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

arxiv（2024）

摘要

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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