We present a more general description of the technique of Tsallis and co-workers, to study the behaviour of dynamical systems. We enlarge to any dimension the power-law generalization of the classical Lyapunov exponents, that was introduced by Tsallis and co-workers in one dimension. We apply the new generalization to the two-dimensional Hénon map and consider some cases.Izlažemo poopćen opis metode Tsallisa i sur. za proučavanje svojstava dinamičkih sustava. Proširujemo poopćenje potencijskog zakona klasičnih Lyapunovih eksponenata na proizvoljan broj dimenzija po uzoru na poopćenje koje su uveli Tsallis i sur. za jednu dimenziju. Primjenjujemo to poopćenje na dvodimenzijsku Henonovu mapu i razmatramo neke slučajeve

Fasanol, Silvia E.

Mesón, Alejandro M.

Salerno, Mirta N.

English

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Printed ISSN 1330–0008Online ISSN 1333–9125CD ISSN 1333–8390CODEN FIZAE4POWER-LAW GENERALIZATIONS FOR STUDYING DYNAMICALSYSTEMSALEJANDRO M. MESÓNa,b, SILVIA E. FASANOa,c and MIRTA N. SALERNOb,caInstituto de F́ısica de Ĺıquidos y Sistemas Biológicos (IFLYSIB)-UNLP-CONICETcc. 565 - (1900) La Plata, Argentina, E-mail address: meson@iflysib.unlp.edu.arbGrupo de Aplicaciones Matemáticas y Estad́ısticas de la Facultad de Ingenieria(GAMEFI), Departamento de Fisicomatemática, Facultad de Ingenieŕıa, UniversidadNacional de La Plata, ArgentinacDepartamento de Matemática, Facultad de Ciencias Exactas Universidad Nacional deLa Plata, ArgentinaReceived 12 January 2006; Revised manuscript received 22 October 2007Accepted 5 December 2007 Online 15 February 2008We present a more general description of the technique of Tsallis and co-workers,to study the behaviour of dynamical systems. We enlarge to any dimension thepower-law generalization of the classical Lyapunov exponents, that was introducedby Tsallis and co-workers in one dimension. We apply the new generalization tothe two-dimensional Hénon map and consider some cases.PACS numbers: 05.45.-a, 05.45.Jn UDC 530.182Keywords: dynamical systems, Tsallis technique, Lyapunov exponents, two-dimensionalHénon map1. IntroductionTo study the behaviour of dynamical systems, there is an important tool: theLyapunov exponents. For a particular case of one-dimensional systems, the meaningof these numbers is the following: let f : I → I be an interval map. A discrete-timedynamical system can be presented by settingx(n + 1) = f(x(n)) , (1)FIZIKA A (Zagreb) 16 (2007) 4, 223–232 223mesón et al.: power-law generalizations for stydunig dynamical systemsand the separation after time N is evaluated byx1(N) − x2(N) = f ′(c)[x1(0) − x2(0)] . (2)The growing rate of the separation obeys the exponential lawDx(0)(fn(δx(0))) ∼ eλn, λ ∈ R , (3)The number λ is the Lyapunov exponent for the dynamical system (1). Anindicator for the occurrence of chaotic behaviour is the “sensitive dependence oninitial conditions”, i.e. there is an ǫ > 0, such that for every neighborhood Uof a point x, there is a y ∈ U with |fN (x) − fN (y)| > ǫ, for some integer N .Therefore, if the Lyapunov exponent is positive, chaotic behaviour occurs. In fact,we consider a change x(0) → x(0)+δx(0), for a variation of the time t, and we haveδx(t) ∼ δx(0)eλt, so λ > 0 indicates a sensitive dependence on initial conditions.In an arbitrary dimension d, the Lyapunov exponents for a differentiable mapf : Rd → Rd are defined as