Bifurcation Analysis of 3D Filippov Systems around Cusp-Fold Singularities

Abstract: This paper investigates the local behavior of 3D Filippov systems $Z=(X,Y)$, focusing on the dynamics around cusp-fold singularities. These singular points, characterized by cubic contact of vector field $X$ and quadratic contact of vector field $Y$ on the switching manifold, are structurally unstable under small perturbations of $Z$, giving rise to significant bifurcation phenomena. We analyze the bifurcations of a 3D Filippov system around an invisible cusp-fold singularity, providing a detailed characterization of its crossing dynamics under certain conditions. We classify the characteristics of the singularity when it emerges generically in one-parameter families (a codimension-one phenomenon), and we show that no crossing limit cycles (CLCs) locally bifurcate from it in this particular scenario. When the vector fields $X$ and $Y$ are anti-collinear at the cusp-fold singularity, we provide conditions for the generic emergence of this point in two-parameter families (a codimension-two phenomenon). In this case, we show that the unfolding of such a singularity leads to a bifurcating CLC, which degenerates into a fold-regular polycycle (self-connection at a fold-regular singularity). Furthermore, we numerically derive the polycycle bifurcation curve and complete the two-parameter bifurcation set for a boost converter system previously studied in the literature. This allows the identification of parameter regions where the boost converter system exhibits a CLC in its phase portrait, providing a understanding of its complex dynamics.