The bicircular model is a periodic time-dependent perturbation of the Earth–Moon restricted three-body problem that includes the direct gravitational effect of the Sun on the infinitesimal particle. In this paper, we focus on the dynamics in the neighbourhood of the $$L_1$$ point of the Earth–Moon system. By means of a periodic time-dependent reduction to the centre manifold, we show the existence of two families of quasi-periodic Lyapunov orbits, one planar and one vertical. The planar Lyapunov family undergoes a (quasi-periodic) pitchfork bifurcation giving rise to two families of quasi-periodic halo orbits. Between them, there is a family of Lissajous quasi-periodic orbits, with three basic frequencies.

中文翻译：

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双圆问题中地月附近$$L_1$$L1点

双圆模型是地月受限三体问题的周期性时间相关扰动，包括太阳对无穷小粒子的直接引力效应。在本文中，我们关注地月系统 $$L_1$$ 点附近的动力学。通过对中心流形的周期性时间相关归约，我们展示了两类准周期李雅普诺夫轨道的存在，一类是平面的，一类是垂直的。平面李雅普诺夫家族经历了一个（准周期）干草叉分支，产生了两个准周期晕轨道家族。在它们之间，有一族 Lissajous 准周期轨道，具有三个基本频率。