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This study explores the web map

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Area of Science:

  • Statistical mechanics
  • Dynamical systems theory
  • Nonlinear dynamics

Background:

  • The web map exhibits complex dynamics.
  • Ergodic behavior is typically described by Boltzmann-Gibbs statistics.
  • Ergodicity breakdown necessitates alternative statistical frameworks.

Purpose of the Study:

  • To investigate the statistical properties of the two-dimensional web map.
  • To explore the transition from Boltzmann-Gibbs to generalized q-statistics.
  • To characterize non-Gaussian distributions arising from fractal dynamics.

Main Methods:

  • Numerical simulations of the two-dimensional, area-preserving web map.
  • Analysis of probability distributions, sensitivity to initial conditions, and entropy production.
  • Application of Boltzmann-Gibbs and generalized q-statistics.

Main Results:

  • For small/large external parameters (K), q-Gaussian (q=1.935...) and Gaussian distributions are observed, respectively.
  • Intermediate K values reveal non-Gaussian distributions due to fractal trajectory structures.
  • Characterization of these distributions using kurtosis and box-counting dimension.

Conclusions:

  • The web map transitions between Boltzmann-Gibbs and generalized q-statistics based on ergodicity.
  • Fractal dynamics in the chaotic sea lead to persistent non-Gaussian behavior.
  • Generalized q-statistics are crucial for understanding complex systems with ergodicity breakdown.