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Unstable periodic orbits and discretization cycles.

P-M Binder1, Nicholas H Okamoto

  • 1Department of Physics and Astronomy, University of Hawaii, Hilo, Hawaii 96720-4091, USA. pbinder@hawaii.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
PubMed
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Discretizing nonlinear maps typically creates limit cycles shadowing unstable periodic orbits (UPOs). However, some cycles require alternative explanations like near-UPOs or UPO crossovers.

Area of Science:

  • Dynamical Systems
  • Nonlinear Dynamics
  • Computational Mathematics

Background:

  • Limit cycles in discrete dynamical systems often arise from discretizing continuous nonlinear maps.
  • These discrete cycles are generally understood to shadow unstable periodic orbits (UPOs) of the parent continuous system.
  • However, discrepancies in this shadowing behavior necessitate further investigation into alternative mechanisms.

Purpose of the Study:

  • To investigate the mechanisms behind limit cycles generated from discretizing nonlinear maps.
  • To explore cases where discretization cycles do not directly shadow individual UPOs.
  • To identify and characterize alternative explanations for observed limit cycle behavior.

Main Methods:

  • Analysis of nonlinear maps and their discretization.

Related Experiment Videos

  • Numerical simulation of discrete dynamical systems.
  • Comparison of discrete limit cycles with unstable periodic orbits of continuous maps.

    • Main Results:

    • Discretization of nonlinear maps generally produces limit cycles that shadow individual unstable periodic orbits (UPOs).
    • In specific instances, discretization cycles cannot be solely explained by shadowing single UPOs.
    • Alternative mechanisms, including the near-occurrence of UPOs or crossovers between multiple UPOs, are identified as explanations.

    Conclusions:

    • The shadowing of UPOs is a common but not exclusive phenomenon for limit cycles arising from discretization.
    • The behavior of discrete limit cycles can be influenced by the proximity and interaction of multiple UPOs.
    • Understanding these alternative mechanisms is crucial for accurately interpreting results from discretized nonlinear systems.