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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Some bifurcation methods of finding limit cycles.

Maoan Han1, Tonghua Zhang

  • 1Department of Mathematics, Shanghai Normal University, Guilin Rd. 100, Shanghai, 200234 PR China. mahan@sjtu.edu.cn.

Mathematical Biosciences and Engineering : MBE
|April 6, 2010
PubMed
Summary

This study presents methods for finding limit cycles in perturbed autonomous systems. It covers Hopf, Poincare, and homoclinic bifurcations, offering new techniques for stability analysis in polynomial systems.

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Area of Science:

  • Dynamical Systems Theory
  • Nonlinear Dynamics
  • Mathematical Analysis

Background:

  • Limit cycles are fundamental in understanding the qualitative behavior of autonomous systems.
  • Parameter perturbations can significantly alter the dynamics, leading to bifurcations.
  • Analyzing bifurcations is crucial for predicting system stability and behavior.

Purpose of the Study:

  • To outline methods for detecting limit cycles in planar autonomous systems with small parameter perturbations.
  • To introduce established techniques for analyzing Hopf and Poincare bifurcations.
  • To present a novel method for studying stability-changing homoclinic bifurcations.

Main Methods:

  • Brief introduction to three methods for studying Hopf bifurcations.
  • Application of Melnikov functions for analyzing Poincare bifurcations.
  • Description of a new stability-changing method for homoclinic bifurcations.

Main Results:

  • The paper provides a framework for identifying limit cycles under parameter variations.
  • It demonstrates the utility of Melnikov functions for Poincare bifurcations.
  • A novel approach to analyzing homoclinic bifurcations and their stability changes is presented.

Conclusions:

  • The outlined methods offer effective tools for analyzing limit cycles and bifurcations in perturbed systems.
  • The new method for homoclinic bifurcations provides valuable insights into stability changes.
  • Applications to polynomial systems highlight the practical relevance of these techniques.