Title: Birth of isolated nested cylinders and limit cycles in 3D piecewise smooth vector fields with symmetry,
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.