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Degeneracy of the action-frequency map: a mechanism for homoclinic bifurcation of invariant tori.

Authors
  • Bridges, Thomas J
Type
Published Article
Journal
Physical Review E
Publisher
American Physical Society (APS)
Publication Date
Jun 01, 2009
Volume
79
Issue
6 Pt 2
Pages
66603–66603
Identifiers
PMID: 19658617
Source
Medline
License
Unknown

Abstract

A family of smooth invariant tori of a Hamiltonian system can be parameterized by the values of the actions or the frequencies. These parameterizations are related by the action-frequency map. The purpose of this paper is to show that when the action-frequency map is degenerate, it signals a homoclinic bifurcation. Remarkably, the nonlinear properties of this homoclinic bifurcation to invariant tori are determined by the curvature of the action-frequency map. A homoclinic angle is also generated which is analogous to a Hannay-Berry phase shift. The theory is constructive and so can usefully be combined with computation. Some implications for quantization, and the generation of solitary waves are also discussed.

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