# Koopman Operator and its Approximations for Systems with Symmetries

## TL;DR

This paper explores how symmetries in nonlinear dynamical systems influence the structure and computation of the Koopman operator, enabling more efficient approximations and revealing hidden organizational features.

## Contribution

It demonstrates how symmetry considerations can simplify the computation of Koopman operator approximations using EDMD and kernel DMD methods, and reveals the block diagonal structure induced by symmetries.

## Key findings

- Symmetries induce a block diagonal structure in Koopman operator approximations.
- Symmetry considerations can improve the efficiency of EDMD and kernel DMD methods.
- The paper discusses the impact of measurement noise on these methods.

## Abstract

Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, including complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and to simplify their analysis. The Koopman operator is an infinite dimensional linear operator that fully captures a system's nonlinear dynamics through the linear evolution of functions of the state space. Importantly, in contrast with local linearization, it preserves a system's global nonlinear features. We demonstrate how the presence of symmetries affects the Koopman operator structure and its spectral properties. In fact, we show that symmetry considerations can also simplify finding the Koopman operator approximations using the extended and kernel dynamic mode decomposition methods (EDMD and kernel DMD). Specifically, representation theory allows us to demonstrate that an isotypic component basis induces block diagonal structure in operator approximations, revealing hidden organization. Practically, if the data is symmetric, the EDMD and kernel DMD methods can be modified to give more efficient computation of the Koopman operator approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out the development, we discuss the effect of measurement noise.