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 epsilon 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.

Automated summary

This work focuses on the bifurcation of periodic orbits in a perturbed piecewise smooth system with a generalized heteroclinic loop connecting a hyperbolic critical point and a quadratic tangential singularity. By constructing displacement functions dependent on the perturbation parameter epsilon and time t, the conditions for the existence of a homoclinic loop and a sliding generalized heteroclinic loop are obtained. A concrete example is provided to demonstrate the occurrence of corresponding phenomena under suitable perturbations of the generalized heteroclinic loop.