Bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems

The generalized homoclinic loop appears in the study of dynamics on piecewise smooth differential systems during the past two decades. For planar piecewise smooth differential systems, there are concrete examples showing that under suitable perturbations of a generalized homoclinic loop one or two limit cycles can appear. But up to now there is no a general theory to study the cyclicity of a generalized homoclinic loop, that is, the maximal number of limit cycles which are bifurcated from it.In this paper, we provide some sufficient conditions on the cyclicity of some homoclinic loops. Especially we prove the existence of one or two limit cycles which are bifurcated from a generalized homoclinic loop of an unperturbed piecewise smooth differential systems. Also we obtain a class of piecewise differential system in which two limit cycles can be bifurcated from a generalized homoclinic loop with multiplicity one. The phenomena cannot happen in the smooth differential systems. Finally we provide five concrete piecewise smooth differential systems showing the applications of our theories on this phenomenon.