Bi-center problem and Hopf cyclicity of a Cubic Liénard system

Min Hu; Tao Li; Xingwu Chen

doi:10.3934/dcdsb.2019187

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2020, Volume 25, Issue 1: 401-414. Doi: 10.3934/dcdsb.2019187

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Bi-center problem and Hopf cyclicity of a Cubic Liénard system

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

^* Corresponding author: Xingwu Chen(xingwu.chen@hotmail.com)
^* Corresponding author: Xingwu Chen(xingwu.chen@hotmail.com)

Received: June 2018

Early access: September 2019

Published: January 2020

The second author is supported by Graduate Student's Research and Innovation Fund 2018YJSY047 and Doctoral Graduate Student's Academic Visit Fund of Sichuan University. The third author is supported by NSFC 11871355

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Abstract

In this paper we investigate the bi-center problem and the total Hopf cyclicity of two center-foci for the general cubic Liénard system which has three distinct equilibria and is equivalent to the general Liénard equation with cubic damping and restoring force. The location of these three equilibria is arbitrary, specially without any kind of symmetry. We find the necessary and sufficient condition for the existence of bi-centers and prove that there is no case of a unique center. On the Hopf cyclicity we prove that there are totally $ 9 $ possible styles of small amplitude limit cycles surrounding these two center-foci and $ 6 $ styles of them can occur, from which the total Hopf cyclicity is no more than $ 4 $ and no less than $ 2 $.

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Mathematics Subject Classification: Primary: 34C07, 34C23, 34D10, 37G10.

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