Resumen de Linearization and Perturbations of Piecewise Smooth Vector Fields with a Boundary Equilibrium
Tao Li, Xingwu Chen
In this paper we study the linearization and perturbations of planar piecewise smooth vector fields that consist of two smooth vector fields separated by the straight line y = 0 and sharing the origin as a non-degenerate equilibrium. In the sense of - equivalence, we provide a sufficient condition for piecewise linearization near the origin, generalizing the classical linearization theorem to piecewise smooth vector fields. This condition is hard to be weakened because there exist vector fields that are not piecewise linearizable when this condition is not satisfied. Then a necessary and sufficient condition for local -structural stability is established when the origin is still an equilibrium of both smooth vector fields under perturbations. In the opposition to this case, we prove that for any piecewise smooth vector field studied in this paper there are perturbations with crossing limit cycles bifurcating from the origin.Moreover, besides the fold-fold type given in previous publications we find some new types of singularities, such as types of center-center, center-saddle and saddle-saddle, to birth any finitely or infinitely many crossing limit cycles.
