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Efficient algorithm for detecting unstable periodic orbits in chaotic systems.

R L Davidchack1, Y C Lai

  • 1Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|April 24, 2002
PubMed
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We developed an efficient method for detecting unstable periodic orbits in chaotic systems. This approach uses an iterative scheme and smart initial point selection for fast, complete, and accurate results.

Area of Science:

  • Dynamical Systems and Chaos Theory
  • Numerical Analysis
  • Computational Physics

Background:

  • Chaotic systems often exhibit unstable periodic orbits (UPOs) that are crucial for understanding system dynamics.
  • Detecting UPOs is challenging due to their inherent instability and the complexity of chaotic attractors.
  • Existing methods may lack efficiency, completeness, or accuracy in UPO identification.

Purpose of the Study:

  • To present a novel, efficient method for the fast, complete, and accurate detection of UPOs in chaotic systems.
  • To provide a robust technique applicable to various dimensional chaotic maps.
  • To facilitate the analysis of complex dynamical behaviors through reliable UPO identification.

Main Methods:

  • An iterative scheme based on the semi-implicit Euler method is employed.

Related Experiment Videos

  • An effective technique for selecting initial points is integrated into the iterative process.
  • The method's convergence properties (fast and global) are leveraged for efficiency.

    • Main Results:

    • The proposed method achieves fast, complete, and accurate detection of UPOs.
    • A small number of initial points are sufficient to identify all UPOs of a given period.
    • Numerical examples in two- and four-dimensional maps demonstrate the method's effectiveness.

    Conclusions:

    • The developed iterative method offers a significant advancement in UPO detection within chaotic systems.
    • Its efficiency and accuracy make it a valuable tool for researchers in nonlinear dynamics.
    • The technique's applicability to higher-dimensional systems is confirmed through demonstrated examples.