scrutinize thenumerical virtues of our approach with alternative methodsthat were employed in 27: it provides estimates as sharpas the analysis of return time statistics by employing how-ever much less initial conditions, while it outperforms directcomputation of correlations.C. Ensemble of modified standard mapsThe last case we present concerns a general issue, namelywhether universality properties are exhibited for Hamiltoniansystems with a hierarchical mixed phase space. This is a10−410−310−2101 102 103 104 105Mλ(n)n~FIG. 2. Color online Decay of M˜n for system 6 withz=2.0, for different cutoffs ˜ , where the red gray curve is a re-gression fit of power-law decay with exponent =1.0.00.10.20.30.40.50.60.70.80.9-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.01Pn(λ n)λnn=2000n=5000n=7000n=1000010-4 10-3 10-2 10-1 100FIG. 3. Color online Distribution of Lyapunov exponentsPnn for map 6 with z=2.0. Inset: a magnification of Pnn isshown in logarithmic scale. The distribution of Lyapunov exponentscloser to zero show a polynomial decay with time.10−410−310−210−11001 2 3 4 5 6 7 8 9 10Mλ(n)103 n~10−810−710−610−510−410−310−210−1100103 104 105Mλ(n)n~(b)(a)FIG. 4. Color online Decay M˜n symbols together with a regression fit full lines for map 8 with a =0.5 and =1 exponentialrate decay 0.870.01 and b =0 and =1 polynomial rate decay 3.050.05. Both fits were done by starting at n=3000.INSTABILITY STATISTICS AND MIXING RATES PHYSICAL REVIEW E 80, 036210 2009036210-3much debated issue we refer to 28,29 for relevant refer-ences over the last twenty years: in particular 28 suggests,on the basis of a Markov tree model 30 with random scal-ing factors for transition probabilities, the existence of anasymptotic algebraic decay for the Poincaré recurrences. Theauthors also discuss in detail the numerical difficulties inprobing such a claim: in particular, following 31,32 theyadvocate that an average over different Hamiltonian systemssignificantly reduces the extremely long times that are re-quired for a single trajectory to sample fine details of thephase-space structure: this procedure allows them to checkfor an ensemble of area-preserving maps their proposal, andthe polynomial decay exponent of Poincaré recurrences isestimated as 1.570.03, which, according to 12–15,suggests for correlations the decay lawCn  1n−1. 11The exponent of algebraic decay of Poincaré recurrences wasalso numerically investigated in 29 with an estimate1.600.05, for a different ensemble of area-preservingmaps, which we also utilize in the present framework.The ensemble of modified standard maps on − ,2considered here are given byPK,K†:pn+1 = pn + K sinxn + K†xn+1 = xn + pn+1 , 12where K and K† are the nonlinear parameter and magneticfield, respectively. They are uniformly chosen in theintervals K  ,1.2· and K† 0.0,0.4·. We have used106 initial conditions taken uniformly in the squarex0 −0.1,0.1, p0 −0.1,0.1, well inside the chaotic re-gion. The behavior we show in Fig. 5 apparently arises aftera few 10 random realizations of the maps: the overalldecay rate with an exponent 0.570.05 is stable, as well asthe oscillations that may be noticed in Fig. 5. Wether suchoscillations are due to finite-size effects, corrections to scal-ing, or multifractality 33, is a feature that in our opiniondeserves further investigations. Again we remark that thepresent method has an important advantage that it needs a“small” number of initial conditions and realizations of themap to show the asymptotic decay of M˜n as comparedwith traditional procedures using direct computation of cor-relation functions or Poincaré recurrences.III. CONCLUSIONSWe have shown how the analysis of large deviations forLyapunov exponents provides a very efficient way to inves-tigate correlation and Poincaré recurrences decay for dy-namical systems: in particular when dealing with weaklychaotic systems, with associated polynomial correlation de-cay, the power-law exponent has been computed from thedistribution function of finite time Lyapunov exponents withremarkable accuracy.We also remark that in all our tests this procedure per-forms quite well numerically: for instance in our last ex-ample the estimate has been obtained with a computationaleffort significantly reduced with respect to former simula-tions 28,29. Another interesting point we plan to investi-gate in the future is to study the decay law of correlationfunction in higher dimensional systems where the nonlinear-ity is not uniformly distributed along different unstable di-rections 34.ACKNOWLEDGMENTSOne of the authors C.M. gratefully acknowledges CNPqBrazil for support and the Center for Nonlinear and ComplexSystems in Como for hospitality. We thank Marcus W. Be-ims, Giulio Casati, Giampaolo Cristadoro, Stefano Luzzatto,Matteo Sala, and Sandro Vaienti for very useful correspon-dence and discussions.1 L. Markus and K. R. Meyer, Mem. Am. Math. Soc. 144, 11974.2 R. Artuso and G. Cristadoro, in Anomalous Transport: Foun-dations and Applications, edited by R. Klages, G. Radons, andI. M. Sokolov Wiley-VCH, New York, 2008.3 M. Falcioni, U. Marini Bettolo Marconi, and A. Vulpiani,Phys. Rev. A 44, 2263 1991.4 S. Tomsovic and A. Lakshminarayan, Phys. Rev. E 76, 0362072007.5 M. W. Beims, C. Manchein, and J. M. Rost, Phys. Rev. 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Shepelyansky, in Proceedings of theIXth International Conference on Nonlinear OscillationsNaukova Dumka, Kiev, UIkraine, 1981, Vol. 2, p. 420;Princeton University Report No. PPPL-TRANS-133, 1983unpublished.32 F. Christiansen and P. Grassberger, Phys. Lett. A 181, 471993.33 V. Afraimovich and G. M. Zaslavsky, Phys. Rev. E 55, 54181997.34 C. Manchein, M. Beims, and J. Rost, e-print arXiv:0907.4181.INSTABILITY STATISTICS AND MIXING RATES PHYSICAL REVIEW E 80, 036210 2009036210-5

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