Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity

Abstract

Power-law sensitivity to initial conditions, characterizing the behaviour of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear 1D logistic-like maps xt+1 = 1 − a |xt| z , (z > 1; 0 < a ≤ 2; t = 0, 1, 2, . . .).The main ingredient of our approach is the generalized deviation law lim ∆x(0)→01−q (equal to e λ 1 t for q = 1, and proportional, for large t, to t 1 1−q for q = 1; q ∈ R is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter q and the fractal dimension d f of the onset-to-chaos attractor is revealed: q appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to −∞ when d f varies from 1 (nonfractal, ergodic-like, limit) to zero. 05.45.+b; 05.20.-y; 05.90.+m

Time evolution of thermodynamic entropy for conservative and dissipative chaotic maps

“…The value of q which allows for a linear growth of S q (t), obtained through this procedure, happens to coincide with the value q * obtained in a completely different method by studying the power-law sensitivity to initial conditions [14]. There exists also another method, based only on the geometrical description of the multifractal attractor existing at a c , which gives exactly the same value of q [14].…”

“…In particular it has been shown that such an entropy covers some types of anomalies due to a possible multifractal structure of the relevant phase space. For example, whenever we have long-range interactions [19], long-range microscopic memory [20], or multifractal boundary conditions [14]. We discuss in the rest of the paper the following results obtained for the logistic map [8]:…”

“…the exponent of the power-law, is given. In particular at a = a c a value q * = 0.2445... is obtained [14]. We now apply to the logistic map the same kind of analysis used for conservative systems.…”

“…Instead at the edge of chaos λ = 0, and the following equation has been proven to be valid in ref. [14] …”

Sensitivity function and entropy increase rates for z-logistic map family at the edge of chaos

“…After these works on the logistic map, numerical evidences supporting this framework came also from the studies of other low-dimensional dynamical systems, such as the z-logistic map family [6,7], the Henon map [8] and the asymmetric logistic map family [9]. Besides these numerical investigations, analytical treatment of the subject is also available recently in a series of paper by Baldovin and Robledo [10].…”