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Pergamon
Chnor, Solrrom & Fracrds. Vol. 8, No. 6, pp. 885%XY I, 1997 0 1997 Elsevier Science Ltd
PII: SO960-0779(%)00167-l
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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)e^l’(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: [email protected]. *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 (^) (
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), (ii) that q = 1 recovers equation (1) and (iii) 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 [ll], 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
t
I 11'
,.,, B :.a. 1'.-.. n. _.. - .. r:m.. I
I/'^ -^ -^.^.^ -^ -^ ..^ -I
I’
of’1 ’ ’ ’ ’ 1 1 I ’
0 10. In N (cl I I I I I I I I I
o 0]‘0°
0 10. In N (e)
- : 1 / I , , , , / ’ .L,
I..\ -I.;
‘
-15.00-15.00 II^ II^ II^ II^ I^ I^ I^ I 00 10.0010. In NIn N
(b)
4 q-600.00 \ !’
0 l.OOe+
\
N (^) i -600.00 I^ I^ I^ I^ I^ I^ I^ I^ I 0 10. In N ( 25.00, , , , , , , , , , , I
_:^. ”2. ..
.. ..:: - .I’.._ ‘..:: :
0 I I I I 1 I I 1 0 10. In N (
0 10. In N
Fig. 1. Log-log plot of lim - W) versus the number of iterations N: (a) for .K,)= 0 at a = a, (the slope of the hx(0)-4 Ax(O) reference straight line is 1.3236); (b) for xg = 0 at a = a, - lo-‘; (c) for x,) = 0 at a = a, + lo-‘; (d) for x,, = -0.0349893 at a = a, : (e) for x,) = 0 at a = 3/4 (the slope of the reference straight line is -3/2); (f) x,, = 0 at a = 7/4 (the slope of the reference straight line is -2).
Power-law sensitivity to initial conditions XX
-ia
I I I I I^ I^ I^ I^ I^ I
In N
Fig. 2. Log-Log plot of lim ~ ~r~O,4A.r(0) versus the number of iterations N for an ensemble of 40 initial conditions uniformly distributed in the interval [ - l,l] (the slope of the reference straight line is 2).
becomes that of the order aP. This gives a good estimate of the exponent of equation (7), i.e. 1 -=In= 1.3236, hence q Z-q In
-0.2445 (^) (16)
In Fig. l(b) we show the typical behavior of L for periodic orbits (a < a,. and A, < 0) and in
Fig. l(c) for the chaotic region (a > a,. and A, >O). In Fig. l(d) we show the behavior associated with an unstable periodic orbit (x,, = - 0.0349893, hence cycle-8) for a = u,. By making the continuous limit approximation, it can be derived that, for the period-doubling bifurcations (e.g. a = 3/4), q = 5/3 and, for the tangent bifurcations (e.g. u = 7/4, i.e. the beginning of the cycle-3 window), q = 3/2. For simplicity, let us illustrate the arguments on the tangent bifurcation at a fixed point x* for some map x,,+, =f(x,) (e.g. the third iterate of the logistic map equation (14) at u = 7/4; notice that t = 3n). Denote by E,, the deviation x,, --x*. For small E,, up to the quadratic term, in general we have
F,,i I = E,,+ A&;,^ (17)
where A =f”(x*)/2. For large enough II, we approximate the discrete equation (17) by the
differential one
d&,, z = AE*,,, (1X)
Power-law sensitivity to initial conditions WI
hand, it is now well-known that certain spatially-extended dynamical systems evolve in a natural and spontaneous way into a critical state [19], in other words, they tune onto their critical point without a particular external action. This phenomenon (self-organized criticality) typically presents power laws and has important applications in physics, biology and even in social sciences, such as economics. It is possible that our unified formalism could be useful in the description of self-organized criticality. In this case, the q # 1 regime would arise in a natural way and would therefore be the generic situation. The application of these concepts to other non-linear dynamical systems presenting chaos or to self-organized criticality, would be very welcome.
Acknowledgements-We gratefully acknowledge enlightening discussions with A. M. Qzorio de Almeida. as well as useful remarks relative to self-organized criticality from D. Stauffer and P. M. C. de Oliveira.
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