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Codimension-4 resonant homoclinic bifurcations with orbit flips and inclination flips

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Abstract

The paper studies a codimension-4 resonant homoclinic bifurcation with one orbit flip and two inclination flips, where the resonance takes place in the tangent direction of homoclinic orbit. Local active coordinate system is introduced to construct the Poincaré returning map, and also the associated successor functions. We prove the existence of the saddle-node bifurcation, the period-doubling bifurcation and the homoclinic-doubling bifurcation, and also locate the corresponding 1-periodic orbit, 1-homoclinic orbit, double periodic orbits and some 2n-homoclinic orbits.

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, P. R. China

    Tian Si Zhang

  2. Department of Mathematics, East China Normal University, Shanghai, 200062, P. R. China

    De Ming Zhu

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  1. Tian Si Zhang
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  2. De Ming Zhu
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Correspondence to Tian Si Zhang.

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Supported by NSFC (Grant No. 11126097)

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Zhang, T.S., Zhu, D.M. Codimension-4 resonant homoclinic bifurcations with orbit flips and inclination flips. Acta. Math. Sin.-English Ser. 31, 1359–1366 (2015). https://doi.org/10.1007/s10114-015-2456-0

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  • DOI: https://doi.org/10.1007/s10114-015-2456-0

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