Periodic orbits in a galactic potential

Abstract

We make a numerical study of the periodic orbits of period-1 and their bifurcations in a galactic type potential V=12[α(x2+y2)+x2y2]\\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=12x2y2\\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 α\\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 Poincaré surfaces of section and the forms of the orbits.