Bifurcation analysis on a class of Z2-equivariant cubic switching systems showing eighteen limit cycles

Abstract

In this paper, the method developed for computing the Lyapunov constants of planar switching systems associated with an elementary singular point is applied to study bifurcation of limit cycles in a cubic switching system. A complete classification on the center conditions and 16 limit cycles of this system are obtained around the two foci (1,0) and (−1,0). Further, with the method, an example of cubic switching systems is constructed to show the existence of 18 small-amplitude limit cycles bifurcating from centers. This is a new lower bound on the maximal number of small-amplitude limit cycles obtained in such cubic switching systems. Finally, a method is present to show the realization of the 18 limit cycles.