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Limit cycle bifurcations in a class of quintic Z 2-equivariant polynomial systems

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Abstract

In this paper, we study a class of cubic Z 2-equivariant polynomial Hamiltonian systems under the perturbation of Z 2-equivariant polynomial of degree 5. First, we consider the unperturbed system and obtain necessary and sufficient conditions for the critical point (0,1) to be a nilpotent saddle, center, or cusp. We show that it can have 14 different phase portraits. Using the methods of Hopf and homoclinic bifurcation theory, we study the bifurcation problem of the perturbed system and prove that there exist 12 limit cycles.

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Acknowledgements

The project was supported by National Natural Science Foundation of China (11271261), a grant from Ministry of Education of China (20103127110001) and FP7-PEOPLE-2012-IRSES-316338.

The authors would like to give thanks to professor Maoan Han for providing us the topic of the paper together with main ideas and for his helpful discussions and useful suggestions during the preparation of the paper. They also would like to thank the reviewers for their valuable comments and suggestions.

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  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Xiaona Kong & Yanqin Xiong

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  1. Xiaona Kong
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  2. Yanqin Xiong
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Correspondence to Yanqin Xiong.

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Kong, X., Xiong, Y. Limit cycle bifurcations in a class of quintic Z 2-equivariant polynomial systems. Nonlinear Dyn 73, 1271–1281 (2013). https://doi.org/10.1007/s11071-013-0861-4

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