We illustrate the chaotic nature of the circular restricted three-body problem from the perspective of the bifurcation diagram with respect to the mass ratio parameter. Moreover, it is shown that when the frequency ratio in different directions of the classical problem is irrational, it has the quasiperiodic characteristics. In addition, a three-dimensional approximate solution to this problem under two time scales is proposed by using the multiple time scales method.

中文翻译：

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圆形受限三体问题的近似解析三维时标

我们从分岔图的角度相对于质量比参数说明了圆形受限三体问题的混沌性质。而且，表明当古典问题在不同方向上的频率比是不合理的时，它具有准周期特征。此外，通过使用多个时间尺度方法，提出了在两个时间尺度下对该问题的三维近似解。