follows: let L be a map defined on Rd which takesvalues in the space of linear transformations of Rd in Rd (or equivalently in thespace d × d matrices). The map L can be taken, for instance, in such a way thatfor any x ∈ Rd, Lx ≡ L(x) : Rd → Rd be the linear approximation (the differentialmap) of f in x. We denoteLnx ≡ Lfn−1(x) . . . Lf(x) ,where the dots are understood as products of matrices. For x, v ∈ Rd, letλ(x) = λ(x, v, f) = limn→∞1nlog ||Lnxv|| (4)if the limit exists, where for x = (x1, x2, . . . , xd) we take ||x|| =√∑di=1 x2i .The number λ(x) = λ(x, v, f) is called the Lyapunov exponent with respect toL = L(f) in (x, v).For a real number λ and x ∈ Rd, we denoteEλ(x) = {v ∈ Rd : λ(x, v, f) ≤ λ} .Notice that if λ1 ≥ λ2, then Eλ1(x) ⊃ Eλ2(x). Furthermore, if x ∈ Rd, there existsan integer m(x) ≤ d, such that there is a collection of numbers λ1, λ2, . . .λm(x)and linear subspaces Rd = Eλ1(x), Eλ2(x), . . . Eλm(x)(x) = {0}, withλ1(x) > λ2(x) > . . . λm(x)(x) ,Eλ1(x) ⊃ Eλ2(x) ⊃ . . . ⊃ Eλm(x)(x) .224 FIZIKA A (Zagreb) 16 (2007) 4, 223–232mesón et al.: power-law generalizations for stydunig dynamical systemsIf v ∈ Eλi(x) − Eλi+1(x), then λ(x, v) = λi(x). The Lyapunov spectrum is nowdefined asSpx(L, f) = {λi(x) : i = 1, 2 . . . ,m(x)} .If it is considered a f−invariant ergodic measure µ, the functions x → m(x),x → Eλi(x) and x → λi(x) are constant except for µ–null sets. For more detailsabout this subject and for the demonstrations of the facts mentioned above see,e.g., Refs. [1] or [2].Tsallis et al. [3] proposed a power-law generalization of the Lyapunov exponentsassociated to one-dimensional discrete dynamical systems in order to study thebehaviour at the edge of chaos, or, more generally, at critical values of controlparameters of this kind of systems. They have been presented in the formx(n + 1) = fα(x(n)) , (5)with fα a family of interval maps depending on a bifurcation parameter α. Ofcourse, there is a dependence of the exponent on the parameter and for the criticalvalues αc, i.e. those for which λ(αc) = 0, the description of the system in thesepoints could not be well understood. Thus in Ref. [3], a one-parameter family {λq}has been introduced, such that the Lyapunov λ is contained as one of its members,specifically when q → 1. From now on, we will refer to λq=1(x) as λ(x), where q = 1actually means q → 1. The aim of the above mentioned paper is to get a parameterq such that λq(αc) /= 0 when λ(αc) = 0. The results of the above quoted paper ofTsallis et al. have raised many developments in several directions, (see Ref. [4]). Inthe preset article, we propose to formulate a presentation of the techniques fromRef. [3] for dynamical systems with any number of dimensions.It should be remarked that the dependence is not continuous in general, andthis fact causes problems in the physical context. Taking into account that Lya-punov exponents measure the stability of the system, it would be good that in anexperiment they cannot be modified substantially by perturbations of the inter-nal parameters of the system. In a purely experimental situation, the quantitiesmeasured for predictions could be “smoothed” by instrumental procedures.Another feature to observe is the following: in the case of having, at least ina neighborhood of αc, a continuous dependence on the bifurcation parameter, wewould have eventually a transition λ(αc − ǫ) → λ(αc + ǫ) from chaotic to stable (orstable to chaotic) state, and αc is a threshold of chaos. In general, this situationis not expected, and so we could not ensure by a numerical computation thattransition occurs (even for a small ǫ), i.e., that αc is a genuine threshold of chaos.We shall use the term “possible threshold of chaos”.In this article we treat dynamical systems in discrete time, hence we denotethem byx(n + 1) = fΛ(x(n)) , (6)where Λ is a bifurcation parameter set (α1, α2, . . . αd), i.e., the behaviour of thesystem can change by varying some αi. The technique proposed in Ref. [3] for theFIZIKA A (Zagreb) 16 (2007) 4, 223–232 225mesón et al.: power-law generalizations for stydunig dynamical systemsone-dimensional systems is applied in our context when the Lyapunov spectrum issuch that for q = 1, one of the exponents vanishes and the others are negative. Sothe behaviour for the vanishing one is not well described.In the study of chaotic behaviour, we frequently find invariant sets with a com-plex mathematical structure. The multifractal analysis essentially deals with de-compositions of these kind of sets. One of the main problems is to evaluate somedimensions, e.g. the Hausdorff dimension, correlation dimension. In this article, weestimate some of these dimensions. Herein we treat discrete dynamical systems, andfor the continuos case the considered dynamics are of the flows {ϕt}t∈R. Lyapunovexponents λ ({ϕt}) for the flow {ϕt} are calculated as λ (ϕt=1).The scheme and goals of this article are: in Sec. 2 we present a power-lawgeneralization in the spirit of Tsallis et al. for abstract dynamical systems andillustrate this procedure for the two-dimensional Hénon map.2. Power-law generalizationsIn Ref. [3], the authors proposed the following power-law generalization of theLyapunov exponents∣∣∣∣∆x(t)∆x(0)∣∣∣∣∼ [(1 − q)λq]1/(1−q) × t1/(1−q) .Let f : Rd → Rd be a differentiable map and as in the introduction, Lx ≡ L(x) :Rd → Rd is the linear approximation of f in x. Inspired in the above formula forthe one-dimensional case, we setλq(x) = λq(x, v, f) = limn→∞[(1/n)(1 − ||Lnxv||(1−q))q − 1], for q /=1 . (7)Now we have a power law: ||Lnxv|| ∼ [(1 − q)λq]1/(1−q) × n1/(1−q), therefore||Lnxv|| and n1/(1−q) change asymptotically in the same way.To compare the notations with those in the paper of Tsallis et al.: ||Lnxv||corresponds to |∆x(t)/∆x(0)| and n to the “time” t.We insist that the objective is try to describe the behaviour of dynamical sys-tems for which in the Lyapunov spectrum occurs when λ1(Λc) vanish at a criticalvalue Λc and the other λ1(Λc) are negative. The proposed power-law generalizationenables us to find an optimal qop such that λqop(Λc) /= 0, and so the behaviour ofthe system can be described.Remark: λ mean the numbers obtained by taking the limit q → 1 in Eq. (7),i.e. the classical Lyapunov exponents.The parameter q may be fitted in the case when the boundary of the possiblyfractal set (n, ||Lx,Λc ||) has a shape that can be fitted with a straight line. It is226 FIZIKA A (Zagreb) 16 (2007) 4, 223–232mesón et al.: power-law generalizations for stydunig dynamical systemshoped that this fractality “decreases” as the bifurcation set is moved away fromΛc. Here Lx,Λc means the linear approximation of f restricted to the set Λc.We illustrate these ideas in the two-dimensional discrete Hénon map,x(n + 1) = y(n) + 1 − α(x(n))2,y(n + 1) = βx(n) .