Communications on Pure and Applied Analysis (CPAA)

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

Pages: 2509 - 2526, Volume 15, Issue 6, November 2016      doi:10.3934/cpaa.2016047

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Fangfang Jiang - School of Science, Jiangnan University, Wuxi, 214122, China (email)
Junping Shi - Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, United States (email)
Qing-guo Wang - Institute for Intelligent Systems, the University of Johannesburg, South Africa (email)
Jitao Sun - Department of Mathematics, Tongji University, Shanghai, 200092, China (email)

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.

Keywords:  Discontinuous system, Liénard system, crossing limit cycle, (non) existence, uniqueness.
Mathematics Subject Classification:  Primary: 34A36, 34C05; Secondary: 37N99.

Received: November 2015;      Revised: May 2016;      Available Online: September 2016.