Abstract: 

Impulsive Dynamical Systems (IDS) can be seen as suitable mathematical models of real world phenomena that display abrupt changes in their behaviour. More precisely, an IDS is described by three objects: a continuous semiflow on a space M; a set D contained in M where the flow undergoes sudden perturbations; and an impulsive map from D to M, which determines the change in the trajectory each time it collides with the impulsive set D.  A key challenge, inherent to the dynamics, is that in general, an impulsive semiflow is not continuous.

In this talk I will present sufficient conditions for the existence of invariant measures. Moreover, I will show that, despite several examples which display the wilderness of impulsive semiflows, a C^1 typical impulsive semiflow presents some features of the ones without discontinuities, namely, the existence of plenty of hyperbolic periodic orbits. The talk is based on works in collaboration with Anfonso (State University of São Paulo, Brazil), Bonotto (University of São Paulo, Brazil), Torres (Minho University, Portugal) and Varandas (Aveiro University, Portugal).