Integrability and Chaos in Vortex Lattice Dynamics

    2019, Volume 24, Number 1, pp.  101-113

    Author(s): Kilin A. A., Artemova E. M.

    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.
    Keywords: vortices on a torus, vortex lattices, point vortices, nonintegrability, chaos, invariant manifold, Poincar̉‘e map, topological analysis, numerical analysis, accuracy of calculations, reduction, reduced system
    Citation: Kilin A. A., Artemova E. M., Integrability and Chaos in Vortex Lattice Dynamics, Regular and Chaotic Dynamics, 2019, Volume 24, Number 1, pp. 101-113

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