PMID: 11909212Mar 23, 2002Paper

Asymmetric unimodal maps at the edge of chaos

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
Ugur TirnakliM L Lyra

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< 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.

References

Jun 1, 1985·Physical Review A: General Physics·R V Jensen, L K Ma
Feb 1, 1986·Physical Review A: General Physics·T C HalseyB I Shraiman
Jan 15, 1987·Physical Review A: General Physics· de Sousa Vieira MCC Tsallis
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Dec 2, 2000·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics· de Moura FAM L Lyra
Jan 4, 2001·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·U Tirnakli
Apr 20, 2001·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·C BeckH L Swinney

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Citations

Oct 10, 2006·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Sumiyoshi Abe, Yutaka Nakada
Dec 31, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Jing HuYinhe Cao
Jul 22, 2020·Physical Review. E·S N SaadatmandJ A Vaccaro
Apr 12, 2006·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Ugur Tirnakli, Constantino Tsallis
Jan 7, 2003·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Ugur Tirnakli

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