The paper deals with the problem of the chaotic behaviour of a tethered system anchored on the Phobos surface directly under the L1 collinear libration point. Two gravitational forces of Mars and Phobos, plus a centrifugal force due to the rotation of the Mars-Phobos system, act on the tether. These forces vary with time due to the small eccentricity of the Mars-Phobos orbit. The basic assumptions are formulated in terms of a planar elliptic restricted three-body problem. The motion equations in Nechvile's variables are derived in polar coordinates relative to the anchor point of the tether. The motion of the tethered system is divided into perturbed and unperturbed when the eccentricity of the Mars-Phobos orbit is zero. The points of unstable equilibrium of the tether are found, which together with periodic perturbations associated with small eccentricity are the cause of chaotic behaviour of the tether. To suppress chaos, the control coefficient is chosen using a known tether length control law. The Melnikov method is used to prove the chaotic nature of the tether and to find approximate the control coefficient needed to suppress the chaos. Verification of the obtained calculations is performed by means of Poincaré portraits for the basic nonlinear tether equation. The results of this study can be used for new Phobos exploration missions using an anchored tethered system and other future missions to study small planetary satellites in the Solar System.

中文翻译：

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抑制锚定在火卫一表面 L1 振动点下的系绳的混沌振荡

本文讨论了锚定在火卫一表面 L1 共线振动点正下方的系留系统的混沌行为问题。火星和火卫一的两种引力，加上火星-火卫一系统旋转产生的离心力作用在系绳上。由于火星-火卫一轨道的偏心率较小，这些力随时间变化。基本假设是根据平面椭圆受限三体问题制定的。Nechvile 变量中的运动方程是在相对于系绳锚点的极坐标中导出的。当火星-火卫一轨道偏心率为零时，系留系统的运动分为扰动和非扰动。找到了系链的不稳定平衡点，这些点与与小偏心率相关的周期性扰动一起是系链混沌行为的原因。为了抑制混沌，使用已知的系绳长度控制定律来选择控制系数。梅尔尼科夫方法用于证明系链的混沌性质并找到抑制混沌所需的近似控制系数。通过基本非线性系链方程的庞加莱肖像来验证所获得的计算结果。这项研究的结果可用于使用锚定系留系统的新火卫一探索任务以及研究太阳系中小型行星卫星的其他未来任务。