Koopman Operator and Phase Space Partition of Chaotic Maps

TL;DR

This paper presents a method using Koopman operator eigenfunctions to partition phase space in chaotic maps, aiding the analysis of complex nonlinear dynamics.

Contribution

It introduces a novel symbolic partitioning approach based on Koopman eigenfunctions, applicable to high-dimensional nonlinear systems.

Findings

Partition boundaries are extrema of Koopman eigenfunctions.

Accuracy improves with more basis functions.

Method validated on 1D and 2D maps.

Abstract

Koopman operator describes evolution of observables in the phase space, which could be used to extract characteristic dynamical features of a nonlinear system. Here, we show that it is possible to carry out interesting symbolic partitions based on properly constructed eigenfunctions of the operator for chaotic maps. The partition boundaries are the extrema of these eigenfunctions, the accuracy of which is improved by including more basis functions in the numerical computation. The validity of this scheme is demonstrated in well-known 1-d and 2-d maps. It seems no obstacle to extend the computation to nonlinear systems of high dimensions, which provides a possible way of dissecting complex dynamics.