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

In this paper, we investigate the existence and uniqueness of crossing limit cycle for a planar nonlinear Líenard system which is discontinuous along a straight line (called a discontinuity line). By using the Poincaŕe 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.