Though the existence of periodic orbits has been proved. it' s a challenging task for us to find the periodic solution with a certain accuracy in a given dynamics system. A method to determine accurate periodic orbit ( also called halo orbit) surround the equilibrium points of asteroids is presented. Firstly, to expend the dynamics model and equation of motion, the right end in equation of motion expanded into third-order power series form. Then the nonlinear equations of motion can be extended to quasi-linear differential equations. Secondly, to solve the extended equations of motion by using Lindstedt-Poincaré method, the periodic solution and its frequency are expanded into third-order power series. After substituting the two power series into the quasi-linear differential equations, linear equations of motion with three different orders are got. Solving the differential equations successively and eliminating the secular teams in the solutions, then, the thirdorder analytical solution for halo orbits is obtained. Finally, the accurate halo orbit in true gravitational field by using differential correction to amend the analytical solution is got.%尽管周期解的存在性已经被证明,但要在给定的动力学系统中寻找到满足一定精度要求的周期解依然是一件极富挑战性的工作.提出如下方法确定小行星平衡点附近精确的周期轨道(halo轨道).首先扩展运动方程:将小行星平衡点附近轨道运动方程的右端项在平衡点处展成三阶幂级数.从而将非线性运动学方程扩展为拟线性微分方程.然后求近似解析解:应用Lindstedt-Poincaré方法求解扩展后的运动方程组,将周期解和其运动频率展开成三阶幂级数,并将二者代人扩展后的拟线性微分方程中.这样就可以得到三个不同阶的线性运动方程,逐次求解三个微分方程并消除解中的永年项即可得到hal.轨道的三阶解析解.最后微分校正:将周期轨道在三阶解析解附近线性化,得到状态转移矩阵,并使用状态转移矩阵和轨道终端状态的偏差修正轨道初值,从而得到满足精度要求的精确引力场中的halo轨道.
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