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arXiv · 2507.06408

A Spectral Contraction Framework for Periodic Solutions in Nonsmooth Dynamical Systems

Pascal Stiefenhofer
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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.

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Pascal Stiefenhofer. 2025-07-08. A Spectral Contraction Framework for Periodic Solutions in Nonsmooth Dynamical Systems. https://arxiv.org/abs/2507.06408

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