There exist transitive piecewise smooth vector fields on $\mathbb{S}^2$ but not robustly transitive

TL;DR

The paper demonstrates the existence of topologically transitive piecewise-smooth vector fields on the sphere, highlights the conditions involving sliding and escaping regions, and shows that such transitivity is not robust or structurally stable.

Contribution

It proves the existence of transitive piecewise-smooth vector fields on , characterizes their phase portrait features, and establishes their lack of robustness and structural stability.

Findings

Transitive piecewise-smooth vector fields exist on .

Transitivity involves trajectories switching between sliding and escaping regions.

Such vector fields are not robustly transitive or structurally stable.

Abstract

It is well known that smooth (or continuous) vector fields cannot be topologically transitive on the sphere $\S^2$. Piecewise-smooth vector fields, on the other hand, may present non-trivial recurrence even on $\S^2$. Accordingly, in this paper the existence of topologically transitive piecewise-smooth vector fields on $\S^2$ is proved, see Theorem \ref{teorema-principal}. We also prove that transitivity occurs alongside the presence of some particular portions of the phase portrait known as {\it sliding region} and {\it escaping region}. More precisely, Theorem \ref{main:transitivity} states that, under the presence of transitivity, trajectories must interchange between sliding and escaping regions through tangency points. In addition, we prove that every transitive piecewise-smooth vector field is neither robustly transitive nor structural stable on $\S^2$, see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem \ref{main:general} addressing non-robustness on general compact two-dimensional manifolds.