Bifurcation of point vortex equilibria: four-vortex translating configurations and five-vortex stationary configurations

Abstract

Point vortices in a plane form a dynamical system. A vortex equilibrium is a solution where all vortices move with a common velocity. The configuration is stationary or translating depending on whether the velocity is zero or not. In this paper, we study the bifurcation for four-vortex translating configurations and five-vortex stationary configurations. The number of distinct four-vortex translating configurations can be 0, 3, 4, 6. Among the nonzero cases, there are 1, 0, and 0, or 2 collinear ones, respectively. The number of distinct five-vortex stationary configurations can be 0, 2, 4, 6. In each case, all smaller or equal even numbers of collinear configurations among them are possible. We provide simple and clear necessary and sufficient conditions on nonzero real circulations for all the possible numbers of collinear and strictly planer equilibrium. How two or three equilibrium merge and then apart is also described. All configurations are studied through the minimal polynomial systems obtained and defined by O’Neil. The 4, 5-vortex systems contain two and three independent circulation parameters, respectively. We count the numbers of zeros by univariate polynomials after triangulating the polynomial systems. Information on the numbers of real or complex zeros and their intersection multiplicities for different components are glued together to study the bifurcation as parameters vary in and for the four-vortex and five-vortex cases, respectively. Many computations are assisted by a computer algebra system.