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Limit Cycles in Two Kinds of Quadratic Reversible Systems with Non-smooth Perturbations

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Abstract

This paper deals with the problem of limit cycle bifurcations for two kinds of quadratic reversible differential systems, when they are perturbed inside all discontinuous polynomials of degree n. The switching lines are \(x=1\) and \(y=0\). Firstly, we derive the algebraic structure of the first order Melnikov function M(h) by computing its generating functions, which is more complicated than the Melnikov function corresponding to the perturbations with one switching line. Then, we obtain the detailed expression of M(h) by solving the Picard–Fuchs equations that the generating functions satisfy. Finally, we derive the upper bounds of the number of limit cycles by using the derivation-division algorithm for \(n\ge 2\) and the lower bounds of the number of limit cycles by linear independence for \(n=2\), counting the multiplicity.

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Acknowledgements

The authors would like to express their sincere appreciation to the referee for his/her valuable suggestions and comments. This work was supported by Scientific Research Program of Higher Education of Ningxia (NGY2020074), National Natural Science Foundation of China (12071037, 11701306), Construction of First-class Disciplines of Higher Education of Ningxia (Pedagogy) (NXYLXK2017B11) and Young Top-notch Talent of Ningxia.

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  1. School of Mathematics and Computer Science, Ningxia Normal University, Guyuan, 756000, People’s Republic of China

    Jihua Yang

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Yang, J. Limit Cycles in Two Kinds of Quadratic Reversible Systems with Non-smooth Perturbations. Qual. Theory Dyn. Syst. 20, 54 (2021). https://doi.org/10.1007/s12346-021-00493-7

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