  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.
