Limit Sets of Discontinuous Vector Fields on Two-Dimensional Manifolds

Abstract

In this paper, limit sets of trajectories of a discontinuous vector field Z defined on a two-dimensional manifold M are studied. As in the classical Poincaré–Bendixson theorem, trajectories of Z are supposed to be confined on some compact invariant set $K\subset V$, where V is a coordinate neighborhood of M, and we require that Z and K fulfill some hypotheses analogous to the referred theorem. More precisely, M is split in an arbitrary number of regions by a set of smooth curves $\Sigma$ so that Z is defined by pieces on those regions, being eventually discontinuous on $\Sigma$. Moreover, it is assumed that K contains finite pseudo-equilibria of Z and at most two pieces of it, each piece having finite equilibria on $K{\setminus }\Sigma$ and at most one tangency point on $K\cap \Sigma$. We require no extra assumption on $\Sigma$ but regularity, therefore the existence of the so-called sliding motion is allowed on $\Sigma$ along with crossing and tangency points. The main results of the paper fully describe the limit sets of trajectories of Z under the previous hypotheses (see Theorem 1) and also state some features of a particular limit set presenting non-empty interior and nondeterministic chaos (see Theorem 2). They also generalize some previous results of the literature when trajectories either slide on $\Sigma$ indefinitely or never slide again after some finite time (see the fundamental lemma). Some examples and classes of systems fitting the hypotheses of the main results are also provided in the paper along with an algorithm to apply Theorem 1 to robust discontinuous vector fields.