On the Existence and Uniqueness of Poincar\'e Maps for Systems with Impulse Effects

The Poincar\'e map is widely used to study the qualitative behavior of dynamical systems. For instance, it can be used to describe the existence of periodic solutions. The Poincar\'e map for dynamical systems with impulse effects was introduced in the last decade and mainly employed to study the existence of limit cycles (periodic gaits) for the locomotion of bipedal robots. We investigate sufficient conditions for the existence and uniqueness of Poincar\'e maps for dynamical systems with impulse effects evolving on a differentiable manifold. We apply the results to show the existence and uniqueness of Poincar\'e maps for systems with multiple domains.