Analytical and Data-Driven Koopman Operator for the Perturbed Kepler and Circular Restricted Three-Body Problems

Abstract

A modified version of the Extended Dynamic Mode Decomposition (EDMD) approach for the Koopman operator theory is presented, utilizing sparse-grid quadrature nodes and weights to generate the training data. Compared to standard random sampling, this method emphasizes certain regions of the state space and improves the accuracy and convergence rate of the Koopman operator while reducing the amount of required data. Additionally, we propose a QR decomposition and Koopman modes reduction to reduce the computational effort for analytical propagation. Applications in astrodynamics are considered, where atmospheric drag is included as an additional perturbation in a recently developed set of regularized orbital elements. An extensive comparison of the analytical Galerkin method and the data-driven standard and quadrature-based EDMD approaches is provided. Numerical examples for the analytical prediction of perturbed Keplerian motion and periodic orbits in the circular restricted three-body problem, and a comparison with an analytic propagator demonstrate the efficiency of the developed methods.