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The Dynamics of a Kind of Liénard System with Sixth Degree and Its Limit Cycle Bifurcations Under Perturbations

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Abstract

In this paper, the different topological types of phase portrait of the unperturbed Liénard system \( {\dot{x}}=y,\ \ {\dot{y}}=-g(x)\) are given, where \(\deg {g(x)}=6\). We find that the expansion of the Melnikov function near any of closed orbits appeared in the above phase portraits, except a heteroclinic loop with a hyperbolic saddle and a nilpotent saddle of order one, has been studied. In this paper, we give the expansion of the Melnikov function near this kind of heteroclinic loop. Further, we present the conditions to obtain limit cycles bifurcated from a compound loop with a hyperbolic saddle and a nilpotent saddle of order one, and apply it to study the number of limit cycles for a kind of Liénard system under perturbations.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (11771296, 11931016). The second author was supported by National Natural Science Foundation of China (11571090), Natural Science Foundation of Hebei Province(A2019205133) and Science Foundation of Hebei Normal University (L2017J01).

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, China

    Maoan Han

  2. Mathematics and Science College, Shanghai Normal University, Shanghai, 200234, China

    Maoan Han

  3. School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

    Junmin Yang

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  1. Maoan Han
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  2. Junmin Yang
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Correspondence to Junmin Yang.

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Han, M., Yang, J. The Dynamics of a Kind of Liénard System with Sixth Degree and Its Limit Cycle Bifurcations Under Perturbations. Qual. Theory Dyn. Syst. 19, 26 (2020). https://doi.org/10.1007/s12346-020-00377-2

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