一种计算三体问题周期轨道的新方法

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

针对三体问题周期轨道计算方法存在计算量大、改变雅可比能量和局限于计算特定周期轨道等不足,本文提出了一种计算周期轨道的新方法。首先建立了一种初始点和投影点关系的改进型庞加莱截面图,能够更直观地反映随着初始点改变周期轨道的演变和分叉;其次基于改进的庞加莱截面图,通过初始点与投影点的对应关系筛选出可能存在周期轨道的候选区间;然后在该候选区间内利用状态转移矩阵给出距离周期轨道初始点真实解非常接近的初始猜想;最后采用打靶法求解能够快速得到周期轨道的数值解。本文方法不需要改变三体系统的雅可比能量,迭代次数少,能够快速计算得到大范围、具有x轴对称性的周期轨道。以地月圆形限制性三体问题为例进行仿真,验证了该方法的快速性和有效性。

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 Poineare section map is created, on which the projection points from every initial point are plotted and the evolution and fork of the periodic orbits hy the initial point could be intuitively reflected. Secondly, based on the modified Poincare 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 Jaeobi 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.