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Analysis of a Shil’nikov Type Homoclinic Bifurcation

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Abstract

The bifurcation associated with a homoclinic orbit to saddle-focus including a pair of pure imaginary eigenvalues is investigated by using related homoclinic bifurcation theory. It is proved that, in a neighborhood of the homoclinic bifurcation value, there are countably infinite saddle-node bifurcation values, period-doubling bifurcation values and double-pulse homoclinic bifurcation values. Also, accompanied by the Hopf bifurcation, the existence of certain homoclinic connections to the periodic orbit is proved.

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Acknowledgements

The authors would like to thank the referees for their helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, P. R. China

    Yan Cong Xu

  2. Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, P. R. China

    Xing Bo Liu

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  1. Yan Cong Xu
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  2. Xing Bo Liu
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Correspondence to Xing Bo Liu.

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Supported by National NSF (Grant Nos. 11371140, 11671114) and Shanghai Key Laboratory of PMMP

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Xu, Y.C., Liu, X.B. Analysis of a Shil’nikov Type Homoclinic Bifurcation. Acta. Math. Sin.-English Ser. 34, 901–910 (2018). https://doi.org/10.1007/s10114-018-5236-9

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  • DOI: https://doi.org/10.1007/s10114-018-5236-9

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