Periodic orbits based on geometric structure of center manifold around Lagrange points

This study proposes an analytical method that determines the center manifold and identifies the reduced system on the center manifold. The proposed method expresses the center manifold through general equations containing only state variables, and not functions with respect to time. This is the so-called geometric structure of the center manifold. The location of periodic or quasi-periodic orbits is identified after the geometric structure of the center manifold is determined. The reduced system on the center manifold is described using ordinary differential equations, so that periodic or quasi-periodic orbits can be computed by numerically integrating the reduced system. The results indicate that the analytical method proposed in this study has higher precision compared with the Lindstedt-Poincaré method of the same order.