A piecewise-linear fixed point theorem
Fuente:
arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909428499873792 |
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| author | Simpson, David J. W. |
| author_facet | Simpson, David J. W. |
| contents | We prove that if a continuous piecewise-smooth map on $\mathbb{R}^n$ is comprised of two linear functions, has a bounded orbit, and satisfies a certain non-degeneracy condition, then it has a fixed point. The result has important consequences to the bifurcation theory of nonsmooth dynamical systems, yet the proof requires only elementary linear algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11118 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A piecewise-linear fixed point theorem Simpson, David J. W. Dynamical Systems Classical Analysis and ODEs 37C25, 39A28 We prove that if a continuous piecewise-smooth map on $\mathbb{R}^n$ is comprised of two linear functions, has a bounded orbit, and satisfies a certain non-degeneracy condition, then it has a fixed point. The result has important consequences to the bifurcation theory of nonsmooth dynamical systems, yet the proof requires only elementary linear algebra. |
| title | A piecewise-linear fixed point theorem |
| topic | Dynamical Systems Classical Analysis and ODEs 37C25, 39A28 |
| url | https://arxiv.org/abs/2412.11118 |