Numerical study of the oscillatory convergence to the attractor at the edge of chaos

Abstract.This paper compares three different types of “onset of chaos” in the logistic and generalized logistic map: the Feigenbaum attractor at the end of the period doubling bifurcations; the tangent bifurcation at the border of the period three window; the transition to chaos in the generalized logistic with inflection 1/2 (xn+1 = 1-μxn1/2), in which the main bifurcation cascade, as well as the bifurcations generated by the periodic windows in the chaotic region, collapse in a single point. The occupation number and the Tsallis entropy are studied. The different regimes of convergence to the attractor, starting from two kinds of far-from-equilibrium initial conditions, are distinguished by the presence or absence of log-log oscillations, by different power-law scalings and by a gap in the saturation levels. We show that the escort distribution implicit in the Tsallis entropy may tune the log-log oscillations or the crossover times.

[1]  V. Latora,et al.  The rate of entropy increase at the edge of chaos , 1999, cond-mat/9907412.

[2]  J. P. van der Weele,et al.  Period doubling in maps with a maximum of order z , 1987 .

[3]  A M Scarfone,et al.  Two-parameter deformations of logarithm, exponential, and entropy: a consistent framework for generalized statistical mechanics. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  A. Politi,et al.  Dynamical Behaviour at the Onset of Chaos , 1988 .

[5]  Time evolution of thermodynamic entropy for conservative and dissipative chaotic maps , 2000, cond-mat/0007302.

[6]  Constantino Tsallis,et al.  Ensemble averages and nonextensivity at the edge of chaos of one-dimensional maps. , 2004, Physical review letters.

[7]  Sumiyoshi Abe,et al.  A note on the q-deformation-theoretic aspect of the generalized entropies in nonextensive physics , 1997 .

[8]  F. Baldovin,et al.  Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: Rigorous nonextensive solutions , 2002 .

[9]  C. Tsallis.  Possible generalization of Boltzmann-Gibbs statistics , 1988 .

[10]  C. Tsallis,et al.  Nonextensive Entropy: Interdisciplinary Applications , 2004 .

[11]  M. Feigenbaum.  Quantitative universality for a class of nonlinear transformations , 1978 .

[12]  D. Sornette.  Discrete scale invariance and complex dimensions , 1997, cond-mat/9707012.

[13]  A. M. Scarfone,et al.  Deformed logarithms and entropies , 2004, cond-mat/0402418.

[14]  Peter Grassberger,et al.  Some more universal scaling laws for critical mappings , 1981 .

[15]  G. Kaniadakis,et al.  Non-linear kinetics underlying generalized statistics , 2001 .

[16]  C. Tsallis,et al.  Nonextensivity and Multifractality in Low-Dimensional Dissipative Systems , 1997, cond-mat/9709226.

[17]  Y. Pesin.  CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY , 1977 .

[18]  C. Beck,et al.  Thermodynamics of chaotic systems , 1993 .

[19]  V. Latora,et al.  Kolmogorov-Sinai Entropy Rate versus Physical Entropy , 1998, chao-dyn/9806006.

[20]  J. Muzy,et al.  Complex fractal dimensions describe the hierarchical structure of diffusion-limited-aggregate clusters. , 1996, Physical review letters.

[21]  C. Tsallis,et al.  Power-law sensitivity to initial conditions—New entropic representation , 1997 .

[22]  M. Coraddu,et al.  Weak insensitivity to initial conditions at the edge of chaos in the logistic map , 2004, cond-mat/0403360.

[23]  G. Kaniadakis,et al.  Statistical mechanics in the context of special relativity. II. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  F Baldovin,et al.  Universal renormalization-group dynamics at the onset of chaos in logistic maps and nonextensive statistical mechanics. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  C. Beck,et al.  Thermodynamics of chaotic systems : an introduction , 1993 .

[26]  de Moura FA,et al.  Convergence to the critical attractor of dissipative maps: log-periodic oscillations, fractality, and nonextensivity , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[27]  M. Feigenbaum.  The universal metric properties of nonlinear transformations , 1979 .

[28]  Dynamic Description of the Critical 2∞ Atrractor and 2m-Band Chaos , 1989 .