Limit Sets of Discontinuous Vector Fields on Two-Dimensional Manifolds

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