Limit cycles of a Li\'enard system with symmetry allowing for discontinuity

TL;DR

This paper extends the analysis of limit cycles in Li'enard systems to include discontinuous cases, providing explicit bounds on amplitude and bifurcation positions, with applications to complex oscillator models.

Contribution

It generalizes previous smooth system results to discontinuous systems, offering explicit amplitude bounds and bifurcation estimates for Li'enard systems with symmetry.

Findings

Explicit upper bounds for the amplitude of two limit cycles.

Identification of the double-limit-cycle bifurcation surface.

Application of results to complex oscillator models.

Abstract

This paper presents new results on the limit cycles of a Li\'enard system with symmetry allowing for discontinuity. Our results generalize and improve the results in [33,34]. The results in [34] 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 Li\'enard 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 Li\'enard oscillator (see [21,32]) in Application 1. Li and Rousseau [21] 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.