A piecewise-linear fixed point theorem

TL;DR

This paper proves a fixed point theorem for certain piecewise-linear maps in ^n, with implications for nonsmooth dynamical systems, using elementary linear algebra.

Contribution

It establishes a new fixed point theorem for continuous piecewise-smooth maps composed of two linear functions, with minimal mathematical prerequisites.

Findings

Existence of fixed points under specified conditions

Implications for bifurcation theory of nonsmooth systems

Elementary proof using linear algebra

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.