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

TL;DR

This paper extends the concept of continuous families of periodic orbits to conservative hybrid dynamical systems, introducing hybrid first integrals and numerical continuation methods, with applications to mechanical systems with impulses.

Contribution

It generalizes the existence of periodic orbit families to hybrid systems and develops a numerical continuation approach using hybrid first integrals.

Findings

Periodic orbits form one-parameter families in cHDSs.

Numerical continuation effectively generates periodic orbits in hybrid systems.

Application to mechanical systems demonstrates practical utility.

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.