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Pergamon PII: SO960-0779(%)00167-l

Power-law Sensitivity to Initial Conditions-New Entropic Representation

C. TSALLIS?, A. R. PLASTINO and W.-M. ZHENGS

Centro Brasileiro de Pesquisas Ffsicas Rua Xavier Sigaud 150. 22290-180, Rio de Janeiro, Brazil (Accepted 26 November 1996)

Abstract-The exponential sensitivity to the initial conditions of chaotic systems (e.g. D = 1) is characterized by the Liapounov exponent A, which is, for a large class of systems, known to equal the Kolmogorov-Sinai entropy K. We unify this type of sensitivity with a weaker, herein exhibited, power-law one through (for a dynamical variable x) lim

(equal to e”” for 9 = 1, and proportional, for large r, to f ti(,t)-.,, [A,@)]@#)] = [I + (1 - q)A,r]+‘) i’(i-q) for 9 # 1; 9 E ?)I). We show that A, = K,

(Vq), where K,, is the generalization of K within the non-extensive thermostatistics based upon the generalized entropic form S, = (1 - xi p3/(9 - 1) (hence, 5, = -xi p, In p,). The well-known theorem A, = K, (Pesin equality) is thus extended to arbitrary 9. We discuss the logistic map at its threshold to chaos. at period doubling bifurcations and at tangent bifurcations, and find 9 50.2445, 9 = 5/3 and 9 = 3/2, respectively. 05.45. + b; 05.20. - y; 05.90. + m. 0 1997 Elsevier Science Ltd

Chaos is a ubiquitous phenomenon appearing in a great variety of systems including turbulence, electronic and optical devices, fluids, magnetism, biology, stock markets and many others (see Ref. [1] for a review). One of its most prominent aspects is the sensitivity ro initial conditions (sensitivity in fact to any numerical rounding at any calculational step, not necessarily at the initial time). It is characterized by the Liapounov exponent A, (the subindex 1 will soon become clear) defined, for say the simple case of a one-dimensional dynamical variable x, through

Ax(t) - Ax(O)el’(Ax(O)-+ 0,r + s) (1)

If A, > 0 (A, < 0) the system is said to be sensitive (insensitive) to the initial conditions, the so-called marginal case being A, = 0. Besides the relatively trivial cases of period doubling

and tangent bifurcation points, the cumulating point of the period doubling bifurcation is precisely the marginal case which corresponds to the threshold to chaos, and which will be the main purpose of the present work. But let us still consider the generic case A, #O. It is clear that, whenever Al > 0, we loose information on the system [more precisely, on its actual value x(t)] along time. To characterize this loss of information, Kolmogorov and Sinai [2] introduced the so-called Kolmogorov-Sinai entropy Kl which is basically the increase, per unit time, of the Boltzmann-Gibbs-Shannon entropy S, = - Cz, pi lnp, where W is the total number of possible configurations and {pi} its associated probabilities (notice that equiprobability yields the well-known expression S, = In W). Many basically equivalent definitions of Kl exist (see, for instance, [3]). We shall use here a very simple one, namely that adopted by Hilbom 141. Following Hilborn, we consider the evolution of an ensemble of identical copies of our system. Let pi now stand for the fractional number of ‘points’ of the

tE-mail: tsallis@cat.clopf.br. *Permanent address: Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, People’s Republic of China.

X86 C. TSALLIS et al

ensemble that are in the ‘i’ cell of a suitable partition of phase space. The size of the cells is characterized by a linear scale 1. The K-entropy can be cast as

K1 = lii kir Jilix $T (S,(N) - S,(O)), (2)

where S,(O) and S,(N) stand, respectively, for the entropies of the ensemble evaluated at the times t = 0 and t = Nr. If we make the simplifying assumption that at time Nz there are W(N) occupied cells (whereas there was just one at t = 0), each one with the same occupation number, then we can write

K, =lii$nlirliZ$rln W(N), (3) where we have taken into account that S,(O) = 0. Consistently with equation (1) we have

W(N) = W(l)eAlN5 (41

which when substituted into equation (3) yields the well-known Pesin equality (for 1D dynamical chaotic systems) [4]

