A Novel Method of Periodic Orbit Computation for Three-Body Problem

Current methods of the periodic orbit computation have the disadvantages of requiring large amount of computation, varying the Jacobi energy, and the limitation to calculate the specific periodic orbits, a novel method of the periodic orbit computation is proposed in this paper. Firstly, a modified Poincaré section map is created, on which the projection points from every initial point are plotted and the evolution and fork of the periodic orbits by the initial point could be intuitively reflected. Secondly, based on the modified Poincaré section map, the candidate interval of the periodic orbits are selected by the relationship between the initial points and the projection points. Thirdly, the initial guesses are generated which are quite close to the true solution in the candidate interval computation using the state transition matrix. Finally, the periodic orbits can be rapidly computed numerically by a single shooting method. The proposed method does not need to change the Jacobi energy, requires less iterations for a given value of the Jacobi energy, and enables a large set of periodic orbits with x-axis symmetry. Examples are presented in the Earth-Moon Circular Restricted Three-Body Problem to verify the efficiency and rapidity of this method.