# American Institute of Mathematical Sciences

November  2016, 15(6): 2509-2526. doi: 10.3934/cpaa.2016047

## 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, China 2 Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795 3 Institute for Intelligent Systems, the University of Johannesburg, South Africa 4 Department of Mathematics, Tongji University, Shanghai, 200092, China

Received  November 2015 Revised  May 2016 Published  September 2016

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.
Citation: Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
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