On the Uniqueness of Limit Cycles in a Generalized Liénard System

Abstract

Kooij and Sun (J Math Anal Appl 208:260-276, 1997) proposed a theorem to guarantee the uniqueness of limit cycles for the generalized Liénard system dx/dt = h(y) − F(x), dy/dt = −g(x). We will give a counterexample to their theorem. Moreover, we shall give some sufficient conditions for the existence, uniqueness and hyperbolicity of limit cycles. Keywords Generalized Liénard systems • Limit cycle • Uniqueness • Hyperbolicity where the functions in (1) are assumed to be continuous and such that uniqueness for solutions of initial value problems is guaranteed. We define, as usual, G(x) := x 0 g(s)ds and H (y) := y 0 h(s)ds. Huang and Sun [8] have shown a theorem to B Ping Yan

A Note on Small Amplitude Limit Cycles of Liénard Equations Theory

“…It represents a very important class of nonlinear systems due to its appearance in some branches of science and engineering as well as in some ecological models, planar physical models, and even in some chemical models, where using a suitable transformation can change these systems into nonlinear Liénard systems. However, an extensive attention has been also devoted to the question of its uniqueness [4][5][6]; this uniqueness can be verified using different ways of methods based on Poincare-Bendixson theorem. In [4], Zhou et al proposed a set of theorems for the limit cycles' uniqueness for the Liénard systems; the proposed theorems represent a guarantee to complete the proof of some previous works' propositions.…”