Stability of the Rhombus Vortex Problem with a Central Vortex

We characterize the type stability of the five-vortex problem in the plane, where it is assumed that configuration is a rhombus vortex problem with a central vortex. More precisely, we suppose that the opposite vortices have the same vorticity Γ1=Γ2=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _1=\Gamma _2=1$$\end{document} and Γ3=Γ4=Γ≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _3=\Gamma _4=\Gamma \ne 0$$\end{document} and the vortex at the center has vorticity Γ0≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma _0\ne 0$$\end{document}. The stability is given in terms of the size of the rhombus and the vorticities.