Bifurcações de campos vetoriais em duas zonas com simetria - Castro, Ubirajara José Gama de (2017)
In this work we study reversible vector fields in two zones and equivariant vector fields in two zones. Our main result is the classification of the symmetric singularities of codimensions 0,1 and 2 of such vector fields. More precisely, in the reversible case in R3, where the dimension of the fixed points variety of the involution associated to the vector field is 2, we present all bifurcation diagram of the codimensions 1 and 2 singularities, describing the changes in the behavior of the symmetric singularities and tangents of the vector field with the transition manifold, S, according to the variation of the bifucartion parameter. We also show the existence of invariant cylinders and, in this case, doing small perturbations we determine invariant manifolds that persisted and we determine the number of limit cycles that were born. When the vector field defined on two zones is equivariant, the dynamic is enriched with the emergence of the sliding vector field and we also do a local study and the classification of singularities (and pseudo-singularities) of codimensions 0,1 and 2. We show the existence of homoclinic sliding orbit and that it is a codimension one phenomenon. Moreover, provided the symmetry we get a double Shilnikov sliding orbit.