Angular values of nonautonomous and random linear dynamical systems: Part I -- Fundamentals

TL;DR

This paper introduces angular values as a way to measure the long-term behavior of subspaces in nonautonomous and random linear systems, establishing foundational theory and computational methods.

Contribution

It defines angular values for deterministic and random systems, proves their existence, and develops algorithms for their computation, extending classical rotation number concepts.

Findings

Angular values exist for random dynamical systems.

In 2D, angular values relate to classical rotation numbers.

A numerical algorithm for computing angular values is proposed.

Abstract

We introduce the notion of angular values for deterministic linear difference equations and random linear cocycles. We measure the principal angles between subspaces of fixed dimension as they evolve under nonautonomous or random linear dynamics. The focus is on long-term averages of these principal angles, which we call angular values: we demonstrate relationships between different types of angular values and prove their existence for random dynamical systems. For one-dimensional subspaces in two-dimensional systems our angular values agree with the classical theory of rotation numbers for orientation-preserving circle homeomorphisms if the matrix has positive determinant and does not rotate vectors by more than $\frac{\pi}{2}$. Because our notion of angular values ignores orientation by looking at subspaces rather than vectors, our results apply to dynamical systems of any dimension and to subspaces of arbitrary dimension. The second part of the paper delves deeper into the theory of the autonomous case. We explore the relation to (generalized) eigenspaces, provide some explicit formulas for angular values, and set up a general numerical algorithm for computing angular values via Schur decompositions.