Order-chaos-order and invariant manifolds in the bounded planar Earth-Moon system

TL;DR

This paper explores the complex dynamical behavior of the Earth-Moon system modeled by the planar circular restricted three-body problem, focusing on invariant manifolds, bifurcations, and transport properties near the Lagrangian point L1.

Contribution

It identifies and characterizes different dynamical regimes, analyzes bifurcations using Monodromy matrix theory, and links phase space structures to invariant manifolds and transport phenomena.

Findings

Three distinct dynamical scenarios near the Moon.

Bifurcations govern transitions between these scenarios.

Invariant manifolds influence stability regions and transport.

Abstract

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.