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September 10, 2025

Canonical Forms of 3D Cusp-Fold Singularities and its Unfolding

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The study identifies 32 distinct canonical forms for 3D cusp-fold singularities, expanding our understanding of their properties.
Each canonical form is topologically unique, providing essential insights into the structure of cusp-fold singularities in vector fields.
Bifurcations of these forms are explored, further detailing the behavior of vector fields near cusp-fold singularities.
This work highlights the importance of piecewise smooth vector fields in understanding topological aspects of singularities.

Resumen

In this paper we obtain 32 canonical forms for 3D piecewise smooth vector fields presenting the so called cusp-fold singularity. All these canonical forms are topologically distinct and collect the main topological aspects of the singularities described. Also, one-parameter bifurcations of these canonical forms are presented and the topologically equivalent piecewise smooth vector fields are obtained.

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synapsesocial.com/papers/68c1ae7754b1d3bfb60e6aa7 https://doi.org/https://doi.org/10.52843/cassyni.cdtz3f

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