Homoclinic Boundary-Saddle Bifurcations in Planar Nonsmooth Vector Fields

In a smooth dynamical system, homoclinic connection is an orbit connecting saddle equilibrium to itself. Under perturbation, homoclinics are associated with bifurcations of periodic orbits, and chaos in higher dimensions. Homoclinic connections nonsmooth systems complicated by their interactions discontinuities vector fields. A may involve regular outside discontinuity set, or pseudo-saddle on segments the cross slide along discontinuity. Even simplest case saddle, which hits as parameter varied, surprisingly complex. this paper, we construct bifurcation diagrams for nonresonant saddles plane, unfolding boundary system. As application, exhibit such model forced pendulum.