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The Cyclicity of a Class of Global Nilpotent Center Under Perturbations of Piecewise Smooth Polynomials with Four \(\hbox {Zones}^*\)

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Abstract

In this paper, we study the bifurcation of limit cycles of near-Hamilton system with four zones separated by nonlinear switching curves. We derive the expression of the first order Melnikov function. As an application, we consider the cyclicity of the system \({\dot{x}}=y, {\dot{y}}=-x^{2m-1}\), where (0, 0) is a global nilpotent center and \(2\le m\in {\mathbb {N}}^{+}\), under the perturbations of piecewise smooth polynomials with four zones separated by \(y=\pm kx^{m}\) with \(k>0\). By analyzing the first order Melnikov function, we obtain the exact bound of the number of limit cycles bifurcating from the period annulus if the first order Melnikov function is not identically zero. We also give some examples to illustrate our results.

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Acknowledgements

The authors would like to express their sincere appreciation to the reviewer for his/her helpful comments which helped with improving the quality of this work.

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  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Li Zou & Liqin Zhao

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  1. Li Zou
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  2. Liqin Zhao
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Correspondence to Liqin Zhao.

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This work was supported by National Natural Science Foundation of China (12071037)

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Zou, L., Zhao, L. The Cyclicity of a Class of Global Nilpotent Center Under Perturbations of Piecewise Smooth Polynomials with Four \(\hbox {Zones}^*\). Qual. Theory Dyn. Syst. 21, 73 (2022). https://doi.org/10.1007/s12346-022-00600-2

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