Evolution of the characteristic curves in the restricted three-body problem in terms of the mass parameter

In this work, we study the evolution of the families of simple symmetric periodic orbits in the restricted three-body problem whatever the value of the mass parameter μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}. To classify these characteristic curves, we introduce a topological characterization of both orbits and families. Starting from the work of Strömgren for the Copenhagen case, we analyze the evolution of these families, when the mass parameter μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} varies in (0, 1/2], focusing on their topological characterization, the existence of asymptotic points and the appearance of certain types of orbits such as horseshoe orbits. Lastly, we consider two samples, the Earth–Moon and Sun–Jupiter systems and classify the different types of orbits for these systems.