On families of periodic orbits in the restricted three-body problem

TL;DR

This paper investigates the existence of continuous families of periodic orbits in the planar circular restricted three-body problem by analyzing invariants that obstruct orbit cylinders, providing insights into the structure of these orbits.

Contribution

It introduces a method to determine whether two periodic orbits are connected by an orbit cylinder using invariants, advancing understanding of orbit families in the three-body problem.

Findings

Invariants can obstruct the existence of orbit cylinders between periodic orbits.

Comparison of invariants helps classify periodic orbits in the three-body problem.

The approach offers a new tool for studying orbit families in dynamical systems.

Abstract

Since Poincar\'e, periodic orbits have been one of the most important objects in dynamical systems. However, searching them is in general quite difficult. A common way to find them is to construct families of periodic orbits which start at obvious periodic orbits. On the other hand, given two periodic orbits one might ask if they are connected by an orbit cylinder, i.e., by a one-parameter family of periodic orbits. In this article we study this question for a certain class of periodic orbits in the planar circular restricted three-body problem. Our strategy is to compare the Cieliebak-Frauenfelder-van Koert invariants which are obstructions to the existence of an orbit cylinder.