Spectral Contraction Framework for Planar Filippov Systems: Existence and Stability of Nonsmooth Periodic Orbits

Year 2025, Volume: 8 Issue: 4, 189 - 209, 08.12.2025

Pascal Stiefenhofer

https://doi.org/10.33434/cams.1729696

Cited By: 1

Abstract

We present a contraction-based framework for analyzing periodic behavior in planar nonsmooth dynamical systems governed by Filippov differential inclusions. The approach combines a time- and state-dependent weighted Riemannian metric with Clarke’s generalized Jacobian, together with a uniform jump condition across switching manifolds, to obtain sufficient conditions for exponential contraction on compact forward-invariant sets. These local estimates are assembled into a global stability result via a Banach fixed-point argument, establishing the existence and uniqueness of an exponentially attracting periodic orbit. The theory extends classical contraction results from the scalar setting to two-dimensional systems with discontinuities, sliding motion, and switching phenomena. Beyond clarifying the role of contraction in nonsmooth dynamics, the framework provides a systematic analytic tool for stability verification, with potential applications to hybrid control, nonsmooth mechanical oscillators, and computational methods for piecewise-smooth systems.

Keywords

Clarke generalized Jacobian , Contraction theory , Filippov differential inclusions , Nonsmooth dynamical systems , Periodic solutions

Ethical Statement

no data used

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