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Limit Cycles of a Class of Cubic Liénard Equations

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Abstract

In this paper, a class of polynomial Liénard systems

$$\begin{array}{ll}&\dot{x}=y-(a_{1}x+a_{2}x^{2}+a_{3}x^{3}),\\&\dot{y}=-(b_{1}x+b_{2}x^2+b_{3}x^3), \end{array}$$

is considered. Some conditions of the existence, non-existence and uniqueness of limit cycles are obtained by using Filippov transformations and Zhang’s theorem. We obtain that the above system has at most one limit cycle surrounding the origin if a 1 a 3 < 0 or b 2 = 0. And, one example is given to illustrate that the system can have three limit cycles.

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Authors and Affiliations

  1. College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Huatao Jin & Shuliang Shui

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  1. Huatao Jin
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  2. Shuliang Shui
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Correspondence to Shuliang Shui.

Additional information

This work was supported by the National Natural Science Foundation of China (Grant No. 10901140), the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y6110195) and the Zhejiang Innovation Project (Grant No. T200905).

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Jin, H., Shui, S. Limit Cycles of a Class of Cubic Liénard Equations. Qual. Theory Dyn. Syst. 10, 317–326 (2011). https://doi.org/10.1007/s12346-011-0045-x

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  • DOI: https://doi.org/10.1007/s12346-011-0045-x

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