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Nongeneric bifurcations near a nontransversal heterodimensional cycle

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Abstract

In this paper bifurcations of heterodimensional cycles with highly degenerate conditions are studied in three dimensional vector fields, where a nontransversal intersection between the two-dimensional manifolds of the saddle equilibria occurs. By setting up local moving frame systems in some tubular neighborhood of unperturbed heterodimensional cycles, the authors construct a Poincaré return map under the nongeneric conditions and further obtain the bifurcation equations. By means of the bifurcation equations, the authors show that different bifurcation surfaces exhibit variety and complexity of the bifurcation of degenerate heterodimensional cycles. Moreover, an example is given to show the existence of a nontransversal heterodimensional cycle with one orbit flip in three dimensional system.

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Authors and Affiliations

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

    Xingbo Liu & Xiaofei Wang

  2. School of Mathematics and Physics, Handan College, Handan, 056005, Hebei, China

    Ting Wang

Authors
  1. Xingbo Liu
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  2. Xiaofei Wang
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  3. Ting Wang
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Corresponding author

Correspondence to Ting Wang.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 11371140) and the Shanghai Key Laboratory of PMMP.

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Liu, X., Wang, X. & Wang, T. Nongeneric bifurcations near a nontransversal heterodimensional cycle. Chin. Ann. Math. Ser. B 39, 111–128 (2018). https://doi.org/10.1007/s11401-018-1055-7

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  • DOI: https://doi.org/10.1007/s11401-018-1055-7

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