Heteroclinic Orbit Transfer (Heteroclinic Orbit Transfer / Homoclinic Connections)

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

Definition

A heteroclinic orbit connects two different invariant sets (equilibrium points or periodic orbits); if the departure and arrival sets are the same, the orbit is called homoclinic. In the CR3BP, heteroclinic connections commonly appear as geometric intersections of the stable manifold of one libration-point periodic orbit with the unstable manifold of another (Koon et al. 1999).

Let $\mathcal{A}$ and $\mathcal{B}$ be two periodic orbits and denote by $\mathcal{H}_{\mathcal{AB}}$ a heteroclinic orbit from $\mathcal{A}$ to $\mathcal{B}$ . Then

\lim_{t\to-\infty}\phi^t(x)\to\mathcal{A},\qquad \lim_{t\to+\infty}\phi^t(x)\to\mathcal{B}.

When both $\mathcal{H}_{\mathcal{AB}}$ and $\mathcal{H}_{\mathcal{BA}}$ exist, they form a heteroclinic cycle. A homoclinic orbit $\mathcal{H}_{\mathcal{AA}}$ is already a cycle by itself.

Intersection via Poincaré Sections

A heteroclinic connection corresponds to a transverse intersection of two manifold tubes. Directly intersecting two 2-D tubes in 6-D phase space is difficult. Using the Jacobi integral reduces the space to 5-D; taking a transverse Poincaré section reduces the intersection of each tube with the section to a 1-D curve, and the intersection of two curves yields candidate heteroclinic connections (Koon et al. 1999).

For example, to construct a connection from an $L_2$ periodic orbit to an $L_1$ periodic orbit:

Compute the unstable manifold of the $L_2$ orbit and integrate it forward to the section;
Compute the stable manifold of the $L_1$ orbit and integrate it backward to the same section;
Locate intersections of the two resulting curves on the section;
Integrate forward and backward from the intersection and differentially correct to obtain the heteroclinic orbit.

This reduction from a surface-surface intersection problem to a curve-curve intersection problem is the core of the space-manifold-dynamics design approach.

Homoclinic / Heteroclinic Phasing

Homoclinic and heteroclinic connections can also be used for indirect phasing. A spacecraft leaves a target orbit along its unstable manifold, evolves along a connecting orbit, and returns along the stable manifold of another (or the same) periodic orbit. By choosing different connection combinations, a desired phase shift can be accumulated while paying only the small impulses needed to enter and exit the libration-point orbits.

Interplanetary Superhighway and Mission Examples

The network formed by libration-point manifold tubes together with their heteroclinic and homoclinic connections is called the Interplanetary Superhighway. Its characteristics include:

Low energy: trajectories follow natural dynamical channels with very small fuel cost;
Networked: manifolds of different systems splice together, forming a solar-system-scale transfer network;
Long time scales: transfer times are often measured in months or years rather than days.

A canonical example is the Genesis return trajectory. The spacecraft moved from a Sun–Earth $L_1$ halo orbit through an $L_2$ -region heteroclinic cycle and back to the $L_1$ neighborhood, covering millions of kilometers with only a few m/s of deterministic $\Delta V$ (Koon et al. 1999; Lo 2002).

References

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.
Lo, M. W. (2002). The Interplanetary Superhighway and the Genesis Mission. JPL.
Gómez, G., et al. (2001). Invariant manifolds, the spatial three-body problem and space mission design.
Ren, Y., et al. (2011). On the mechanisms of natural transport in the solar system.
Guo, J. (2020). Libration-point orbit design and maintenance based on double-baseline invariant manifolds. Beijing University of Technology.