Angular Values of Nonautonomous Linear Dynamical Systems: Part II -Reduction Theory and Algorithm

TL;DR

This paper develops a reduction theory and an efficient algorithm for computing angular values in nonautonomous linear dynamical systems, linking these values to spectral properties and enabling analysis of systems up to dimension four.

Contribution

It introduces a reduction theorem connecting angular values to spectral bundles and provides an algorithm for calculating angular values in low-dimensional systems.

Findings

The reduction theorem simplifies the computation of angular values.

The algorithm efficiently computes angular values for systems up to dimension four.

Application demonstrates detection of fastest rotating subspaces beyond dominant dynamics.

Abstract

This work focuses on angular values of nonautonomous dynamical systems which have been introduced for general random and (non)autonomous dynamical systems in a previous publication [W.-J. Beyn, G. Froyland, and T. H\"uls, SIAM J. Appl. Dyn. Syst., 21 (2022), pp. 1245--1286]. The angular value of dimension $s$ measures the maximal average rotation which an $s$-dimensional subspace of the phase space experiences through the dynamics of a discrete-time linear system. Our main results relate the notion of angular value to the well-known dichotomy (or Sacker--Sell) spectrum and its associated spectral bundles. In particular, we prove a reduction theorem which shows that instead of maximizing over all subspaces, it suffices to maximize over so-called trace spaces which have their basis in the spectral fibers. The reduction leads to an algorithm for computing angular values of dimensions one and two. We apply the algorithm to several systems of dimension up to 4 and demonstrate its efficiency to detect the fastest rotating subspace even if it is not dominant under the forward dynamics.