On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line

Fangfang  Jiang; Junping Shi; Qing-guo  Wang; Jitao  Sun

doi:10.3934/cpaa.2016047

Article Contents

2016, Volume 15, Issue 6: 2509-2526. Doi: 10.3934/cpaa.2016047

This issue Previous Article Positive solutions for Robin problems with general potential and logistic reaction Next Article

On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line

1.
School of Science, Jiangnan University, Wuxi, 214122
2.
Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795
3.
Institute for Intelligent Systems, the University of Johannesburg
4.
Department of Mathematics, Tongji University, Shanghai, 200092

Received: November 2015

Revised: May 2016

Published: November 2016

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Abstract

In this paper, we investigate the existence and uniqueness of crossing limit cycle for a planar nonlinear Liénard system which is discontinuous along a straight line (called a discontinuity line). By using the Poincaré mapping method and some analysis techniques, a criterion for the existence, uniqueness and stability of a crossing limit cycle in the discontinuous differential system is established. An application to Schnakenberg model of an autocatalytic chemical reaction is given to illustrate the effectiveness of our result. We also consider a class of discontinuous piecewise linear differential systems and give a necessary condition of the existence of crossing limit cycle, which can be used to prove the non-existence of crossing limit cycle.

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Mathematics Subject Classification: Primary: 34A36, 34C05; Secondary: 37N99.

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