Canard explosion in Filippov system with a cusp-fold singularity via regularization

In this paper, we reveal the completely dynamical process of canard explosion in planar Filippov system with a cusp-fold singularity via Sotomayor-Teixeira regularization. It is found that the cusp-fold singularity is the organization center responsible for the birth and the death of canard explosion in planar Filippov system. By unfolding the cusp-fold singularity, we obtain a suitable topology in the resulting regularized system to describe canard explosion from small-amplitude cycle via the first supercritical Hopf bifurcation to canard cycle without head, the maximal canard, canard cycle with head, and finally relaxation oscillation happening quickly. In the current setting, the visible-invisible fold-fold singularity and the invisible fold singularity unfolded from the cusp-fold singularity respectively play the roles analogous to the canard point and the jump point in smooth singular perturbation system. After the occurrence of canard explosion, the relaxation oscillation will then disappear via the bifurcation of homoclinic-like connection and the second Hopf bifurcation. All the bifurcation curves are determined explicitly, and all the theoretical findings are verified by numerical experiments.