A Generalization of Poincaré’s Theorem to Periodic Hybrid and Impulsive Dynamical Systems

Poincaré’s method is well known for analyzing the stability of continuous-time periodic dynamical systems by studying the stability properties of a fixed point as an equilibrium point of a discrete-time system. In this paper we generalize Poincaré’s method to dynamical systems possessing left-continuous flows to address the stability of limit cycles and periodic orbits of left-continuous, hybrid, and impulsive dynamical systems. It is shown that resetting manifold (which gives rise to the state discontinuities) provides a natural hyperplane for defining a Poincaré return map. In the special case of impulsive dynamical systems, we show the Poincaré map replaces an nth-order impulsive dynamical system by an (n − 1)th-order discrete-time system for analyzing the stability of periodic orbits.