Integrability and Chaos in Vortex Lattice Dynamics
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This paper is concerned with the problem of the interaction of vortex lattices, which is equivalent to the problem of the motion of point vortices on a torus. It is shown that the dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate configurations are found and their stability is investigated. For two vortex lattices it is also shown that, in absolute space, vortices move along closed trajectories except for the case of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero total strength are considered. For three vortices, a reduction to the level set of first integrals is performed. The nonintegrability of this problem is numerically shown. It is demonstrated that the equations of motion of four vortices on a torus admit an invariant manifold which corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices on this invariant manifold and on a fixed level set of first integrals are obtained and their nonintegrability is numerically proved.
Keywordsvortices on a torus vortex lattices point vortices nonintegrability chaos invariant manifold Poincaré map topological analysis numerical analysis accuracy of calculations reduction reduced system
MSC2010 numbers:76B47 70H05 37Jxx 34Cxx
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- 2.Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., and Vainchtein, D., Vortex Crystals, in Advances in Applied Mechanics: Vol. 39, E. van der Giessen, H. Aref (Eds.), San Diego: Acad. Press, 2003, pp. 1–79.Google Scholar
- 9.Borisov, A.V. and Mamaev, I. S., Mathematical Methods in the Dynamics of Vortex Structures, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).Google Scholar
- 14.Fridman, A.A. and Polubarinova, P.Ya., On Moving Singularities of a Flat Motion of an Incompressible Fluid, Geofiz. Sb., 1928, vol. 5, no. 2, pp. 9–23 (Russian).Google Scholar
- 15.Gromeka, I. S., On Vortex Motions of a Liquid on a Sphere, Uchen. Zap. Imp. Kazan. Univ., 1885, no. 3, pp. 202–236 (Russian).Google Scholar
- 20.Lamb, H., Hydrodynamics, 6th ed., New York: Dover, 1945.Google Scholar
- 22.Novikov, E.A., Dynamics and Statistics of a System of Vortices, JETP, 1975, vol. 41, no. 5, pp. 937–943; see also: Zh. Éksper. Teoret. Fiz., 1975, vol. 68, no. 5, pp. 1868–1882.Google Scholar
- 26.Tkachenko, V.K., Stability of Vortex Lattices, JETP, 1966, vol. 23, no. 6, pp. 1049–1056; see also: Zh. Èksper. Teoret. Fiz., 1966, vol. 50, no. 6, pp. 1573–1585.Google Scholar