# Koopman Spectrum for Cascaded Systems

## TL;DR

This paper analyzes the Koopman spectrum of cascaded dynamical systems, showing conditions under which the systems' evolutions are asymptotically equivalent and their eigenfunctions relate, applicable to both linear and nonlinear cases.

## Contribution

It establishes the relationship between component subsystems' Koopman eigenfunctions and the cascaded system, including stability and eigenvalue properties, for both linear and nonlinear systems.

## Key findings

- Cascaded systems with stable components are asymptotically equivalent to decoupled systems.
- Koopman eigenvalues of components are preserved in the cascaded system.
- Eigenfunctions can be extended and composed to form eigenfunctions of the cascaded system.

## Abstract

This paper considers the evolution of Koopman principal eigenfunctions of cascaded dynamical systems. If each component subsystem is asymptotically stable, the matrix norms of the linear parts of the component subsystems are strictly increasing, and the component subsystems have disjoint spectrums, there exist perturbation functions for the initial conditions of each component subsystem such that the orbits of the cascaded system and the decoupled component subsystems have zero asymptotic relative error. This implies that the evolutions are asymptotically equivalent; cascaded compositions of stable systems are stable. These results hold for both cascaded systems with linear component subsystem dynamics and linear coupling terms and nonlinear cascades topologically conjugate to the linear case. We further show that the Koopman principal eigenvalues of each component subsystem are also Koopman eigenvalues of the cascaded system. The corresponding Koopman eigenfunctions of the cascaded system are formed by extending the domain of definition of the component systems' principal eigenfunctions and then composing them with the perturbation function.