Linearization and Perturbations of Piecewise Smooth Vector Fields with a Boundary Equilibrium

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 E 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 s-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.

Automated summary

This paper investigates the linearization and perturbations of planar piecewise smooth vector fields composed of two smooth vector fields separated by the line y=0 and sharing the origin as a non-degenerate equilibrium. A sufficient condition is provided for piecewise linearization near the origin in terms of E equivalence, generalizing the classical linearization theorem to piecewise smooth vector fields. A necessary and sufficient condition for local s-structural stability is established when the origin remains an equilibrium of both smooth vector fields under perturbations. In contrast, it is proved that for any piecewise smooth vector field studied in this paper, there are perturbations with crossing limit cycles bifurcating from the origin. Besides the fold-fold type, new types of singularities such as center-center, center-saddle, and saddle-saddle are found to give rise to finitely or infinitely many crossing limit cycles.