2 research outputs found
There exist transitive piecewise smooth vector fields on but not robustly transitive
It is well known that smooth (or continuous) vector fields cannot be
topologically transitive on the sphere . Piecewise-smooth vector fields,
on the other hand, may present non-trivial recurrence even on .
Accordingly, in this paper the existence of topologically transitive
piecewise-smooth vector fields on 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 ,
see Theorem \ref{main:no-transitive}. We finish the paper proving Theorem
\ref{main:general} addressing non-robustness on general compact two-dimensional
manifolds