Continuation of Periodic Orbits in Conservative Hybrid Dynamical Systems and its Application to Mechanical Systems with Impulsive Dynamics

Abstract

In autonomous differential equations where a single first integral is present, periodic orbits are well-known to belong to one-parameter families, parameterized by the first integral's values. This paper shows that this characteristic extends to a broader class of conservative hybrid dynamical systems (cHDSs). We study periodic orbits of a cHDS, introducing the concept of a hybrid first integral to characterize conservation in these systems. Additionally, our work presents a methodology that utilizes numerical continuation methods to generate these periodic orbits, building upon the concept of normal periodic orbits introduced by Sepulchre and MacKay (1997). We specifically compare state-based and time-based implementations of an cHDS as an important application detail in generating periodic orbits. Furthermore, we showcase the continuation process using exemplary conservative mechanical systems with impulsive dynamics.