Periodic orbits in a galactic potential

We make a numerical study of the periodic orbits of period-1 and their bifurcations in a galactic type potential V = 1 2 [ alpha ( x 2 + y 2 ) + x 2 y 2 ] \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=\frac{1}{2}[\alpha (x<^>2+y<^>2)+x<^>2y<^>2]$$\end{document} , which tends to the Yang-Mills potential V = 1 2 x 2 y 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V=\frac{1}{2}x<^>2y<^>2$$\end{document} , when alpha \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} tends to zero. We consider their stability diagrams, the corresponding Poincar & eacute; surfaces of section and the forms of the orbits.