A rational ansatz for the approximation of Koopman eigenfunctions

• Published in: Proceedings in applied mathematics and mechanics Vol. 24; no. 3
• Main Authors: Römer, Ulrich J., Breitenhuber, Margarete
• Format: Journal Article
• Language: English
• Published: 01.10.2024
• ISSN: 1617-7061; 1617-7061
• DOI: 10.1002/pamm.202400187

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.