A piecewise-linear fixed point theorem
Dynamical Systems
2024-12-17 v1 Classical Analysis and ODEs
Abstract
We prove that if a continuous piecewise-smooth map on 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.
Cite
@article{arxiv.2412.11118,
title = {A piecewise-linear fixed point theorem},
author = {David J. W. Simpson},
journal= {arXiv preprint arXiv:2412.11118},
year = {2024}
}