Heteroclinic Connection

Author: Tianjiang Shuo
Contributing Institution: School of Astronautics, Harbin Institute of Technology, National Key Laboratory of Rapid Design and Intelligent Swarm of Small Spacecraft

Definition

A Heteroclinic Connection is a path formed by the intersection of invariant manifolds between two different periodic orbits in a dynamical system. Specifically, when the unstable manifold of one orbit coincides with the stable manifold of another orbit, a heteroclinic connection is formed — the spacecraft can naturally transition from one periodic orbit to another along this path without applying control thrust. In the Circular Restricted Three-Body Problem (CR3BP), heteroclinic connections widely exist between periodic orbits near the collinear libration points (L1, L2), serving as the core dynamical mechanism for designing low-energy transfer trajectories.

Core Elements

Mathematical Description

Let $P_1$ and $P_2$ be two distinct periodic orbits in the CR3BP, $W^u(P_1)$ be the unstable manifold of $P_1$ , and $W^s(P_2)$ be the stable manifold of $P_2$ . If a non-empty intersection exists:

W^u(P_1) \cap W^s(P_2) \neq \emptyset

then this intersection constitutes a heteroclinic connection from $P_1$ to $P_2$ . Along this connection, the orbit approaches $P_1$ as $t \to -\infty$ and approaches $P_2$ as $t \to +\infty$ .

Dynamical Characteristics

Heteroclinic connections have the following dynamical properties:

Zero velocity increment: In the ideal CR3BP, a heteroclinic connection is an exact natural dynamical path requiring no propellant expenditure during transfer
Sensitivity to perturbations: Heteroclinic connections exist at low-dimensional intersections within high-dimensional phase space, making them extremely sensitive to initial conditions and model parameters
Energy matching constraint: The two periodic orbits must have identical Jacobi constants (energy); otherwise, the manifolds will not intersect

Typical Heteroclinic Connections

In the Earth-Moon system, typical heteroclinic connections include:

Connection Type	Connected Objects	Characteristics
L1-L2 connection	L1 Halo ↔ L2 Halo	Connects collinear libration point regions on both sides of the Moon
L1-Lyapunov ↔ L2-Lyapunov	L1 planar orbit ↔ L2 planar orbit	Low-energy transfer channel within the orbital plane
Butterfly orbit internal connection	L1 Halo ↔ L2 Halo (large amplitude)	Butterfly orbits themselves embody heteroclinic connections

Applications in Cislunar Space

Heteroclinic connections have significant application value in cislunar space mission design:

Low-energy transfer trajectories: By concatenating heteroclinic connection segments, low-energy transfer trajectories from near-Earth orbit to orbits near the Moon can be designed, significantly reducing propellant requirements
L1-L2 region interconnection: Heteroclinic connections provide natural dynamical channels between the Lunar Gateway and the far side of the Moon, supporting diverse mission topologies
Formation reconfiguration: Using heteroclinic connections, formation spacecraft can collectively transfer from one periodic orbit to another, enabling global transformation of formation configurations

References

Koon W S, Lo M W, Marsden J E, et al. Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics[J]. Chaos, 2000, 10(2): 427-469.
Haapala A, Vaquero M, Pavlak T A, et al. Trajectory selection strategy for tours in the Earth-Moon system[C]. AAS/AIAA Astrodynamics Specialist Conference, 2013.