Dimension Reduction Near Periodic Orbits of Hybrid Systems

TL;DR

This paper proves that hybrid systems with periodic orbits can be simplified to low-dimensional models near the orbit, facilitating analysis and understanding of complex systems like legged locomotion.

Contribution

It establishes the existence of a finite-dimensional invariant subsystem near periodic orbits in hybrid systems with constant-rank Poincaré maps, enabling reduced-order analysis.

Findings

Existence of low-dimensional invariant subsystems near periodic orbits.

Finite-time attraction of nearby trajectories to the invariant subsystem.

Applicability of smooth system tools like Floquet theory to hybrid systems.

Abstract

When the Poincar\'{e} map associated with a periodic orbit of a hybrid dynamical system has constant-rank iterates, we demonstrate the existence of a constant-dimensional invariant subsystem near the orbit which attracts all nearby trajectories in finite time. This result shows that the long-term behavior of a hybrid model with a large number of degrees-of-freedom may be governed by a low-dimensional smooth dynamical system. The appearance of such simplified models enables the translation of analytical tools from smooth systems-such as Floquet theory-to the hybrid setting and provides a bridge between the efforts of biologists and engineers studying legged locomotion.