Power-law sensitivity to initial conditions - New entropic representation

The exponential sensitivity to the initial conditions of chaotic systems (e.g. D=1) is characterized by the Liapounov exponent lambda, 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(Delta x(0)-->0)[Delta(x)(t)]/[(Delta(x)(0)] = [1 + (1 - q)lambda(q)t](1/(1-q)) (equal to e(lambda 1f) for q = 1, and proportional, for large t, to t(1/(1-q)) for q not equal 1; q is an element of R). We show that lambda(q) = K-q (For All q), where K-q is the generalization of K within the non-extensive thermostatistics based upon the generalized entropic form S-q = (1 - Sigma(i)p(i)(q))/(q - 1) (hence, S-1 = -Sigma(1)p(i)lnp(i)). The well-known theorem lambda(1) = K-1 (Pesin equality) is thus extended to arbitrary q. We discuss the logistic map at its threshold to chaos, at period doubling bifurcations and at tangent bifurcations, and find q approximate to 0.2445, q = 5/3 and q = 3/2, respectively. 05.45. + b; 05.20. - y; 05.90. + m. (C) 1997 Elsevier Science Ltd.