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arXiv · 2412.11118

A piecewise-linear fixed point theorem

David J. W. Simpson
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Abstract

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.

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David J. W. Simpson. 2024-12-15. A piecewise-linear fixed point theorem. https://arxiv.org/abs/2412.11118

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