LAUSR

In this paper, we investigate some qualitative properties of crossing limit cycles for a discontinuous symmetric Liénard system with two zones separated by a straight line. In each zone, it is a smooth Liénard system. Firstly, by Poincaré mapping method and geometrical analysis, we provide two criteria concerning the existence, uniqueness and stability of a crossing limit cycle. Secondly, we consider the position problem of the unique crossing limit cycle. Several lemmas are given to obtain an explicit upper bound of amplitude of the limit cycle. Finally, an application to van der Pol model with discontinuous vector field is given, and Matlab simulations are presented to illustrate the obtained theoretical results.