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 \(V=\frac{1}{2}[\alpha (x^2+y^2)+x^2y^2]\) $V=\frac{1}{2}[\alpha (x^2+y^2)+x^2{y}^2]$ , which tends to the Yang–Mills potential \(V=\frac{1}{2}x^2y^2\) $V=\frac{1}{2}{x}^2{y}^2$ , when \(\alpha \) $\alpha$ tends to zero. We consider their stability diagrams, the corresponding Poincaré surfaces of section and the forms of the orbits.