Invariant Torus & Quasi-Periodic Tori (Invariant Torus & Quasi-Periodic Tori)

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

An invariant torus is a closed higher-dimensional surface in phase space spanned by two or more independent frequencies. If the frequency ratio in the two angular coordinates is irrational, the orbit never closes and is called quasi-periodic; it densely covers the torus. In the CR3BP, invariant tori often surround a reference periodic orbit and provide the geometric foundation for quasi-periodic Lissajous and quasi-periodic halo orbit families (Gómez et al. 2001; Meyer & Offin 2017).

Two-Dimensional Invariant Tori and Lissajous / Halo Orbits

The two center frequencies in the center manifold of a collinear libration point can be parameterized by action-angle variables. For given amplitudes, an irrational frequency ratio yields a 2-D invariant torus. When one amplitude vanishes, the torus degenerates to a planar Lyapunov orbit; when the frequencies satisfy a resonance relation, periodic solutions such as halo orbits can bifurcate. Thus periodic orbits can be viewed as resonant “slices” or bifurcation products of invariant torus families.

Quasi-Periodic Invariant Tori (QPT)

QPTs are bounded closed surfaces covered by quasi-periodic non-resonant orbits in the CRTBP. Unlike strictly periodic halo orbits, QPT orbits do not repeat and naturally drift around the reference periodic orbit, forming a natural “enclosing” structure. Because motion on the torus is bounded, QPTs are useful for designing long-term stable relative trajectories for formation flying: spacecraft placed on different sections of the same torus family remain naturally bounded relative to each other (Capannolo et al. 2023).

Torus Manifolds in the Bicircular Four-Body Problem

In the bicircular restricted four-body problem (BCR4BP), the Sun appears as a fourth body that periodically perturbs the Earth–Moon system. Invariant tori can still exist approximately on the Earth–Moon side, and their stable/unstable manifolds are called torus manifolds. Computationally, one propagates the state-transition matrix along an invariant curve and uses its eigenvalues and eigenvectors to determine the local stable/unstable directions of the torus, which then serve as initial guesses for continuation into the four-body model (Ren et al. 2012).

Application Highlights

• Formation flying: use different initial phases on the same QPT family to build long-term passive deputy trajectories;

• Station-keeping: quasi-periodic orbits are more “flexible” than periodic orbits and can reduce station-keeping cost in some missions;

• Model transition: tori and their manifolds provide initial structures for transitioning solutions from the CR3BP to high-fidelity ephemeris models.

Related Concepts

• Center Manifold

• KAM Theorem

• Circular Restricted Three-Body Problem (CR3BP)

• Halo Orbit

• Lyapunov Orbit

• Invariant Manifold

• Weak Stability Boundary (WSB)

References

• Gómez, G., et al. (2001). Dynamics and Mission Design Near Libration Points, Vol. I/II.

• Meyer, K. R., & Offin, D. C. (2017). Introduction to Hamiltonian Dynamical Systems and the N-Body Problem.

• Capannolo, L., et al. (2023). Model predictive control for formation reconfiguration exploiting quasi-periodic tori in the cislunar environment.

• Ren, Y., et al. (2012). Manifolds of quasi-periodic orbits in the bicircular restricted four-body problem. Celestial Mechanics and Dynamical Astronomy.