Convergence to the critical attractor of dissipative maps: log-periodic oscillations, fractality, and nonextensivity

For a family of logisticlike maps, we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space. We found that the phase-space volume occupied by the ensemble W(t) depicts a power-law decay with log-periodic oscillations reflecting the multifractal character of the critical attractor. We explore the parametric dependence of the power-law exponent and the amplitude of the log-periodic oscillations with the attractor's fractal dimension governed by the inflection of the map near its extremal point. Further, we investigate the temporal evolution of W(t) for the circle map whose critical attractor is dense. In this case, we found W(t) to exhibit a rich pattern with a slow logarithmic decay of the lower bounds. These results are discussed in the context of nonextensive Tsallis entropies.

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

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

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

[4]  Mogens H. Jensen,et al.  Global universality at the onset of chaos: Results of a forced Rayleigh-Benard experiment. , 1985 .

[5]  Jensen,et al.  Erratum: Fractal measures and their singularities: The characterization of strange sets , 1986, Physical review. A, General physics.

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

[7]  Universal scaling of the stress field at the vicinity of a wedge crack in two dimensions and oscillatory self-similar corrections to scaling. , 1990, Physical review letters.

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

[9]  Newman,et al.  log-periodic behavior of a hierarchical failure model with applications to precursory seismic activation. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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

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

[12]  Renio S. Mendes,et al.  SPECIFIC HEAT ANOMALIES ASSOCIATED WITH CANTOR-SET ENERGY SPECTRA , 1997 .

[13]  C. Tsallis,et al.  Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity , 1997, cond-mat/9701096.

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

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

[16]  CONNECTION BETWEEN ENERGY SPECTRUM, SELF-SIMILARITY, AND SPECIFIC HEAT LOG-PERIODICITY , 1998, cond-mat/9803265.

[17]  Weak Chaos: Power-Law Sensitivity to Initial Conditions and Nonextensive Thermostatistics , 1999 .

[18]  Imprints of log-periodic self-similarity in the stock market , 1999, cond-mat/9901025.

[19]  C. Tsallis.  Nonextensive statistics: theoretical, experimental and computational evidences and connections , 1999, cond-mat/9903356.

[20]  M. Lyra,et al.  Low-dimensional non-linear dynamical systems and generalized entropy , 1999 .

[21]  C. Tsallis,et al.  Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity , 1999 .