Stability and Perturbations of Generalized Heteroclinic Loops in Piecewise Smooth Systems

We investigate a class of planar piecewise smooth systems with a generalized heteroclinic loop (a closed curve composed of hyperbolic saddle points, generalized singular points and regular orbits). We give conditions for the stability of the generalized heteroclinic loop and provide some sufficient conditions for the maximum number of limit cycles that bifurcate from the heteroclinic connection. The discussions rely on the approximation of the Poincaré map, which is constructed near the generalized heteroclinic loop. To obtain it, we introduce the Dulac map and use Melnikov method. By analyzing the fixed point of the Poincaré map, we get the number of limit cycles, which can be produced from the generalized heteroclinic loop. As applications to our theories, we give an example to show that two limit cycles can appear.