In this work, we investigate the Earth-Moon system, as modeled by the planar circular restricted three-body problem, and relate its dynamical properties to the underlying structure associated with specific invariant manifolds. We consider a range of Jacobi constant values for which the neck around the Lagrangian point $L_1$ is always open but the orbits are bounded due to Hill stability. First, we show that the system displays three different dynamical scenarios in the neighborhood of the Moon: two mixed ones, with regular and chaotic orbits, and an almost entirely chaotic one in between. We then analyze the transitions between these scenarios using the Monodromy matrix theory and determine that they are given by two specific types of bifurcations. After that, we illustrate how the phase space configurations, particularly the shapes of stability regions and stickiness, are intrinsically related to the hyperbolic invariant manifolds of the Lyapunov orbits around $L_1$ and also to the ones of some particular unstable periodic orbits. Lastly, we define transit time in a manner that is useful to depict dynamical trapping and show that the traced geometrical structures are also connected to the transport properties of the system.

中文翻译：

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有界平面地月系统中的有序-混沌-有序流形和不变流形

在这项工作中，我们研究了由平面圆形受限三体问题建模的地月系统，并将其动力学特性与与特定不变流形相关的基础结构相关联。我们考虑一系列 Jacobi 常数值，其中拉格朗日点 $L_1$ 周围的颈部始终是开放的，但由于希尔稳定性，轨道是有界的。首先，我们展示了该系统在月球附近显示了三种不同的动力学场景：两种混合的，具有规则和混乱的轨道，以及介于两者之间的几乎完全混乱的。然后我们使用 Monodromy 矩阵理论分析这些场景之间的转换，并确定它们是由两种特定类型的分叉给出的。之后，我们说明相空间配置如何，特别是稳定区和粘性的形状，与 L_1 附近的李雅普诺夫轨道的双曲不变流形以及一些特定的不稳定周期轨道的双曲不变流形有着内在的关系。最后，我们以一种有助于描述动态捕获的方式定义传输时间，并表明所追踪的几何结构也与系统的传输特性有关。