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

TL;DR

The paper tackles stabilising unstable periodic orbits in the chaotic, volume-preserving three-body problem by learning local Poincaré maps with SINDy using a novel augmentation strategy around known UPOs. It then computes low-energy, state-dependent control impulses via convex linear matrix inequalities to drive trajectories toward the target UPO along its local stable manifold. The key contributions are a data-efficient mapping approach (as few as 55 data points), a verifiable control design, and successful stabilization of both planar Lyapunov and non-planar halo UPOs, with broader implications for celestial mechanics and other conservative or dissipative systems. The work offers a practical and interpretable framework for energy-efficient chaos control with potential extensions to robotics, plasma, and fluid dynamics.

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 Poincaré map in as few as 55 data points. We suggest that the Poincaré 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.

Figures (12)

• Figure 1: Our method for stabilising a Lyapunov UPO: 1) Initial conditions of known UPOs are selected and small velocity perturbations added. The states $(x,\Dot{x},\Dot{y})$ are recorded at successive crossings of the surface of section $\Sigma = \{ (x,\Dot{x},\Dot{y})| y = 0, x < 0.84\}$. A Poincaré map is discovered using SINDy and linearised at the desired UPO $\mathbf{\bar{x}}$. 2) A controller $\mathbf{K}$ is found through solving a pole-placement problem using linear matrix inequalities with a constraint on the decision variables. 3) Constraint causes the magnitude of the eigenvalues to increase and aligns the impulse $\Delta v$ along the local stable manifold, a locally-optimal solution.
• Figure 2: Augmentation Strategy - We take the initial conditions of the desired UPO and $m$ UPOs centred around it in energy level, concatenating them into $\mathbf{\Bar{X}}$. Next, we add small positive and negative velocity perturbations $\delta v{}$ in $x$ and $y$ separately, to each UPO. These augmented initial conditions and the original $\mathbf{\Bar{X}}$ are concatenated into one initial conditions vector $\mathbf{X}_{ICs}$.
• Figure 3: Location of the unstable $L_1, L_2, L_3$ and stable $L_4,L_5$ Lagrange points of the Earth-Moon CR3BP in the orbital plane.
• Figure 4: Various $L_1$ Lyapunov orbits shown in the $x$-$y$ plane intersecting with the two surfaces of section, $\Sigma_{1,L}$ and $\Sigma_{2,L}$, we sample data from.
• Figure 5: Nonlinear inverse relationship between orbital period $T$ and Jacobi constant $C$ for the $L_1$ Lyapunov family, where all UPOs sit below the $L_1$ value of $C=3.1883$.