The Limit Cycles of the Higgins–Selkov Systems

Abstract

In this paper, we investigate the problem of limit cycles for general Higgins–Selkov systems with degree $$n+1$$ . In particular, we first prove the uniqueness of limit cycles for a general Liénard system, which allows for discontinuity. Then, by changing the Higgins–Selkov systems into Liénard systems, theorems and some techniques for Liénard systems can be applied. After, we prove the nonexistence of limit cycles if the bifurcation parameter is outside an open interval. Finally, we complete the analysis of limit cycles for the Higgins–Selkov systems showing its uniqueness.