Studying network of symmetric periodic orbit families of the Hill problem via symplectic invariants

In the framework of the spatial circular Hill three-body problem, we illustrate the application of symplectic invariants to analyze the network structure of symmetric periodic orbits families. The extensive collection of families within this problem constitutes a complex network, fundamentally comprising the so-called basic families of periodic solutions, including the orbits of the satellite g, f, the libration (Lyapunov) a, c, and collision B0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {B}}_0$$\end{document} families. Since the Conley-Zehnder index leads to a grading on the local Floer homology and its Euler characteristics, a bifurcation invariant, the computation of those indices facilitates the construction of well-organized bifurcation graphs depicting the interconnectedness among families of periodic solutions. The critical importance of the symmetries of periodic solutions in comprehending the interaction among these families is demonstrated.