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Advanced space travel relies on a fundamental understanding of the restricted three-body problem (RTBP), in which one of the three bodies—typically a spacecraft—is so small that its gravity doesn't affect the other two, such as a planet and its moon.

"In RTBP systems, Lagrange points between the two celestial bodies provide specific locations around which a spacecraft can orbit," explains Mingpei Lin, a member of an AIMR research team. "The ability to model how complex orbits—such as halo, and quasi-halo orbits—emerge around the unstable collinear Lagrange points enables better trajectory design using these points."

However, a persistent challenge has been the lack of a unified analytical method for describing all types of Lagrange point orbits. Numerical simulations can model individual trajectories but are computationally intensive and system-specific. Existing analytical methods offer only fragmented solutions—handling Lissajous or halo orbits separately, and failing to capture quasi-halo orbits altogether.

In published in Journal of Guidance, Control, and Dynamics, Lin and Chiba developed a unified analytical framework to describe the center manifolds of collinear Lagrange points in the RTBP. Their method introduces a coupling mechanism that explains how quasi-halo orbits bifurcate from Lissajous orbits—without requiring frequency resonance.

"Previous analytical models treated frequency resonance as the main mechanism behind the emergence of complex orbits," says Lin. "But this approach couldn't account for the bifurcation of quasi-halo orbits from Lissajous orbits, and numerical observations also pointed to a different behavior. Based on this, we proposed that nonlinear coupling—not resonance—is the true cause of orbit bifurcations, a dynamic no existing model had captured adequately."

The novelty of the team's approach lies in introducing a coupling coefficient, η, and a bifurcation equation, Δ = 0, into the RTBP equations. This modification preserves the nonlinear coupling between in-plane and out-of-plane motions, allowing bifurcations to emerge naturally—and without relying on resonance conditions.

The result is a high-order series solution that analytically describes Lissajous, halo, and quasi-halo orbits within the same formalism: when η = 0, the solution yields Lissajous orbits; when η ≠ 0, it captures quasi-halo orbits, with halo orbits appearing as special cases.

"This breakthrough transforms the understanding of orbital dynamics near Lagrange points," concludes Lin. "Our work enables precise analytical modeling of all known center manifold orbit types—greatly benefiting space mission design and bifurcation theory alike."

The team is now extending this coupling-induced bifurcation framework to other dynamical systems, including modeling symmetry-breaking phenomena such as the evolution of right-handedness in humans.

Lin says, "The challenge of explaining why bifurcated two-dimensional tori resist semi-analytical solutions has intrigued me since the early days of my Ph.D., but other projects delayed a focused investigation. At Chiba Laboratory in AIMR, I finally had the time and space to explore it deeply.

"After many failed attempts using conventional approaches, we began questioning the half-century-old resonance mechanism. This shift led to a key insight: coupling interactions—not resonance—drive this local bifurcation. Developing the coupling-induced bifurcation mechanism and explaining halo/quasi-halo orbits resolved a seven-year puzzle for me and boosted my confidence to explore the unknown."