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, $z_i >1$, $0<a\leq 2$, $t=0,1,2,...$) 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 $f(α)$, specially near its extremal points. Such feature prevents to perform a fine check of the accuracy of the scaling relation between $f(α)$ 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.