Crossing periodic orbits of nonsmooth Lienard systems and applications*

Continuing the investigation for the number of crossing periodic orbits of nonsmooth Lienard systems in (2008Nonlinearity212121-42) for the case of a unique equilibrium, in this paper we allow the considered system to have one or multiple equilibria. By constructing two control functions that are decreasing in a much narrower interval than the one used in the above work in the estimate of divergence integrals, we overcome the difficulty of comparing the heights of orbital arcs caused by the multiplicity of equilibria and give results about the existence and uniqueness of crossing periodic orbits, which hold not only for a unique equilibrium but also for multiple equilibria. Moreover, we find a sufficient condition for the existence of periodic annuli formed by crossing periodic orbits. Applying our results to planar piecewise linear systems with a line of discontinuity and without sliding sets, we prove the uniqueness of crossing limit cycles and hence give positive answers to conjectures 1 and 2 of Freireet al's work (2013Planar Filippov Systems with Maximal Crossing Set and Piecewise Linear Focus Dynamics (Progress and Challenges in Dynamical Systemsvol 54) (Heidelberg: Springer) pp 221-32).