Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions

Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The. zeta-component motion is treated as the dominant motion and the xi and eta-component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the zeta-position and zeta-velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on zeta-direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.