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Detecting unstable periodic orbits in chaotic continuous-time dynamical systems.

D Pingel1, P Schmelcher, F K Diakonos

  • 1Theoretical Chemistry, Institute for Physical Chemistry, Im Neuenheimer Feld 229, University of Heidelberg, Germany. detlef.pingel@tc.pci.uni-heidelberg.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 11, 2001
PubMed
Summary

This study presents a new method to find unstable periodic orbits in continuous dynamical systems by reducing them to a Poincaré map. The algorithm demonstrates global convergence and works for diverse systems without prior knowledge.

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Area of Science:

  • Dynamical Systems Theory
  • Chaos Theory
  • Computational Physics

Background:

  • Detecting unstable periodic orbits is crucial for understanding chaotic dynamics.
  • Existing methods often require specific knowledge of the system or lack global convergence.
  • Time-discrete methods for detecting periodic points exist but need adaptation for continuous systems.

Purpose of the Study:

  • To extend a method for detecting unstable periodic points from discrete to continuous dynamical systems.
  • To develop a robust algorithm for finding unstable periodic orbits in ordinary differential equations.
  • To demonstrate the method's applicability to both dissipative and Hamiltonian systems.

Main Methods:

  • Reduction of the continuous flow of ordinary differential equations to a Poincaré map.

Related Experiment Videos

  • Application of an existing algorithm for detecting periodic points to the derived Poincaré map.
  • Testing the method on the Lorenz system and the hydrogen atom in a strong magnetic field.

    • Main Results:

    • Successfully adapted a method for detecting unstable periodic points in time-discrete systems to time-continuous systems.
    • The developed algorithm exhibits global convergence properties.
    • The method is effective for both dissipative (Lorenz system) and Hamiltonian (hydrogen atom) dynamical systems.

    Conclusions:

    • The proposed approach provides a general and robust method for identifying unstable periodic orbits in continuous dynamical systems.
    • The algorithm's global convergence and independence from a priori knowledge offer significant advantages.
    • This method facilitates a deeper analysis of chaotic behavior in various physical systems.