(8)where we set Λ = (α, β). Some features about this map are well known: for thespecial values α = 1.4, β = 0.3, there is positive Lyapunov exponent λ1(α, β), andthe other one is λ2 = log |J | − λ1, where J is the Jacobian of the map, |J | = β.Numerically it is known that λ1(1.4, 0.3) ≃ 0.42, and so λ2(1.4, 0.3) ≃ 1.62 [5].We fix now β = 0.3 and sweep α in order to find a critical value Λc = (αc, 0.3),such that λ1(αc, 0.3 = 0).In Fig. 1 we see a plot of λ1 versus α for β fixed at 0.3 and 0.0 ≤ α ≤ 1.4.The first positive λ1 value occurs at α = 1.058047982596 . . .. To the right of thisα, although many λ1 > 0 appear, one also finds a scattering of subintervals onwhich λ1 < 0, and each such λ1 corresponds to a periodic attractor. The fact thata period 7 attractor occurs for α = 1.3 had been already noticed [6].Fig. 1. Lyapunov exponents for the Hénon map, with β = 0.3, vs. α ∈ [0, 1.4].We shall focus our attention at the values αc = 1.058047982596 . . . and αc =1.26182278611 . . . which yield values of λ1 = 0.95 × 10−10 and λ1 = 0.43 × 10−10,respectively.Recall that ||Lnx || =∏n−1i=0 ||Lfi−1(x)||, thus we consider the estimationL(n) = log ||Ln(x(n),y(n))Λcv|| =n−1∑i=0log ||Lfi−1(x(i),y(i))Λcv|| , (9)FIZIKA A (Zagreb) 16 (2007) 4, 223–232 227mesón et al.: power-law generalizations for stydunig dynamical systemswith the initial coordinate values x(0) = y(0) = 0 and the initial vector v0 = (1, 0).However, the same result is obtained with different unitary vectors, e.g. (0, 1),(1/√2, 1/√2), etc.Fig. 2. log-log of∥∥LnxΛc∥∥ vs. the number of iterations n, for x(0) = y(0) = 0,α = αc = 1.26182278611.A plot of L(n) vs. log n (Fig. 2) with n = 8000 and αc = 1.26182278611 . . .,reveals the fractal nature of this set, as expected, and besides that the points forlarge values of n are placed on a line with the slope m; it suggests that the optimalchoice for the parameter q, taking into account the asymptotic behaviour expressedbelow Eq. (7), is11 − qop= m ≃ 1.4 . (10)which leads to qop ≃ 0.285714, and therefore λqop1 = 0.035 . . ., which indicates thatthere is a chaotic behaviour, and Λc is a possible threshold of chaos.In Fig. 3 is shown how the fractality “decreases” as the bifurcation parameteris moved away from the threshold of chaos, i.e. the value αc = 1.26182278611 . . ..Indeed this is a genuine threshold of chaos, at least locally. Investigations about thebehaviour of Lyapunov exponents of the two-dimensional Hénon map, were doneby Feit [6].228 FIZIKA A (Zagreb) 16 (2007) 4, 223–232mesón et al.: power-law generalizations for stydunig dynamical systemsFig. 3. log-log of∥∥LnxΛc∥∥ vs the number of iterations n, for x(0) = y(0) = 0,αc = 1.26182278611; (a) α = αc + 10−3; (b) α = αc + 10−5;(c) α = αc − 10−5; (d)α = αc − 10−3.With a similar procedure applied to αc = 1.058047982596 . . . (Fig. 4), we ob-tained11 − qop= m ≃ −4.285 .which leads to qop ≃ 1.23333 and λqop1 (Λc) = −0.000909 . . ., which indicates non-chaotic behaviour. In Fig. 5 we observe the same behaviour as in Fig. 3 while onemoves away from the threshold of chaos.In the last two columns of the tables above, we calculate the Hausdorff dimension(d) and the correlation dimension (ν) [7] for the fractal set in Fig. 2.The Hausdorff dimension d is a purely geometric measure, because it is indepen-dent of the frequency with which a trajectory visits various parts of the attractor.The correlation dimension ν is obtained from the correlations between randompoints on the attractor. As it can be observed, ν < d for all values of the α param-eter.FIZIKA A (Zagreb) 16 (2007) 4, 223–232 229mesón et al.: power-law generalizations for stydunig dynamical systemsFig. 