Invariant Manifold (Invariant Manifold)

Author: Tianjiang Shuo
Reference: 钱霙婧(2014)《地月空间拟周期轨道上航天器自主导航与轨道保持研究》
Website: https://cislunarspace.cn

Definition

An invariant manifold is a core concept in dynamical systems theory, referring to sets that remain unchanged during system evolution. For Hamiltonian systems, invariant manifolds are geometric structures in phase space that preserve system properties.

In the restricted three-body problem, invariant manifolds describe the manifold structure around periodic and quasi-periodic orbits near libration points, serving as key tools for understanding libration point dynamics and designing low-energy transfer orbits.

Stable and Unstable Manifolds

Stable Manifold

The stable manifold $W^s$ refers to trajectories that, starting from any point on this manifold, asymptotically converge to the target periodic orbit or equilibrium point as $t \to +\infty$ :

W^s(x_0) = \{ x \in \mathcal{M} : \lim_{t \to +\infty} \phi^t(x) = x_0 \}

For collinear libration points (L₁, L₂, L₃), stable manifolds correspond to trajectories converging to the libration point along the unstable characteristic direction.

Unstable Manifold

The unstable manifold $W^u$ refers to trajectories that, starting from any point on this manifold, asymptotically converge to the target periodic orbit or equilibrium point as $t \to -\infty$ :

W^u(x_0) = \{ x \in \mathcal{M} : \lim_{t \to -\infty} \phi^t(x) = x_0 \}

For collinear libration points, unstable manifolds correspond to trajectories diverging from the libration point along the unstable characteristic direction.

Dynamic Characteristics

Manifold Structure of Libration Points

Collinear libration points (L₁, L₂, L₃) have a saddle×center×center dynamical structure:

Direction	Stability	Corresponding Manifold
Crossing direction	Saddle (unstable)	One-dimensional unstable manifold
In-plane perpendicular to line	Center (stable)	Two-dimensional stable/unstable manifold
Perpendicular to orbital plane	Center (stable)	Two-dimensional stable/unstable manifold

Geometric Form of Manifolds

Stable and unstable manifolds form "tube-like" structures in phase space:

Stable manifold tubes: Starting near periodic orbit, spiraling inward to periodic orbit
Unstable manifold tubes: Starting near periodic orbit, diverging outward

These tube structures constitute the main body of the Interplanetary Superhighway.

Application in Orbit Design

Low-Energy Transfer Orbit Design

Invariant manifolds can be used to design low-energy transfer orbits:

Stable manifold transfer: Starting from target orbit, propagate outward along stable manifold to find intersection with departure orbit
Unstable manifold transfer: Starting from departure orbit, converge inward along unstable manifold to target orbit

Libration Point Orbit Generation

When designing periodic orbits near libration points:

Calculate initial estimate of periodic orbit in CR3BP model
Use stable/unstable manifolds to verify orbit stability
Use invariant manifolds as convergence directions for orbit design

Non-intersecting manifold tubes: Orbit may be stable
Intersecting manifold tubes: Orbit exhibits chaotic behavior

Control-Point Method

The control-point method for orbit keeping is essentially controlling spacecraft to return to periodic orbit along stable manifolds.

References

Koon W S, Lo M W, Marsden J E, et al. Dynamical systems, the three-body problem and space mission design[M]. 2011.
钱霙婧. 地月空间拟周期轨道上航天器自主导航与轨道保持研究[D]. 哈尔滨工业大学, 2014.