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This article is cited in 18 scientific papers (total in 18 papers)
Stability and bifurcations for the $N+1$ vortex problem on the sphere
H. E. Cabrala, K. R. Meyerb, D. S. Schmidtc a Departamento de Matemática,
Universidade Federal de Pernambuco,
Recife, Pernambuco, Brazil
b Department of Mathematical Sciences,
University of Cincinnati,
Cincinnati, Ohio 45221-0025
c Department of Electrical & Computer,
Engineering and Computer Science,
University of Cincinnati, Cincinnati,
Ohio 45221-0030
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 semi-numerical form. As expected the new configurations look very similar to those which had been found previously for the planar case.
Received: 23.05.2003
Citation:
H. E. Cabral, K. R. Meyer, D. S. Schmidt, “Stability and bifurcations for the $N+1$ vortex problem on the sphere”, Regul. Chaotic Dyn., 8:3 (2003), 259–282
Linking options:
https://www.mathnet.ru/eng/rcd781 https://www.mathnet.ru/eng/rcd/v8/i3/p259
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