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Homoclinic bifurcation at resonant eigenvalues

  • Shui -Nee Chow
  • Bo Deng
  • Bernold Fiedler
Article

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.

Key words

homoclinic orbit period doubling pathfollowing global bifurcation resonance 

AMS classification

34C15 34C35 58F14 

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Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • Shui -Nee Chow
    • 1
  • Bo Deng
    • 2
  • Bernold Fiedler
    • 3
  1. 1.Center for Dynamical Systems and Nonlinear StudiesGeorgia Institute of TechnologyAtlanta
  2. 2.Department of Mathematics and StatisticsUniversity of NebraskaLincoln
  3. 3.Institute of Applied MathematicsUniversity of HeidelbergHeidelbergGermany

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