Solving nonlinear ordinary differential equations using the invariant manifolds and Koopman eigenfunctions

TL;DR

This paper introduces a novel method to construct global Koopman eigenfunctions from invariant manifolds, enabling analytical solutions for certain nonlinear ODEs previously unsolvable analytically.

Contribution

The paper presents a new approach to derive exact, globally valid Koopman eigenfunctions from invariant manifolds, extending the applicability of Koopman theory to solve nonlinear ODEs analytically.

Findings

Successfully applied to 1D and 2D ODEs

Produced analytical solutions for previously unsolved equations

Established a connection between invariant manifolds and eigenfunctions

Abstract

Nonlinear ordinary differential equations can rarely be solved analytically. Koopman operator theory provides a way to solve nonlinear systems by mapping nonlinear dynamics to a linear space using eigenfunctions. Unfortunately, finding such eigenfunctions is difficult. We introduce a method for constructing eigenfunctions from a nonlinear ODE's invariant manifolds. This method, when successful, allows us to find analytical solutions for constant coefficient nonlinear systems. Previous data-driven methods have used Koopman theory to construct local Koopman eigenfunction approximations valid in different regions of phase space; our method finds analytic Koopman eigenfunctions that are exact and globally valid. We demonstrate our Koopman method of solving nonlinear systems on 1-dimensional and 2-dimensional ODEs. The nonlinear examples considered have simple expressions for their invariant manifolds which produce tractable analytical solutions. Thus our method allows for the construction of analytical solutions for previously unsolved ordinary differential equations. It also highlights the connection between invariant manifolds and eigenfunctions in nonlinear ordinary differential equations and presents avenues for extending this method to solve more nonlinear systems.