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Unstable dimension variability in coupled chaotic systems.

Y C Lai1, D Lerner, K Williams

  • 1Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
Summary

Coupled chaotic systems can exhibit unstable dimension variability, where periodic orbits have varying unstable directions. This phenomenon hinders deterministic modeling of chaotic systems, impacting their accurate quantification and physical interpretation.

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Area of Science:

  • Complex Systems Dynamics
  • Nonlinear Science
  • Mathematical Physics

Background:

  • Coupled chaotic maps and flows are prevalent in physical and biological systems.
  • Understanding the behavior of these systems is crucial for accurate modeling.

Purpose of the Study:

  • To analyze and provide numerical evidence for unstable dimension variability in coupled chaotic systems.
  • To investigate the impact of this behavior on deterministic modeling.

Main Methods:

  • Numerical analysis of coupled chaotic systems.
  • Examination of unstable periodic orbits within dynamical invariant sets.

Main Results:

  • Unstable dimension variability is a common nonhyperbolic behavior in these systems.

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  • Unstable periodic orbits often possess differing numbers of unstable directions.
  • This variability can arise even with small coupling parameters.

    • Conclusions:

    • Unstable dimension variability severely compromises deterministic modeling of chaotic systems.
    • Significant modeling difficulties emerge with appreciable unstable dimension variability.
    • Potential physical consequences of this phenomenon warrant further investigation.