On the Number of Limit Cycles for Discontinuous Generalized Liénard Polynomial Differential Systems

Abstract

In this paper, we investigate the number of limit cycles for a class of discontinuous planar differential systems with multiple sectors separated by many rays originating from the origin. In each sector, it is a smooth generalized Liénard polynomial differential system x = −y + g 1 (x) + f 1 (x)y and y = x + g 2 (x) + f 2 (x)y, where f i (x) and g i (x) for i = 1, 2 are polynomials of variable x with any given degree. By the averaging theory of first-order for discontinuous differential systems, we provide the criteria on the maximum number of medium amplitude limit cycles for the discontinuous generalized Liénard polynomial differential systems. The upper bound for the number of medium amplitude limit cycles can be attained by specific examples.

On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line

“…From (14) in Lemma 3.3, then the functions H(p z (y)) − H(y) (where z ∈ {L, R}) are strictly decreasing for all orbits crossing the curve Σ 0 F for x > x 0 and x < x 1 . So one has that…”

“…For the continuous or smooth differential systems, there have been many achievements, see for example [8,12,6,21,20,27,29,17,1] and references therein. In recent years, much progress has been made in studying relevant problems for the discontinuous differential systems, see for example [2,3,4,13,14,15,5,7,9,10,19,22] and references therein. But most 2510 FANGFANG JIANG, JUNPING SHI, QING-GUO WANG AND JITAO SUN of these existing papers focus on the existence, uniqueness or multiplicity of limit cycle for the piecewise linear discontinuous differential systems [2,7,9,10].…”

“…For the nonlinear discontinuous planar Liénard systems, there have been several recent studies, for example [14,15,19,22]. In [22], the uniqueness of crossing limit cycle for a nonlinear Liénard system with a discontinuity line was shown, provided that periodic orbit exists.…”