Bifurcation mechanism of quasihalo orbit from Lissajous orbit in the restricted three-body problem


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This paper presents a general analytical method to describe the center manifolds of collinear libration points in the Restricted Three-body Problem (RTBP). It is well-known that these center manifolds include Lissajous orbits, halo orbits, and quasihalo orbits. Previous studies have traditionally tackled these orbits separately by iteratively constructing high-order series solutions using the Lindstedt-Poincaré method. Instead of relying on resonance between their frequencies, this study identifies that halo and quasihalo orbits arise due to intricate coupling interactions between in-plane and out-of-plane motions. To characterize this coupling effect, a novel concept, coupling coefficient η, is introduced in the RTBP, incorporating the coupling term ηΔ x in the z-direction dynamics equation, where Δ represents a formal power series concerning the amplitudes. Subsequently, a uniform series solution for these orbits is constructed up to a specified order using the Lindstedt-Poincaré method. For any given paired in-plane and out-of-plane amplitudes, the coupling coefficient η is determined by the bifurcation equation Δ = 0. When η = 0, the proposed solution describes Lissajous orbits around libration points. As η transitions from zero to non-zero values, the solution describes quasihalo orbits, which bifurcate from Lissajous orbits. Particularly, halo orbits bifurcate from planar Lyapunov orbits if the out-of-plane amplitude is zero. The proposed method provides a unified framework for understanding these intricate orbital behaviors in the RTBP.
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