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Secondary homoclinic bifurcation theorems.

Vered Rom-Kedar1

  • 1The Department of Applied Mathematics and Computer Science, The Weizmann Institute of Science, P.O.B. 26, Rehovot 76100, Israel.

Chaos (Woodbury, N.Y.)
|June 1, 1995
PubMed
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We developed criteria to detect manifold intersections in perturbed Hamiltonian systems using the Secondary Melnikov Function (SMF). This function predicts intersection types, rotary and transition numbers, and scaling near bifurcations.

Area of Science:

  • Dynamical Systems
  • Chaos Theory
  • Mathematical Physics

Background:

  • Hamiltonian systems with one degree of freedom are fundamental in classical mechanics.
  • Perturbations can lead to complex dynamics, including chaos.
  • Understanding manifold intersections is crucial for analyzing chaotic behavior.

Purpose of the Study:

  • To develop criteria for detecting secondary intersections and tangencies of stable and unstable manifolds.
  • To introduce and analyze the Secondary Melnikov Function (SMF).
  • To predict key dynamical properties related to homoclinic tangles.

Main Methods:

  • Construction of the Secondary Melnikov Function (SMF).
  • Analysis of simple and degenerate zeros of the SMF to identify intersection types.

Related Experiment Videos

  • Theoretical prediction of intersection properties like rotary and transition numbers.

    • Main Results:

    • Simple zeros of the SMF correspond to transverse manifold intersections.
    • Degenerate zeros of the SMF correspond to tangent manifold intersections.
    • The SMF predicts rotary number, transition number, tangencies, and scaling of intersection angles.

    Conclusions:

    • The developed theory provides a robust framework for analyzing complex dynamics in perturbed Hamiltonian systems.
    • The SMF is a powerful tool for predicting chaotic transport and topological entropy.
    • Further research is needed to explore implications for dissipative systems.