Homoclinic bifurcation at resonant eigenvalues

Shui Nee Chow, Bo Deng, Bernold Fiedler

Research output: Contribution to journalArticle

115 Scopus citations

Abstract

We consider a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. This bifurcation can occur in generic two parameter vector fields when a homoclinic orbit is attached to a stationary point with resonant eigenvalues. The resonance condition requires the eigenvalues with positive/negative real part closest to zero to be real, simple, and equidistant to zero. Under an additional global twist condition, an exponentially flat bifurcation of double homoclinic orbits from the primary homoclinic branch is established rigorously. Moreover, associated period doublings of periodic orbits with almost infinite period are detected. If the global twist condition is violated, a resonant side switching occurs. This corresponds to an exponentially flat bifurcation of periodic saddle-node orbits from the homoclinic branch.

Original languageEnglish (US)
Pages (from-to)177-244
Number of pages68
JournalJournal of Dynamics and Differential Equations
Volume2
Issue number2
DOIs
StatePublished - Apr 1990

Keywords

  • AMS classification: 34C15, 34C35, 58F14
  • global bifurcation
  • homoclinic orbit
  • pathfollowing
  • period doubling
  • resonance

ASJC Scopus subject areas

  • Analysis

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