3 research outputs found
Power-Law Sensitivity to Initial Conditions within a Logistic-like Family of Maps: Fractality and Nonextensivity
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 The main ingredient of our approach is the generalized deviation
law \lim_{\Delta x(0) -> 0} \Delta x(t) / \Delta x(0)} = [1+(1-q)\lambda_q
t]^{1/(1-q)} (equal to for q=1, and proportional, for large
t, to for 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 -infinity when d_f varies from 1
(nonfractal, ergodic-like, limit) to zero.Comment: LaTeX, 6 pages , 5 figure