2025/07/14 by Oscar A.R. Cespedes, Rony Cristiano, Cespedes, Oscar A. R. +3
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Bifurcation Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Filippov systems #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #boost converter #crossing limit cycle #cusp-fold singularity #polycycle
paper · pdf · doi:10.48550/arxiv.2507.10514
openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.