Hildeberto Cabral
Universidade Federal de Pernambuco, Recife, Pernambuco
Publications:
Carvalho A., Cabral H. E.
Lyapunov Orbits in the $n$Vortex Problem on the Sphere
2015, vol. 20, no. 3, pp. 234246
Abstract
In the phase space reduced by rotation, we prove the existence of periodic orbits of the $(n + 1)$vortex problem emanating from a relative equilibrium formed by $n$ unit vortices at the vertices of a regular polygon at a fixed latitude and an additional vortex of intensity κ at the north pole when the ideal fluid moves on the surface of a sphere.

Carvalho A., Cabral H. E.
Lyapunov Orbits in the $n$Vortex Problem
2014, vol. 19, no. 3, pp. 348362
Abstract
In the reduced phase space by rotation, we prove the existence of periodic orbits of the $n$vortex problem emanating from a relative equilibrium formed by $n$ unit vortices at the vertices of a regular polygon, both in the plane and at a fixed latitude when the ideal fluid moves on the surface of a sphere. In the case of a plane we also prove the existence of such periodic orbits in the $(n+1)$vortex problem, where an additional central vortex of intensity κ is added to the ring of the polygonal configuration.

Cabral H. E., Meyer K. R., Schmidt D. S.
Stability and bifurcations for the $N + 1$ vortex problem on the sphere
2003, vol. 8, no. 3, pp. 259282
Abstract
The equations of motion for $N$ vortices on a sphere were derived by V.A.Bogomolov in 1977. References to related work can be found in the book by P.K.Newton. We use the equations of motion found there to discuss the stability of a ring of $N$ vortices of unit strength at the latitude $z$ together with a vortex of strength $\kappa$ at the north pole. The regions of stability are bounded by curves $\kappa = \kappa(z)$. These curves are computed explicitly for all values of $N$.
When the stability of a configuration changes, for example by varying the strength of the vortex at the north pole, bifurcations to new configurations are possible. We compute the bifurcation equations explicitly for $N = 2, 3$ and $4$. For larger values of $N$ the complexity of the formal computations becomes too great and we use a numerical value for the latitude instead. We thus derive the bifurcation equations in a seminumerical form. As expected the new configurations look very similar to those which had been found previously for the planar case. 