Orbital-Elements-Based Koopman Operator Applied to Investigating Invariant Manifolds Around Asteroids

Abstract

Koopman operator theory enables global linear representations of nonlinear dynamic systems and has gained substantial prominence in orbital dynamics. This paper focuses on a fundamental nonlinear problem in orbital dynamics: the initial value problem of motion around a central celestial body. In practice, an approximate invariant subspace of the Koopman operator should be constructed using a dictionary to realize its finite-dimensional approximation. However, traditional dictionaries are prone to the curse of dimensionality when high precision is required. To address this issue, a novel dictionary based on orbital elements is proposed for the initial value problem. By exploiting the intrinsic dynamic structure of orbital elements, the proposed dictionary effectively mitigates the curse of dimensionality while ensuring high-precision continuous orbital propagation. Numerical simulations demonstrate the feasibility and superiority of the proposed dictionary. On this basis, a new Koopman-operator-based framework is developed for computing and analyzing invariant manifolds of large-scale periodic orbits around asteroids. More significantly, the stickiness effect near invariant manifolds is numerically validated for the first time under a high-precision asteroid gravitational field model. This work provides an efficient dictionary for Koopman operator approximation in orbital dynamics and establishes a critical foundation for invariant manifold studies and low-energy trajectory design in future space missions.