Data-Driven Stabilisation of Unstable Periodic Orbits of the Three-Body Problem

Abstract

Many different models of the physical world exhibit chaotic dynamics, from fluid flows and chemical reactions to celestial mechanics. The study of the three-body problem (3BP) and its many different families of unstable periodic orbits (UPOs) have provided fundamental insight into chaotic dynamics as far back as the 19th century. The 3BP, a conservative system, is inherently challenging to sample due to its volume-preservation property. In this paper we present an interpretable data-driven approach for the state-dependent control of UPOs in the 3BP, through leveraging the inherent sensitivity of chaos and the local manifold structure. We overcome the sampling challenge by utilising prior knowledge of UPOs and a novel augmentation strategy. This enables sample-efficient discovery of a verifiable and accurate local Poincar\'{e} map in as few as 55 data points. We suggest that the Poincar\'{e} map is best discovered at a surface of section where the norm of the monodromy matrix, i.e. the local sensitivity to small perturbations, is the smallest. To stabilise the UPOs, we apply small velocity impulses once each period, determined by solving a convex system of linear matrix inequalities based on the linearised map. We constrain the norm of the decision variables used to solve this system, resulting in locally optimal velocity impulses directed along the local stable manifold. Critically, this behaviour is achieved in a computationally efficient manner. We demonstrate this sample-efficient and low-energy method across several orbit families in the 3BP, with potential applications ranging from robotics and spacecraft control to fluid dynamics.