Thomson’s Heptagon: A case of bifurcation at infinity

Vortex modeling has a long history. Descartes (1644) used it as a model for the solar systems. J. J. Thomsom (1883) used it as a model for the atom. We consider point-vortex systems, which can be regarded as “discrete” solutions of the Euler equation. Their dynamics is described by a Hamiltonian system of equations. We are interested in polygonal configurations and how their stability depends upon various dynamical variables. In the plane a polygon with seven vortices has been shown to be a special boundary case: polygons with N<7N<7 vortices are (linearly and nonlinearly) stable while polygons with N>7N>7 vortices are unstable. Why should N=7N=7 be special? Celestial Mechanics helped us to simplify a problem that has been studied for over a century, and to show that the case of Thomson’s Heptagon is actually a case of bifurcation at infinity. This becomes particularly clear when considering the corresponding problem of a ring on a sphere with two polar vortices of variable intensities ΓN and ΓS, at the North and South Pole, respectively.