Asymmetric unimodal maps at the edge of chaos

Abstract

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, $z_i>1$, $0<a\le 2$, $t=0,1,2,...$) 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 nonextensive entropies $S_q$ and multifractal analysis. The first two methods provide consistent estimates for the exponent governing the power-law sensitivity. In addition to this, we verify that the multifractal analysis does not provide precise estimates of the singularity spectrum $f(\alpha )$, specially 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 multifractal analysis for studing the sensitivity to initial conditions in this class of asymmetric maps.

8 pages, 6 figs in eps format, submitted for publication

References in corpus (1)

• Dissipative maps at the chaos threshold: Numerical results for the single-site map

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