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Heterodimensional cycle bifurcation with two orbit flips

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Abstract

In this paper, we consider a heteroclinic cycle consisting of two hyperbolic equilibria with different indices, one robust heteroclinic connection and a heteroclinic connection within a codimension-2 intersection of the corresponding manifolds of the equilibria, which is called the heterodimensional cycle. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, we construct a Poincaré return map under the nongeneric conditions two orbit flips and further obtain the bifurcation equations. By the bifurcation equations, different bifurcation phenomena are discussed under small perturbations. New features produced by the degeneracy that heterodimensional cycles and periodic orbits coexist on the same bifurcation surface are shown. Some known results are extended. An example is given to show the existence of the system which has a heterodimensional with two orbit flips.

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Acknowledgments

The authors would like to thank the reviewers for their careful reading on the paper and very helpful comments and suggestions to improve the content of this paper. This work is supported by NNSFC (No. 11371140), Shanghai Key Laboratory of PMMP, and Zhejiang Province (LY13A010020) and Program for Excellent Young Teachers in HNU (HNUEYT2013)

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Authors and Affiliations

  1. Department of Mathematics, East China Normal University, Shanghai, 200241, China

    Xingbo Liu & Sisi Wang

  2. Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China

    Yancong Xu

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  1. Xingbo Liu
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  2. Yancong Xu
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  3. Sisi Wang
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Correspondence to Xingbo Liu.

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Liu, X., Xu, Y. & Wang, S. Heterodimensional cycle bifurcation with two orbit flips. Nonlinear Dyn 79, 2787–2804 (2015). https://doi.org/10.1007/s11071-014-1846-7

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  • DOI: https://doi.org/10.1007/s11071-014-1846-7

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