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Global Dynamics of a Planar Filippov System with Symmetry

Hongjie Pan

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116, P. R. China

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Xiaofeng Chen

http://orcid.org/0000-0002-1967-9223

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116, P. R. China

E-mail Address: xiaofeng@fzu.edu.cn

Corresponding author.

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Jiao Pu

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116, P. R. China

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, and

Xiaoxing Chen

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350116, P. R. China

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https://doi.org/10.1142/S0218127420501059Cited by:1 (Source: Crossref)

Abstract

Chen [2016a, 2016b] studied global dynamics of the Filippov systems $ẍ + (α + β x^{2}) ẋ + x \pm sgn (x) = 0$ , respectively. To study the global dynamics of $ẍ + (α + β x^{2}) ẋ \pm x \pm sgn (x) = 0$ completely, since the dynamics of $ẍ + (α + β x^{2}) ẋ - x - sgn (x) = 0$ is very simple, we are only interested in the global dynamics of $ẍ + (α + β x^{2}) ẋ - x + sgn (x) = 0$ in this paper. Firstly, we use Briot–Bouquet transformations and normal sector methods to discuss these degenerate equilibria at infinity. Secondly, we discuss the number of limit cycles completely. Then, the sufficient and necessary conditions of existence of the heteroclinic loop are found. To estimate the upper bound of the heteroclinic loop bifurcation function on parameter space, a result on the amplitude of a unique limit cycle of a discontinuous Liénard system is given. Finally, the complete bifurcation diagram and all global phase portraits are presented. The global dynamic property of system $ẍ + (α + β x^{2}) ẋ - x + sgn (x) = 0$ is totally different from systems $ẍ + (α + β x^{2}) ẋ + x \pm sgn (x) = 0$ .

Keywords:

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