Asymmetric unimodal maps at the edge of chaos.
We numerically investigate the sensitivity to initial conditions of asymmetric unimodal maps x(t+1)=1-a/x(t)/(z(i)) (i=1,2 correspond to x(t)>0 and x(t)<0, respectively, z(i)>1, 0<a< or =2, t=0,1,2,...) at the edge of chaos. We employ three distinct algorithms to characterize the power-law sensitivity to initial conditions at the edge of chaos, namely: direct measure of the divergence of initially nearby trajectories, the computation of the rate of increase of generalized non-extensive entropies S(q), and multi-fractal analysis. The first two methods provide consistent estimates for the exponent governing the power-law sensitivity. In addition to this, we verify that the multi-fractal analysis does not provide precise estimates of the singularity spectrum f(alpha), especially near its extremal points. Such feature prevents to perform a fine check of the accuracy of the scaling relation between f(alpha) and the entropic index q, thus restricting the applicability of the multi-fractal analysis for studying the sensitivity to initial conditions in this class of asymmetric maps.
Tirnakli
Ugur
U
Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey. tirnakli@sci.ege.edu.tr
Tsallis
Constantino
C
Lyra
Marcelo L
ML