Dynamics of planar vector fields near a non-smooth equilibrium

TL;DR

This paper analyzes the local behavior of planar piecewise smooth vector fields near a non-smooth equilibrium, providing conditions for linearization, stability, and bifurcation phenomena including limit cycles.

Contribution

It offers new criteria for linearization and structural stability of non-smooth equilibria, and characterizes bifurcations leading to multiple limit cycles.

Findings

Provided a sufficient condition for linearization of the vector fields.

Established a necessary and sufficient condition for local structural stability.

Proved the existence of bifurcating limit cycles, including arbitrarily many.

Abstract

In this paper we contribute to qualitative and geometric analysis of planar piecewise smooth vector fields, which 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 $\Sigma$-equivalence, we provide a sufficient condition for linearization and give phase portraits and normal forms for these linearizable vector fields. This condition is hard to be weakened because there exist vector fields which are not linearizable when this condition is not satisfied. Regarding perturbations, a necessary and sufficient condition for local $\Sigma$-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 is a limit cycle bifurcating from the origin, and there are some piecewise smooth vector fields such that for any positive integer $m$ there is a perturbation having exactly $m$ limit cycles bifurcating from the origin. Here $m$ maybe infinity.