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On Jacobian matrices for flows.

B Doyon1, L J Dubé

  • 1Département de Physique, de Génie Physique, et d'Optique Université Laval, Cité Universitaire, Québec, Canada G1K 7P4. bdoyon@phy.ulaval.ca

Chaos (Woodbury, N.Y.)
|April 20, 2005
PubMed
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We developed a numerical method to find periodic orbits in dynamical systems using Jacobian matrices on a Poincaré surface of section, accounting for energy conservation in Hamiltonian flows.

Area of Science:

  • Numerical analysis
  • Dynamical systems theory
  • Computational physics

Background:

  • Numerical methods are crucial for analyzing complex dynamical systems.
  • Poincaré surfaces of section are effective for visualizing and analyzing phase space.
  • Hamiltonian systems require special consideration due to energy conservation.

Purpose of the Study:

  • To present a general method for constructing numerical Jacobian matrices.
  • To adapt this method for Hamiltonian flows with energy conservation.
  • To apply the technique for detecting periodic orbits.

Main Methods:

  • Discretization of flows on a Poincaré surface of section.
  • Construction of numerical Jacobian matrices.
  • Incorporation of energy conservation constraints for Hamiltonian flows.

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Main Results:

  • A generalizable numerical approach for Jacobian matrix construction was developed.
  • The method successfully accounts for energy conservation in Hamiltonian systems.
  • The technique enables robust detection of periodic orbits.

Conclusions:

  • The presented method offers a versatile tool for analyzing dynamical flows.
  • It provides a robust framework for identifying periodic orbits in conservative systems.
  • This approach enhances the study of complex dynamical behaviors.