# Birth of isolated nested cylinders and limit cycles in 3D piecewise smooth vector fields with symmetry

## TL;DR

This paper investigates the bifurcation phenomena in symmetric 3D piecewise smooth vector fields, demonstrating how to generate nested cylinders and limit cycles through perturbations of a symmetric fold model.

## Contribution

It introduces a normal form for symmetric 3D piecewise smooth vector fields and shows how to produce specific numbers of invariant planes and limit cycles via perturbations.

## Key findings

- Existence of a continuum of nested cylinders with centers.
- Perturbation methods to create exactly $ ext{L}$ invariant planes with centers.
- Construction of models with exactly $k imes ext{L}$ limit cycles.

## Abstract

Our start point is a 3D piecewise smooth vector field defined in two zones and presenting a shared fold curve for the two smooth vector fields considered. Moreover, these smooth vector fields are symmetric relative to the fold curve, giving raise to a continuum of nested topological cylinders such that each orthogonal section of these cylinders is filled by centers. First we prove that the normal form considered represents a whole class of piecewise smooth vector fields. After we perturb the initial model in order to obtain exactly $\mathcal{L}$ invariant planes containing centers. A second perturbation of the initial model also is considered in order to obtain exactly $k$ isolated cylinders filled by periodic orbits. Finally, joining the two previous bifurcations we are able to exhibit a model, preserving the symmetry relative to the fold curve, and having exactly $k.\mathcal{L}$ limit cycles.