Lyapunov Orbit

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

A Lyapunov orbit is a planar periodic orbit around a collinear libration point of the circular restricted three-body problem (CR3BP), one family each at L1, L2, and L3 (Gómez 2001, Parker & Anderson 2014). Under linearization, the center subspace of a collinear point decomposes into two normal oscillations: in-plane (frequency ωp) and out-of-plane (frequency ωv); the periodic solution that excites only the in-plane center mode is the planar Lyapunov orbit (Belló 2010). The family's existence is guaranteed by Lyapunov's center theorem (hence the name; Alessi 2009) and has nothing to do with Lyapunov stability theory.

What is usually meant by Lyapunov orbit is the planar family; the vertical family is its sibling family of the same rank (see Vertical Orbit). Earth–Moon L1/L2 family periods run about 2–4 weeks (closer to 2 weeks the nearer the libration point), and the L3 family about 4 weeks (Parker & Anderson 2014).

Geometry and Symmetry

The orbit is symmetric about the rotating-frame x-axis and crosses it perpendicularly twice per revolution: a perpendicular crossing means zero x-velocity at the crossing point, velocity along y, and an initial state of the form [x₀, 0, 0, ẏ₀] (Parker & Anderson 2014, Tan 2014). The motion is prograde on the near-Moon side and retrograde on the far side (Oshima 2019, citing Lam & Whiffen 2005).

Stability and Manifolds

All family members are unstable (Parker & Anderson 2014): the monodromy matrix eigenvalues are λ₁=λ₂=1 and λ₃=1/λ₄, giving the stable and unstable directions (Tan 2014). The manifolds form smooth tubes ahead of the Moon. Besides the in-plane hyperbolic mode, some members also carry a vertical instability mode: their out-of-plane stable/unstable manifolds are natural transport channels between in-plane and out-of-plane states, and can raise inclination through multiple lunar flybys (Oshima 2019).

Two important corollaries:

• At equal Jacobi constant, free (zero-maneuver) heteroclinic transfers exist between members of the LL1 and LL2 families, findable on Poincaré maps (Parker & Anderson 2014).
• The stable manifolds of L1/L2 planar Lyapunov orbits mark the boundary of the WSB region, while the unstable manifolds govern temporary capture by the Moon (Fantino 2010); patching Sun–Earth EL1/EL2 Lyapunov manifolds to Earth–Moon LL2 ones is the classic construction of a weak stability boundary transfer (Xu 2010).

Relation to Other Families

• Bifurcating the halo family: when the in-plane and out-of-plane frequencies become equal (1:1 resonance), the family member whose vertical stability turns critical bifurcates into the halo family (Belló 2010, Gómez 2001, Alessi 2009).
• Bifurcating the axial family, at a location different from the halo bifurcation (He 2026).
• Sibling of the vertical family: the planar Lyapunov is the Lissajous limit as out-of-plane amplitude → 0, the vertical orbit the limit as in-plane amplitude → 0; the two are connected by families of quasi-periodic tori (Belló 2010, Guzzetti 2016).

Parameterization and Notation

Family members can be labeled by x₀, ẏ₀, or the Jacobi constant C, and are commonly presented as x₀–C characteristic curves (Parker & Anderson 2014). Literature notation: ELi denotes an orbit at point Li of the Sun–Earth system, LLi one of the Earth–Moon system (Xu 2010, Parker & Anderson 2014).

Large-Amplitude Members and Earth–Moon Transfers

Members whose amplitude grows large enough to be tangent to lunar orbits have special transfer value: their stable manifolds no longer have tube topology but reach directly into near-Earth space: a single stable manifold connects a near-Earth orbit straight to a near-Moon orbit (first shown by Tan 2014; Zheng & Zhao 2023). Two-impulse Earth–Moon transfers designed on this basis save about 100 m/s over Hohmann transfers and take less time than WSB transfers (Tan 2014); the Zheng & Zhao 2023 example totals about 3921 m/s ΔV over 62 days. Xu 2010 gives the parameters of Lyapunov orbits tangent to large-amplitude DROs (for a DRO with Ax=90867 km, the tangent L1 Lyapunov has Ax=32828 km), used respectively for fast and low-energy entry into the DRO.

Lyapunov-like orbits: in NRHO→DRO outbound transfers, the arc after the lunar flyby often takes a large-amplitude Lyapunov shape: tangent to the DRO on one side and close to the Moon on the other, naturally accompanied by a lunar flyby, which favors low-impulse DRO insertion (Wang 2021, the term traced back to Zhang 2020). Wang 2025 (the WSB paper) further classifies the L1/L2 Lyapunov-like segments in DRO capture trajectories by arrival Jacobi energy: high energy enters directly on the near-Earth side; low energy goes through a more inclined, larger-amplitude Lyapunov-like segment to the far side.

Terminology Variants

Related Concepts

• Vertical Orbit
• Axial Orbit
• Halo Orbit
• Lissajous Orbit
• Manifold Connection
• Weak Stability Boundary Transfer Trajectory

References

• Gómez et al., 2001, Dynamics and Mission Design Near Libration Points, Vol. I
• Alessi et al., 2009, Leaving the Moon by means of invariant manifolds of libration point orbits
• Belló et al., 2010, Invariant manifolds, Lagrangian trajectories and space mission design
• Fantino et al., 2010, A note on libration point orbits, temporary capture and low-energy transfers
• Xu, 2010, Onset conditions and trajectory construction of Earth–Moon low-energy transfers
• Parker & Anderson, 2014, Low-Energy Lunar Trajectory Design
• Tan et al., 2014, Earth–Moon transfer trajectory design based on large-amplitude Lyapunov orbits
• Guzzetti et al., 2016, Rapid trajectory design in the Earth–Moon ephemeris system via an interactive catalog of periodic and quasi-periodic orbits
• Oshima, 2019, The use of vertical instability of L1 and L2 planar Lyapunov orbits for transfers from near rectilinear orbits
• Wang et al., 2021, Transfers between NRHOs and DROs in the Earth-Moon system
• Zheng & Zhao, 2023, Earth–Moon transfer method based on the stable manifolds of large-amplitude Lyapunov orbits
• Wang et al., 2025, Mechanism and characteristics analysis of weak stability boundary transfers to the 2:1 DRO
• He et al., 2026, A review of cislunar constellation design and optimization