A Spectral Contraction Framework for Periodic Solutions in Nonsmooth Dynamical Systems

TL;DR

This paper introduces a contraction-based framework to prove the existence and exponential stability of periodic solutions in planar nonsmooth dynamical systems with discontinuities, extending classical methods to more complex systems.

Contribution

It generalizes contraction theory to two-dimensional nonsmooth systems using a weighted metric and Clarke's Jacobian, providing a robust stability analysis tool.

Findings

Establishes global exponential contraction in nonsmooth systems

Guarantees existence and uniqueness of attracting periodic orbits

Applicable to hybrid control and nonsmooth mechanics

Abstract

We develop a contraction-based framework to establish the existence and exponential stability of periodic solutions in planar nonsmooth dynamical systems governed by Filippov differential inclusions. The method integrates a time- and state-dependent weighted metric with Clarke's generalized Jacobian and a uniform jump condition across switching manifolds to guarantee global exponential contraction on compact, forward-invariant sets. This work generalizes classical contraction results from smooth one-dimensional systems to two-dimensional systems with discontinuities and sliding behavior. A fixed-point argument ensures the existence and uniqueness of an attracting periodic orbit. The framework offers a robust analytic tool for stability analysis in piecewise-smooth systems, with applications in hybrid control, nonsmooth mechanics, and computational dynamics.