Talk 8

Title: Bifurcations and Enumeration of Relative Equilibria of the Restricted Rhombus+1 Vortex Problem

Speaker:  Juscelino Grigório Lopes

Affiliation: Universidade de Pernambuco - Campus Nazaré da Mata

Abstract: We consider some basic issues on the relative equilibria of the  restricted five-vortex problem in which the nonzero vorticities form a rhombus relative equilibrium. Firstly, we elicit a basic structure underlying the relative equilibrium equations, an involutive diffeomorphism which, as in the case of the Newtonian five-body problem, has proved very useful as a simplifying tool throughout our work. Next, we exhibit a bounded neighborhood of the rhombus family containing all relative equilibria, determine all the solutions for the values of the vortex ratio 𝜞 of the vorticities either sufficiently large or close to zero, and enumerate the symmetric classes of relative equilibria for any allowable choice of values 𝜞 for the rhombus semi-diagonal length d. Degeneracy of symmetric classes is also examined, and the related bifurcations are discussed. The presence of a nonlinear constraint to be satisfied by the parameters 𝜞 and d represents a significant challenge to a rigorous analysis of relative equilibria.