A nonlinear autonomous ODE's invariant manifolds are described by generating functions M(x) whose time derivative factors as dM/dt = M(x)N(x). When a linear combination of the N-functions collapses to a constant, the corresponding product of powers of M-functions is an exact Koopman eigenfunction with that constant as its eigenvalue, valid over the whole phase space rather than only near a fixed point. Students learn to identify invariant manifolds, derive their N-functions, choose weight vectors that are linearly independent so the eigenfunctions lie in different equivalence classes, and then invert the eigenfunction map to recover closed-form analytical solutions for planar nonlinear systems that had none.
This paper develops techniques for solving nonlinear ordinary differential equations by leveraging Koopman operator theory, which linearizes nonlinear dynamics through eigenfunction mappings. The aut…