The first part of the present paper reveals interplays among invariant manifolds emanating from planar Lyapunov orbits around the three collinear Lagrange points \(L_1, L_2\), and \(L_3\) for high energies. Once the energetically forbidden region vanishes, the invariant manifolds together form closed separatrices bounding transit orbits in the phase space, deviating from the low-energy picture of invariant manifold tubes. Though the qualitatively different behavior of invariant manifolds emerges for high energies, associated transit orbits possess a common feature generalized from that of low-energy transit orbits. The second part extends our previous proposal of using singular collision orbits associated with the secondary to find trajectories reaching the vicinity of the secondary to low energies. Statistical analyses indicate that singular collision orbits are useful to find such transfer trajectories except for the very low-energy regime. These results are numerically obtained in the Earth–Moon and Sun–Jupiter planar circular restricted three-body problems.

中文翻译：

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将平面圆形受限三体问题中的不变流形、过渡轨道和奇异碰撞轨道的低能动力学与高能动力学联系起来

本文的第一部分揭示了从围绕三个共线拉格朗日点 \(L_1, L_2\) 和 \(L_3\) 的平面李雅普诺夫轨道发出的不变流形之间的相互作用，以获得高能。一旦能量禁区消失，不变流形就会在相空间中形成封闭的分离轨道，从而偏离不变流形管的低能量图。尽管不变流形在高能量下出现了性质不同的行为，但相关的凌日轨道具有从低能量凌日轨道概括出来的共同特征。第二部分扩展了我们之前的提议，即使用与次级相关的奇异碰撞轨道来寻找到达次级到低能量附近的轨迹。统计分析表明，除了非常低能量的区域外，奇异碰撞轨道对于找到这种转移轨迹很有用。这些结果是在地球-月球和太阳-木星平面圆形受限三体问题中数值获得的。