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Related Experiment Videos

Multiple returns for some regular and mixing maps.

N Haydn1, E Lunedei, L Rossi

  • 1Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113, USA. nhaydn@math.usc.edu

Chaos (Woodbury, N.Y.)
|October 29, 2005
PubMed
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We analyzed visit distributions in dynamical systems, finding power-law decays for skew maps and coupled systems, unlike the typical Poissonian behavior. This reveals hidden dynamics in mixed regular and chaotic regions.

Area of Science:

  • * Dynamical systems theory
  • * Statistical mechanics
  • * Chaos theory

Background:

  • * Limit distributions of visit counts are known to follow Poissonian behavior in highly mixing systems.
  • * Understanding these distributions is crucial for characterizing system dynamics, especially in regions with mixed ergodic and regular motions.

Purpose of the Study:

  • * To investigate limit laws for the number of visits in various dynamical systems, particularly non-mixing ones.
  • * To analyze visit distributions in skew integrable maps, irrational rotations, coupled systems, and the Henon map.
  • * To explain observed deviations from Poissonian behavior and explore finite-size effects.

Main Methods:

  • * Numerical analysis of visit distributions for shrinking domains around points in phase space.

Related Experiment Videos

  • * Theoretical explanation for observed power-law decay in skew maps.
  • * Superposition principle for coupled systems to analyze boundary effects.
  • * Statistical analysis of first return times for periodic points in the Henon map.

    • Main Results:

    • * Irrational rotations on a circle suggest limit laws under specific domain choices.
    • * Skew integrable maps exhibit power-law decay in visit distributions, with a theoretical explanation provided.
    • * Coupled systems show linear superposition of distributions at boundaries; finite-size effects can mask true limit laws.
    • * The Henon map shows Poissonian behavior for generic points but a different law for periodic points.

    Conclusions:

    • * Deviations from Poissonian behavior are observed in non-uniformly hyperbolic and integrable systems.
    • * Power-law distributions arise in specific dynamical systems and coupled systems, offering insights into complex dynamics.
    • * The study provides a framework for understanding visit distributions in diverse dynamical systems, including those with mixed regular and chaotic behaviors.