# 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.

Access to the full text on the Springer website |