Bifurcation of Nonhyperbolic Limit Cycles in Piecewise Smooth Planar Systems with Finitely Many Zones

Abstract

Like for smooth systems, it is very important to discuss the stability and bifurcation of limit cycles in a piecewise smooth planar system. Most of the previous works focus only on hyperbolic limit cycles. Few works have considered nonhyperbolic limit cycles. In fact, to date, no concrete examples of piecewise smooth planar system with nonhyperbolic limit cycles have been given in literature. In this paper, we consider for the first time the bifurcation of nonhyperbolic limit cycles in piecewise smooth planar systems with discontinuities on finitely many straight lines intersecting at the origin. We present a method of Melnikov type to derive two quantities which can be used to determine the stability and the number of limit cycles that can bifurcate from a nonhyperbolic limit cycle of a piecewise smooth planar system. As applications, we present two examples of piecewise smooth systems with two and three zones respectively whose unperturbed system has a nonhyperbolic limit cycle.