2001/09/18 by Uǧur Tırnaklı, U. Tirnakli, Constantino Tsallis +2
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Algorithm #Chaos control and synchronization #Complex Systems and Time Series Analysis #Computation #Computer science #Divergence (linguistics) #Edge of chaos #Exponent #Fractal #Geometry #Mathematical analysis #Mathematics #Measure (data warehouse) #Physics #Power law #Scaling #Sensitivity (control systems) #Singularity #Statistical Mechanics and Entropy #Statistical physics #Statistics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.65.036207
8 pages, 6 figs in eps format, submitted for publication
arxiv created 2001/09/18 · openalex publication_date 2002/02/11 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.