In this paper, we present a new ansatz for solving equations of motion for the trapped orbits of the infinitesimal mass m, which is moving near the primary M₃ in case of bi-elliptic restricted problem of four bodies (BiER4BP), where three primaries M₁, M₂, M₃ are rotating around their common center of mass on elliptic orbits with hierarchical configuration M₃ ≪ M₂ ≪ M₁. A new type of the solving procedure is implemented here to obtain the coordinates \( \vec{r} = \;\{ x,y,z\} \) of the infinitesimal mass m with its orbit located near the primary M₃. Meanwhile, the system of equations of motion has been successfully explored with respect to the existence of analytical or semi-analytical (approximated) way for presentation of the solution. We obtain as follows: (1) the solution for coordinate x is described by the key nonlinear ordinary differential equation of fourth order at simplifying assumptions, (2) solution for coordinate y is given by the proper analytical expression, depending on coordinate x and true anomaly f, (3) the expression for coordinate z is given by the equation of Riccati-type—it means that coordinate z should be quasi-periodically oscillating close to the fixed plane \( \{ x,y,\,0\} \).

中文翻译：

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BiER4BP中无穷小运动的求解程序

在本文中，我们提出了一种新Ansatz方法求解运动方程式为截留轨道的微小质量米，其被邻近所述主移动中号₃中的情况下，双-椭圆的受限问题4个机构（BiER4BP），其中三原色中号₁，中号₂，中号₃在它们的上质公共中心旋转围绕椭圆轨道与分层配置中号₃  « 中号₂  « 中号₁。在此实现一种新型的求解过程，以获得其轨道位于主M ₃附近的无穷小质量m的坐标\（\ vec {r} = \; \ {x，y，z \} \）。同时，已经针对存在于解析或半解析（近似）方法中的解决方案的存在，成功地探索了运动方程组。我们得到如下结果：（1）坐标x的解由简化假设下的四阶关键非线性常微分方程描述；（2）坐标y的解由适当的解析表达式给出，具体取决于坐标x和真正的异常f，（3）坐标z的表达式由Riccati类型的方程式给出-这意味着坐标z应该是准的-周期性地在固定平面\（\ {x，y，\，0附近振荡\} \）。