Limit cycles of a Liénard system with symmetry allowing for discontinuity

Abstract

This paper presents new results on the limit cycles of a Liénard system with symmetry allowing for discontinuity. Our results generalize and improve the results in [32, Theorem 1 and 2] or the monograph [33, Chapter 4, Theorem 5.2]. The results in [33] are only valid for the smooth system. We emphasize that our main results are valid for discontinuous systems. Moreover, we show the presence and an explicit upper bound for the amplitude of the two limit cycles, and we estimate the position of the double-limit-cycle bifurcation surface in the parameter space. Until now, there is no result to determine the amplitude of the two limit cycles. The existing results on the amplitude of limit cycles guarantee that the Liénard system has a unique limit cycle. Finally, some applications and examples are provided to show the effectiveness of our results. We revisit a co-dimension-3 Liénard oscillator (see [22,34]) in Application 1. Li and Rousseau [22] studied the limit cycles of such a system when the parameters are small. However, for the general case of the parameters (in particular, the parameters are large), the upper bound of the limit cycles remains open. We completely provide the bifurcation diagram for the one-equilibrium case. Moreover, we determine the amplitude of the two limit cycles and estimate the position of the double-limit-cycle bifurcation surface for the one-equilibrium case. Application 2 is presented to study the limit cycles of a class of the Filippov system.