A rational ansatz for the approximation of Koopman eigenfunctions

Koopman operator theory offers a basis for the systematic transformation and linearization of complex dynamical systems. We propose a method to approximate eigenfunctions of the Koopman operator for sufficiently smooth, deterministic and autonomous dynamical systems with hyperbolic fixed points in an equation-based context. Approximations of the eigenfunctions are obtained in form of a rational ansatz whose coefficients are determined by minimizing a residual through a bi-quadratic optimization problem. In addition, we consider an extension of the Hartman-Grobman theorem, which was first proposed by Lan and Mezić in 2013, as a linear constraint. The implementation for a damped pendulum shows that the approach works in general, however, the optimization problem is non-convex and thus sensible w. r. t. initial conditions, and increases proportional to the number of ansatz functions to the power of four.