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Variational method for finding periodic orbits in a general flow.

Yueheng Lan1, Predrag Cvitanović

  • 1Center for Nonlinear Science, School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA. gte158y@prism.gatech.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 5, 2004
PubMed
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This study introduces a novel variational principle to find unstable periodic orbits in dynamical systems. The method effectively locates complex solutions in systems like the three-body problem.

Area of Science:

  • Dynamical Systems Theory
  • Computational Physics
  • Nonlinear Dynamics

Background:

  • Determining unstable periodic orbits is crucial for understanding complex dynamical systems.
  • Existing methods often struggle with the inherent instability and complexity of these solutions.
  • Extended systems exhibit challenging spatiotemporally periodic behaviors.

Purpose of the Study:

  • To propose and implement a robust variational principle for finding unstable periodic orbits.
  • To extend the method for identifying unstable spatiotemporally periodic solutions in extended systems.
  • To demonstrate the method's applicability across diverse and complex physical systems.

Main Methods:

  • A variational principle is formulated based on minimizing local errors along a loop approximating a periodic solution.

Related Experiment Videos

  • The evolution is governed by a "Newton descent" partial differential equation, an iterative refinement process.
  • The method employs a damped Newton-Raphson iteration approach for stability and convergence.

    • Main Results:

    • The variational principle successfully determined unstable periodic orbits in benchmark systems.
    • The method was applied to the Hénon-Heiles system and the circular restricted three-body problem.
    • The Kuramoto-Sivashinsky system in a weakly turbulent regime also yielded valid solutions.

    Conclusions:

    • The proposed variational principle offers a feasible and effective approach for locating unstable periodic orbits.
    • The method's successful application to diverse systems highlights its broad applicability in dynamical systems research.
    • This technique provides a powerful new tool for analyzing complex and unstable dynamics.