On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

Abstract

In this paper we study the non-existence and the uniqueness of limit cycles for the Liénard differential equation of the form x'' − f(x)x' + g(x) = 0 where the functions f and g satisfy xf(x) > 0 and xg(x) > 0 for x ≠ 0 but can be discontinuous at x = 0. In particular, our results allow us to prove the non-existence of limit cycles under suitable assumptions, and also prove the existence and uniqueness of a limit cycle in a class of discontinuous Liénard systems which are relevant in engineering applications.