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Homoclinic orbits in three-dimensional continuous piecewise linear generalized Michelson systems.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • Generalized Michelson systems are a class of continuous piecewise linear systems.
  • Homoclinic orbits are significant in understanding the complex dynamics of such systems.
  • Saddle-focus equilibria are critical points in the phase space of these systems.

Purpose of the Study:

  • To investigate the existence and properties of homoclinic orbits in three-dimensional continuous piecewise linear generalized Michelson systems.
  • To analyze the behavior of trajectories that approach and return to a saddle-focus equilibrium point.
  • To contribute to the understanding of chaotic dynamics in piecewise linear systems.

Main Methods:

  • Analytical investigation using Poincaré map theory.
  • Application of invariant manifold theory.
  • Numerical simulations for validation and illustration.

Main Results:

  • The existence of homoclinic orbits connecting the saddle-focus equilibrium is analytically demonstrated.
  • The study provides a theoretical framework for understanding these complex orbits.
  • Numerical simulations corroborate the analytical findings, visualizing the homoclinic behavior.

Conclusions:

  • Homoclinic orbits exist in the studied generalized Michelson systems.
  • The combination of analytical and numerical methods is effective for analyzing such dynamical systems.
  • This research enhances the comprehension of complex dynamics in piecewise linear systems.