Analysis of period-multiplying bifurcations of distant retrograde orbits in the Hill three-body problem

Abstract

In this study, the period-multiplying bifurcations of distant retrograde orbits (DROs) in the Hill three-body problem are analyzed. Simple planar DRO can be accurately approximated as elliptical motion by Fourier expansion, and the linearized equations describing motion around the approximate analytical solution of the DRO can be separated into in-plane and out-of-plane motions. An approximate analytical solution for the out-of-plane monodromy matrix can be obtained by Magnus expansion, while the in-plane monodromy matrix is calculated quasi-analytically by approximating the DRO as a generalized elliptic motion with non-constant angular velocity. The period-multiplying bifurcation conditions of the DRO family can be analyzed by examining the eigenvalues of the monodromy matrix. The relationship between the direction of the bifurcation and the derivative of the period at the bifurcation point is also analyzed. Finally, the branching family is calculated numerically.