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Characteristic times in the standard map.

Mirella Harsoula1, Kostas Karamanos2, George Contopoulos1

  • 1Research Center for Astronomy, Academy of Athens Soranou Efesiou 4, GR-115 27 Athens, Greece.

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|April 20, 2019
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Summary

This study compares characteristic times in the standard map, finding stickiness time significantly impacts diffusion, leading to normal diffusion (μ=1) outside stability islands. Anomalous diffusion (μ=2) occurs within accelerator mode islands.

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Area of Science:

  • Dynamical Systems
  • Statistical Physics
  • Nonlinear Dynamics

Background:

  • The standard map is a fundamental model for studying chaotic dynamics.
  • Characteristic times like Lyapunov time, Poincaré recurrence time, and stickiness time offer insights into system behavior.
  • Understanding diffusion processes in chaotic systems is crucial for various scientific fields.

Purpose of the Study:

  • To compare three characteristic times: Lyapunov time (tL), Poincaré recurrence time (tr), and stickiness time (tst) in the standard map.
  • To establish empirical relations for these times based on system parameters like nonlinearity (K) and chaotic area (A).
  • To investigate the influence of stickiness time on diffusion dynamics and the diffusion exponent (μ).

Main Methods:

  • Calculation of Lyapunov characteristic number (L) and its relation to Lyapunov time (tL).
  • Empirical determination of Poincaré recurrence time (tr) as a function of K, A, and initial condition box size (ε).
  • Comparison of stickiness time distributions with Poincaré recurrence time distributions, particularly in sticky regions.

Main Results:

  • Empirical relations were found for L and tr concerning K and A.
  • Stickiness time (tst) in sticky regions is orders of magnitude smaller than Poincaré recurrence time (tr).
  • Diffusion exponent (μ) consistently converges to 1 due to stickiness, indicating normal diffusion outside stability islands, while anomalous diffusion (μ=2) is observed within accelerator mode islands.

Conclusions:

  • Stickiness time significantly influences diffusion dynamics, simplifying the diffusion exponent to μ=1 in most regions.
  • The standard map exhibits distinct diffusion behaviors: normal diffusion (μ=1) and anomalous diffusion (μ=2) within specific stability islands.
  • Extreme stickiness cases reveal complex hierarchical structures of islands within islands.