Stability and bifurcations for the \(N+1\) vortex problem on the sphere
The equations of motion for \(N\) vortices on a sphere were derived by \textit{V. A. Bogomolov} [Fluid Dyn. 12(1977), 863--870 (1978; Zbl 0442.76020)]. References to related work can be found in the book by \textit{P. K. Newton} [The \(N\)-vortex problem. Analytical techniques. New York, NY: Springer (2001; Zbl 0981.76002)]. The authors 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. The authors 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 the authors use a numerical value for the latitude instead. They thus derive the bifurcation equations in a semi-numerical form. As expected the new configurations look very similar to those which had been found previously for the planar case.
scientific article; zbMATH DE number 2097039