A Spectral Contraction Framework for Periodic Solutions in Nonsmooth Dynamical Systems
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909681070374912 |
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| author | Stiefenhofer, Pascal |
| author_facet | Stiefenhofer, Pascal |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06408 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Spectral Contraction Framework for Periodic Solutions in Nonsmooth Dynamical Systems Stiefenhofer, Pascal Dynamical Systems 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. |
| title | A Spectral Contraction Framework for Periodic Solutions in Nonsmooth Dynamical Systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2507.06408 |