We consider the motion of a 2π-periodic in time two-degree-of-freedom Hamiltonian system in a neighborhood of the equilibrium position. It is assumed that the system depends on a small parameter e and other parameters and is autonomous at e = 0. It is also assumed that in the autonomous case there is a set of parameter values for which a 1:1 resonance occurs, and the matrix of the linearized equations of perturbed motion is reduced to a diagonal form. The study is carried out using an example of the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on an elliptical orbit with small eccentricity in the neighborhood of the cylindrical precession. The character of the motions of the reduced two-degree-of-freedom system in the vicinity of the resonance point in the three-dimensional parameter space is studied. Stability regions of the unperturbed motion (the cylindrical precession) and two types of parametric resonance regions corresponding to the case of zero frequency and the case of equal frequencies in the transformed approximate system of the linearized equations of perturbed motion are considered. The problem of the existence, number and stability of 2π-periodic motions of the satellite is solved, and conclusions on the existence of two- and three-frequency conditionally periodic motions are obtained.

中文翻译：

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关于一个近似1：1：1共振的哈密顿系统的运动

我们认为的2运动π为周期的两个学位的自由时间哈密顿系统在平衡位置的附近。假设系统依赖于小参数e和其他参数，并且在e处是自治的=0。还假设在自主情况下，存在一组参数值，会发生1：1共振，并且扰动运动的线性化方程的矩阵简化为对角线形式。这项研究以一个动态对称的刚体（卫星）相对于其质心在一个椭圆形轨道附近，偏心率小的椭圆形轨道上的中心牛顿重力场中的运动相对于质心的运动问题为例。研究了三维参数空间中共振点附近的简化二自由度系统的运动特性。考虑了扰动运动的线性化方程的变换近似系统中的零扰动的稳定区域（圆柱进动）和对应于零频率情况和等频率情况的两种类型的参数共振区域。2的存在，数目和稳定性问题解决了卫星的π周期运动，并得出了关于两频和三频条件周期运动存在性的结论。