4. log-log of∥∥LnxΛc∥∥ vs the number of iterations n, for x(0) = y(0) = 0,α = αc = 1.058047982596.Fig. 5. log-log of∥∥LnxΛc∥∥ vs the number of iterations n, for x(0) = y(0) = 0,αc = 1.058047982596; (a) α = αc + 10−3; (b) α = αc + 10−5;(c) α = αc − 10−5; (d)α = αc − 10−3.230 FIZIKA A (Zagreb) 16 (2007) 4, 223–232mesón et al.: power-law generalizations for stydunig dynamical systemsTABLE 1. Characteristic dimensions for the fractal set of Fig. 2 with αc =1.2618.....limn→∞ ||Lnxv||Hénon 2Dx(0) = 0y(0) = 0HausdorffdimensionCorrelationdimensionλ1 > 0 eλ1n α = αc + 10−5 0.58424 0.53737λ1 < 0 eλ1n α = αc − 10−5 0.52221 0.45721λ1 = 0λ1q > 0[(1 − q)λ1q n]1/(1−q) αc = 1.2618...q ∼= 0.285714 0.53861 0.50034TABLE 2. Characteristic dimensions for the fractal set of Fig. 2 with αc = 1.05904.limn→∞ ||Lnxv||Hénon 2Dx(0) = 0y(0) = 0HausdorffdimensionCorrelationdimensionλ1 > 0 eλ1n α = αc + 10−5 0.59302 0.55878λ1 < 0 eλ1n α = αc − 10−5 0.41428 0.40987λ1 = 0λ1q > 0[(1 − q)λ1q n]1/(1−q) αc = 1.05804...q ∼= 1.23333 0.44412 0.44400Fig. 6. log-log of∥∥LnxΛc∥∥ vs the number of iterations n, for x(0) = y(0) = 0,α = αc = 1.06328909597577.Another interesting value of the parameter α for which λ1(α, 0.3) ≃ 0 is α ≃1.063289095. For this value we calculate λ1qop(Λc) = 0.66× 10−9 (Fig. 6). It can benumerically shown that period-doubling can occur, leading to the onset of chaos[8].FIZIKA A (Zagreb) 16 (2007) 4, 223–232 231mesón et al.: power-law generalizations for stydunig dynamical systems3. ConclusionThe results for the model studied here show that the techniques introduced byTsallis can be applied to analyze chaotic behaviour in dynamical systems with anynumber of dimensions.AcknowledgementsOne of the authors (A.M.) wishes to thank to the “Centro Brasileiro dasPesquisas Fisicas” and Professor C. Tsallis for the hospitality and fruitful discus-sions.References[1] P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, Berlin (1982).[2] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems,Cambridge Univ. Press (1995).[3] C. Tsallis, A. R. Plastino and W. H. Zheng, Chaos, Solitons and Fractals 8 (1997) 885.[4] J. P. Boon and C. Tsallis, Eds. Europhysics News (Special Issue) 36 (Novem-ber/December 2005).[5] D. Ruelle, Chaotic Evolution and Stange Attractors, Notes prepared by S. Isola, Cam-brige Univ. Press, London (1987).[6] D. S. Feit, Commun. Math. Phys. 61 (1978) 249.[7] P. Grassberg, and I. Proccacia, Physica D 9 (1983) 189.[8] S. Strogatz, Nonlinear Dynamics and Chaos, Addison-Wesley, TOWN?, Canada(1984).POOPĆENJE POTENCIJSKOG ZAKONA ZA PROUČAVANJE DINAMIČKIHSUSTAVAIzlažemo poopćen opis metode Tsallisa i sur. za proučavanje svojstava dinamičkihsustava. Proširujemo poopćenje potencijskog zakona klasičnih Lyapunovih ekspo-nenata na proizvoljan broj dimenzija po uzoru na poopćenje koje su uveli Tsallis isur. za jednu dimenziju. Primjenjujemo to poopćenje na dvodimenzijsku Hénonovumapu i razmatramo neke slučajeve.232 FIZIKA A (Zagreb) 16 (2007) 4, 223–232

Poopćenje potencijskog zakona za proučavanje dinamičkih sustava

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