Solving Nonlinear Ordinary Differential Equations Using the Invariant Manifolds and Koopman Eigenfunctions

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 924-960, March 2024.
Abstract. Nonlinear ODEs can rarely be solved analytically. Koopman operator theory provides a way to solve two-dimensional nonlinear systems, under suitable restrictions, by mapping nonlinear dynamics to a linear space using Koopman eigenfunctions. Unfortunately, finding such eigenfunctions is difficult. We introduce a method for constructing Koopman eigenfunctions from a two-dimensional nonlinear ODE’s one-dimensional invariant manifolds. This method, when successful, allows us to find analytical solutions for autonomous, 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 one-dimensional and two-dimensional ODEs. The nonlinear examples considered have simple expressions for their codimension-1 invariant manifolds which produce tractable analytical solutions. Thus our method allows for the construction of analytical solutions for previously unsolved ODEs. It also highlights the connection between invariant manifolds and eigenfunctions in nonlinear ODEs and presents avenues for extending this method to solve more nonlinear systems.