Angular spectra of linear dynamical systems in discrete time

TL;DR

The paper introduces the angular spectrum for linear discrete-time nonautonomous systems, formalizing the outer angular spectrum as the set of accumulation points of long-time averages of principal angles between successive subspaces driven by the dynamics. It develops a reduction to trace spaces, connects the spectrum to the dichotomy (Sacker–Sell) framework, and establishes invariance under kinematic similarity and under summable perturbations for systems with a complete exponential dichotomy (CED). A suite of perturbation and roughness results ties GED and CED concepts to spectral invariants, and a numerical framework based on finite-time spectra and trace-space reduction enables practical computation for models like the Lorenz and Henon systems as well as homoclinic orbits. The findings provide a robust, computable characterization of rotational behavior in nonautonomous linear dynamics and offer insights into spectral stability under perturbations, with implications for understanding long-time rotational stress in complex systems.

Abstract

In this work we introduce the notion of an angular spectrum for a linear discrete time nonautonomous dynamical system. The angular spectrum comprises all accumulation points of longtime averages formed by maximal principal angles between successive subspaces generated by the dynamical system. The angular spectrum is bounded by angular values which have previously been investigated by the authors. In this contribution we derive explicit formulas for the angular spectrum of some autonomous and specific nonautonomous systems. Based on a reduction principle we set up a numerical method for the general case; we investigate its convergence and apply the method to systems with a homoclinic orbit and a strange attractor. Our main theoretical result is a theorem on the invariance of the angular spectrum under summable perturbations of the given matrices (roughness theorem). It applies to systems with a so-called complete exponential dichotomy (CED), a concept which we introduce in this paper and which imposes more stringent conditions than those underlying the exponential dichotomy spectrum.

Figures (9)

• Figure 1: \newlabelwinkel20 Computation of the angle between two planes in $\mathbbm{R}^3$.
• Figure 1: \newlabelHenon0 Center part of a homoclinic orbit of \ref{['Hmap']} (left). The same orbit in phase space (right) with approximations of unstable (red) and stable manifolds of $\xi$.
• Figure 1: Distance of a homoclinic orbit$(\bar{x}_n)_{n\in\{0,1,\dots,2000\}}$ of the $3$D-Hénon system \ref{['Hmap']} to the fixed point $\xi$ in a semilogarithmic scale. \newlabelOrbit_norm0
• Figure 2: \newlabelexplicit0 Outer angular spectrum of the autonomous system $u_{n+1}=A(\rho,\varphi)u_n$ for $A(\rho,\varphi)$ from \ref{['rhomatrix']} as a function of the parameters $\rho \in (0,1]$, $\varphi\in (0,\frac{\pi}{2}]$.
• Figure 2: Construction of multi-humped orbits. \newlabelMulti0

Theorems & Definitions (73)

• Proposition 2.1
• Example 2.2
• Lemma 2.3
• Lemma 2.4
• Definition 2.5
• Lemma 2.6
• Proof 1
• Definition 2.7
• Proposition 2.8
• Proof 2