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Homoclinic bifurcation at resonant eigenvalues

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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.

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Authors and Affiliations

  1. Center for Dynamical Systems and Nonlinear Studies, Georgia Institute of Technology, 30332, Atlanta, Georgia

    Shui -Nee Chow

  2. Department of Mathematics and Statistics, University of Nebraska, 68588-8323, Lincoln, Nebraska

    Bo Deng

  3. Institute of Applied Mathematics, University of Heidelberg, Im Neuenheimer Feld 294, D-6900, Heidelberg, Germany

    Bernold Fiedler

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  1. Shui -Nee Chow
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Chow, S.N., Deng, B. & Fiedler, B. Homoclinic bifurcation at resonant eigenvalues. J Dyn Diff Equat 2, 177–244 (1990). https://doi.org/10.1007/BF01057418

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