Crossing limit cycles of nonsmooth Li\'enard systems and applications

TL;DR

This paper extends the analysis of crossing limit cycles in nonsmooth Li'enard systems to multiple equilibria, providing conditions for their existence, uniqueness, and nonexistence, with applications to piecewise linear systems.

Contribution

It generalizes previous results by considering multiple equilibria and offers new criteria for the existence and uniqueness of crossing limit cycles in nonsmooth systems.

Findings

Established conditions for the existence of crossing limit cycles.

Proved the uniqueness of crossing limit cycles in certain systems.

Provided a sufficient condition for the nonexistence of crossing limit cycles.

Abstract

Continuing the investigation for the number of crossing limit cycles of nonsmooth Li\'enard systems in [Nonlinearity 21(2008), 2121-2142] for the case of a unique equilibrium, in this paper we consider the case of any number of equilibria. We give results about the existence and uniqueness of crossing limit cycles, which hold not only for a unique equilibrium but also for multiple equilibria. Moreover, we find a sufficient condition for the nonexistence of crossing limit cycles. Finally, applying our results we prove the uniqueness of crossing limit cycles for planar piecewise linear systems with a line of discontinuity and without sliding sets.