Lissajous and halo orbits in the restricted three-body problem by normalization method

We perform an analytical study of the Lissajous and halo orbits around collinear points L₁ and L₂ in a spatial circular restricted three-body problem of an arbitrary value of the mass ratio. Using a canonical transformation procedure, we generate complete and resonant normal forms through reduction to center manifolds. The coefficients in the normal forms are explicitly expressed as functions of mass ratio for the first time so that one can evaluate the energy level at which bifurcation of halo orbit takes place. Another contribution of this paper is giving the analytical solutions of Lissajous and halo orbits in the initial synodic reference system through the inverse transformation of normalization. The analytical results are the series form of normalized action-angle variables, and their coefficients are also explicitly expressed as functions of mass ratio. Finally, comparison results demonstrate that the solutions for a Lissajous orbit derived through normalization method and Lindstedt–Poincaré method are completely the same, while the solutions for a halo orbit derived through these two methods are different but have the roughly equal accuracy.