Analytical computation of bifurcation of orbits near collinear libration point in the restricted three-body problem

Abstract

A unified analytical solution is presented for constructing the phase space near collinear libration points in the Circular Restricted Three-body Problem (CRTBP), encompassing Lissajous orbits and quasihalo orbits, their invariant manifolds, as well as transit and non-transit orbits. Traditional methods could only derive separate analytical solutions for the invariant manifolds of Lissajous orbits and halo orbits, falling short for the invariant manifolds of quasihalo orbits. By introducing a coupling coefficient {\eta} and a bifurcation equation, a unified series solution for these orbits is systematically developed using a coupling-induced bifurcation mechanism and Lindstedt-Poincar\'e method. Analyzing the third-order bifurcation equation reveals bifurcation conditions for halo orbits, quasihalo orbits, and their invariant manifolds. Furthermore, new families of periodic orbits similar to halo orbits are discovered, and non-periodic/quasi-periodic orbits, such as transit orbits and non-transit orbits, are found to undergo bifurcations. When {\eta} = 0, the series solution describes Lissajous orbits and their invariant manifolds, transit, and non-transit orbits. As {\eta} varies from zero to non-zero values, the solution seamlessly transitions to describe quasihalo orbits and their invariant manifolds, as well as newly bifurcated transit and non-transit orbits. This unified analytical framework provides a more comprehensive understanding of the complex phase space structures near collinear libration points in the CRTBP.