Invariant Manifold (Invariant Manifold / Stable & Unstable Manifolds)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

An invariant manifold is a set of states that remains invariant under the flow of a dynamical system: if a state lies on the set at one time, it lies on the set for all time (Gómez et al. 2001; Koon et al. 1999). In the circular restricted three-body problem (CR3BP), the most widely used invariant manifolds are the stable manifold $W^s$ and unstable manifold $W^u$ of libration-point periodic orbits such as halo and Lyapunov orbits.

Let $\Gamma$ denote a periodic orbit, $x$ a nearby state, and $\phi^t$ the flow map. Then

W^s(\Gamma)=\{x:\lim_{t\to+\infty}\mathrm{dist}(\phi^t(x),\Gamma)=0\},

W^u(\Gamma)=\{x:\lim_{t\to-\infty}\mathrm{dist}(\phi^t(x),\Gamma)=0\}.

Trajectories on the stable manifold approach the target orbit in forward time and are used for capture/arrival; trajectories on the unstable manifold depart from the target orbit in forward time and are used for departure/escape.

Monodromy Matrix and Local Linearization

The variational equation of the CR3BP state equation $\dot{x}=f(x)$ is

\dot{\Phi}(t,0)=A(t)\Phi(t,0),\quad A(t)=\frac{\partial f}{\partial x}\bigg|_{x(t)},\quad \Phi(0,0)=I.

Integrating along a periodic orbit $\Gamma$ over one period $T$ yields the monodromy matrix $M=\Phi(T,0)$ . Its eigenvalues determine local stability. In Hamiltonian systems eigenvalues occur in reciprocal and complex-conjugate pairs. Periodic orbits near collinear libration points typically have one real pair $\lambda_s<1$ , $\lambda_u=1/\lambda_s>1$ (hyperbolic directions) and two complex-conjugate pairs on the unit circle (center directions), i.e. a saddle×center×center structure (Koon et al. 1999; Szebehely 1967).

The eigenvectors $v_s$ and $v_u$ associated with $\lambda_s$ and $\lambda_u$ give the local stable and unstable directions at each point of the orbit. If $x_p(\tau)$ is the state on the orbit at phase $\tau\in[0,T)$ , a manifold initial state is

x(0)=x_p(\tau)\pm\varepsilon\,\Phi(\tau,0)v_{s/u},\quad 0<\varepsilon\ll1,

where the stable direction is integrated forward and the unstable direction backward to obtain the stable manifold, and vice versa for the unstable manifold. The $\pm$ sign produces the two branches of the same manifold, which often extend into different regions of configuration space.

Manifold Tubes, Branches, and Direction Conventions

The collection of all stable (or unstable) trajectories of a periodic orbit forms an invariant manifold tube. In the planar problem the tube is a separatrix on the 3-D energy surface, separating transit orbits from non-transit orbits; in the spatial problem it remains the fundamental topological channel for low-energy transfer design (Gómez et al. 2001; Howell & Kakoi 2006).

For the Earth–Moon $L_2$ halo orbit the stable manifold is commonly divided into:

Interior branch: extends toward the Moon, useful for low-energy transfers from lunar orbits to the halo orbit;
Exterior branch: extends away from the Moon into the exterior region, useful for transfers from near-Earth orbits.

Unstable manifolds are divided similarly by their forward-time evolution. Out-of-plane branches are sometimes called vertically stable/unstable manifolds and are used to study the evolution of out-of-plane deviations.

Computation and Engineering Approximations

Invariant manifolds are usually generated numerically as follows:

Compute the target periodic orbit with differential correction;
Integrate the state-transition matrix to obtain the monodromy matrix and its eigenvectors;
Apply small perturbations along the eigenvectors at discrete points on the orbit to obtain manifold initial states/starting points;
Integrate in the appropriate time direction to produce the manifold propagation.

To speed up optimization, a manifold interpolation database is often precomputed: states are stored on a grid of orbit phase $\tau$ and manifold integration time $t$ , then retrieved by 2-D interpolation during optimization (Pontani & Teofilatto 2016).

Natural manifolds rarely satisfy mission constraints exactly (e.g. perilune altitude or arrival time). Two common approximations are:

Pseudo-manifold: a CR3BP manifold slightly modified by a small $\Delta v$ to meet constraints, extending the feasible design space (Davis, Born & Butcher 2013);
Disturbed manifold: a natural unstable manifold with a single impulse applied at a selected point to redirect the trajectory onto the target orbit.

A piercing point is the intersection of a manifold with a reference plane; in Earth–Moon transfer design the plane $x=-\mu$ passing through the Earth is often used. Geocentric distance, inclination, and eccentricity of piercing points are key criteria for selecting transfer initial conditions.

Earth–Moon / Sun–Earth Manifolds and Cross-System Splicing

Libration-point orbits in the Earth–Moon and Sun–Earth systems each possess invariant manifolds. When the position projections of the two manifold tubes overlap on a common reference plane (a Poincaré section), a small maneuver at the overlap region can splice the two systems, enabling low-energy Sun–Earth $L_2$ ↔ Earth–Moon $L_1/L_2$ transfers (Howell & Kakoi 2006). This overlap is the geometric basis of the interplanetary superhighway in the Earth neighborhood.

Application Highlights

Low-energy transfer: depart on an unstable manifold and arrive on a stable manifold to reduce $\Delta V$ ;
Station-keeping: target-point station-keeping essentially steers the spacecraft back along the stable manifold;
Mission design workflow: manifolds provide good initial guesses that are then differentially corrected into high-fidelity ephemeris models.

References

Gómez, G., Koon, W. S., Lo, M. W., Marsden, J. E., Masdemont, J., & Ross, S. D. (2001). Invariant manifolds, the spatial three-body problem and space mission design.
Koon, W. S., Lo, M. W., Marsden, J. E., & Ross, S. D. (1999). The Genesis trajectory and heteroclinic connections.
Koon, W. S., Lo, M. W., Marsden, J. E., & Ross, S. D. (2006/2011). Dynamical systems, the three-body problem and space mission design.
Howell, K. C., & Kakoi, M. (2006). Transfers between the Earth–Moon and Sun–Earth systems using manifolds and transit orbits.
Szebehely, V. (1967). Theory of Orbits: The Restricted Problem of Three Bodies.
Vallado, D. A. (2022). Fundamentals of Astrodynamics and Applications.
Davis, K., Born, G., & Butcher, E. (2013). Transfers to Earth-Moon L3 Halo orbits. Acta Astronautica, 88, 116–128.
Pontani, M., & Teofilatto, P. (2016). Polyhedral representation of invariant manifolds applied to orbit transfers in the Earth–Moon system.
Qian, Y. (2014). Autonomous navigation and station-keeping of spacecraft on quasi-periodic orbits in cislunar space. Harbin Institute of Technology.
Peng, K., et al. (2016). Halo-orbit transfer design to Earth–Moon L2 based on invariant manifolds.