Limit sets in impulsive semidynamical systems

Abstract

In this paper, we establish several fundamental properties in impulsive semidynamical systems. First, we formulate a counterpart of the continuous dependence on the initial conditions for impulsive dynamical systems, and also establish some equivalent properties. Second, we present several theorems similar to the Poincaré-Bendixson theorem for two-dimensional impulsive systems, i.e. if the omega limit set of a bounded infinite trajectory (with an infinite number of impulses) contains no rest points, then there exists an almost recurrent orbit in the limit set. Further, if the omega limit set contains an interior point, then it is a chaotic set; otherwise, if the limit set contains no interior points, then the limit set contains a periodic orbit or a Cantor-type minimal set in which each orbit is almost recurrent.