Bifurcations of a Generalized Heteroclinic Loop in a Planar Piecewise Smooth System with Periodic Perturbations

This work deals with the periodic orbit bifurcations of a T-periodic perturbed piecewise smooth system whose unperturbed part has a generalized heteroclinic loop connecting a hyperbolic critical point and a quadratic tangential singularity. By constructing several displacement functions that depend on perturbation parameter ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} and time t, sufficient conditions of the existence of a homoclinic loop and a sliding generalized heteroclinic loop (that is a generalized heteroclinic loop a part of which lies on the switching manifold) are obtained. As the application, we give a concrete example to show that under suitable perturbations of the generalized heteroclinic loop the corresponding phenomena can appear.