The stability of fixed points on switching manifolds of piecewise-smooth continuous maps

TL;DR

This paper investigates the stability of fixed points on switching manifolds in continuous, piecewise-smooth maps, introducing measure-theoretic stability concepts and providing conditions for exponential stability using invariant measures.

Contribution

It develops a measure-theoretic framework for stability analysis of fixed points on switching manifolds and derives conditions for exponential stability in piecewise-linear maps.

Findings

Fixed points can be Milnor attractors despite being unstable.

Stability can be analyzed via invariant probability measures on the sphere.

Fixed points may be stable even with area-expanding components.

Abstract

This paper concerns piecewise-smooth maps on $\mathbb{R}^d$ that are continuous but not differentiable on switching manifolds (where the functional form of the map changes). The stability of fixed points on switching manifolds is investigated in scenarios for which one-sided derivatives are locally bounded. The lack of differentiability allows fixed points to be Milnor attractors despite being unstable. For this reason a measure-theoretic notion of stability is considered in addition to standard notions of stability. Locally the map is well approximated by a piecewise-linear map that is linearly homogeneous when the fixed point is at the origin. For the class of continuous, linearly homogeneous maps, and $o({\bf x})$ perturbations of these maps, a sufficient condition for the exponential stability of the origin is obtained. It is shown how the stability of the origin can be determined by analysing invariant probability measures of a map on $\mathbb{S}^{d-1}$. The results are illustrated for the two-dimensional border-collision normal form. The fixed point may be asymptotically stable even if both smooth components of the map are area-expanding, and unstable even if it is the $\omega$-limit set of almost all points in $\mathbb{R}^d$.