Bifurcation Analysis of 3D Filippov Systems around Cusp-Fold Singularities
Fuente:
arXiv
Guardado en:
| Autores principales: | , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908449203290112 |
|---|---|
| author | Cespedes, Oscar A. R. Cristiano, Rony Gomide, Otávio M. L. |
| author_facet | Cespedes, Oscar A. R. Cristiano, Rony Gomide, Otávio M. L. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_10514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bifurcation Analysis of 3D Filippov Systems around Cusp-Fold Singularities Cespedes, Oscar A. R. Cristiano, Rony Gomide, Otávio M. L. Dynamical Systems Filippov systems, Bifurcation Theory, cusp-fold singularity, boost converter, crossing limit cycle, polycycle 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. |
| title | Bifurcation Analysis of 3D Filippov Systems around Cusp-Fold Singularities |
| topic | Dynamical Systems Filippov systems, Bifurcation Theory, cusp-fold singularity, boost converter, crossing limit cycle, polycycle |
| url | https://arxiv.org/abs/2507.10514 |