K, = A, (5)

which quantitatively relates the sensitivity to initial conditions with the rhythm of loss of information. To prove equation (5), we have reproduced Hilborn’s arguments. However, more sophisticated proof of this equality can be found in Ref. [3]. Let us now focus on the marginal case h, = 0. As we shall soon illustrate, this is a very rich and complex case, strongly reminiscent of what happens at the critical point of all thermal equilibrium (or geometrical, such as percolation) critical phenomena. To just say that A, = 0 is a very poor description of its richness, intimately connected to fractality. To show how this poverty can be overcome is the precise aim of the present note. Let us first generalize equation (1) as follows:

Ax(t) - AX(O)[l + (1 - q)h,t]“’ -yx(o)-+ 0, q E 8) (6)

We verify (i) that this equation is identically satisfied for t = 0 (Vq), (i) that q = 1 recovers equation (1) and (i) that q f 1 yields, for large times, the power-fuw

Ax(t) - [(l - qp,p t”‘-yAx(O)~o, t-9 30) (7)

The particular generalization of equation (1) introduced in equation (6) totally follows along the lines of the non-extensive thermostatistics introduced by one of us [5] to describe systems which involve long-range interactions or long-range microscopic memory or fractal space-

time, and which has been recently applied to a variety of physical problems, like self-gravitating systems [6], anomalous diffusions of the Levy [7] and correlated [8] types, two-dimensional turbulence in pure-electron plasma confined in a conducting cylinder in the presence of an external 507 G magnetic field [9], the solar neutrino problem [lo], cosmology

[l], simulated-annealing optimization techniques [12], among others (see also [13] for a recent connection to the dynamic linear response theory). Within this generalized formalism, the Boltzmann-Gibbs equilibrium distribution emp” is extended into [l - (1 - q)PH]““-q’, which explains the form adopted in equation (6) to unify exponential and power-law

sensitivities to the initial non-extensive formalism, S1

Power-law sensitivity to initial conditions 8X7 conditions. Let us now turn to the entropy. Within the is generalized into [5] which, for equiprobability, becomes WI-9 - 1

s, = 1-q (9)

The use of equation (9), instead of S, = In W, yields (along Zanette’s lines [14]) the following generalization of the Kolmogorov-Sinai entropy

which, under the assumption of equiprobability, reduces to

1 [W(N)]“-Y’ - 1 K, = li+q~~i;yzNr l-q (1)

We must remark that our generalizations K, of the KS entropy are different from the generalizations K(P) based upon Renyi information, usually called ‘Renyi entropies’ in the literature of the thermodynamics of chaotic systems [15] (sometimes the parameter characterizing these generalizations of the KS entropies is called q instead of p [16]. This parameter q should not be confused with our q). Consistently with the behavior indicated in equation (6), we have

W(N) = W(l)[l + (1 - q)A,Nz]h (12) which, when substituted into equation (l), immediately yields (for 1D dynamical systems)

K, = A, (13)

This result generalizes that expressed in equation (5) and unifies (within a single scenario for both exponential and power-law sensitivities to initial conditions) the connection between sensitivity and rythm of loss of information. Let us now illustrate these concepts with a simple example, namely with the logistic map

The bifurcation road to chaos accumulates on the critical point a,. = 1.40115518909 | [17]. In |

X r+,=l-ax: (x,E[-l,l];aE[0,2];t=0,1,2,3...) (14) the A(O)-+ 0 limit we have

L = In c lim ““‘) = ($: ) In 12a,x, 1

Ax(O)43 Ax(O)

The results concerning the time evolution of L are shown in Fig. 1 for a single initial condition x,, and in Fig. 2 for an ensemble, i.e. for a great number of initial conditions (uniformly distributed in the interval [ - 1, 1) superimposed.

The straight line in Fig. l(a) with the slope In cy /ln 2 very satisfactorily fits the upper edge points, thus exhibiting a power law. This slope has been obtained as follows. The scale of trajectory splitting from a 2”- to a 2(“+‘) -cycle is described by the universal scaling constant a = 2.5029... [17]. After 2” steps of iterations, the smallest splitting of the order ay-2n

_, .,_, B :.a. 1'. n m | - . . . _. . - . . r: |

I 1'

- - | - - . . |

of’1 ’ ’ ’ ’ 1 1 I ’ 0 10.0

In N

(cl

o 0]‘0°

0 10.0 In N

(e) 0. : 1 / I , , , , /

I. -I. .\ ;\ ‘1

(Parte **1** de 3)