Alexander Gudimenko

Qualitative structure of relative motion of three point vortices on the unbounded plain is studied. A classification of phase portraits is proposed.

Motion of three point vortices in a perturbed singular configuration is studied numerically and analytically. Several cases of the motion are analyzed according to location of the vorticity center to the orbit of one of the vortices. For each of these cases the trajectories of vortices are calculated.

Dynamics of perturbed stable equilateral and collinear configurations of three point vortices in an incompressible ideal fluid is studied. The asymptotics of the perturbed motion to the unperturbed one is obtained. It is shown that in the first approximation in a appropriate coordinate system the vortices rotate about their undisturbed positions in elliptical orbits. The velocity of the rotation is calculated. It is shown that the eccentricities of the orbits are coincide. The ratio of major axes of any two orbits is calculated. In case of equilateral configuration this ratio is equal to the ratio of inverse intensities of the corresponding vortices. The angle between major axes of any two orbits of the vortices is calculated. In case of equilateral configuration this angle is ±120 degrees.

A Hamiltonian dynamical system describing a rotating incompressible two-dimensional flow, disturbed by an oscillating point vortex, is studied numerically and analytically. It is shown numerically that under perturbation the region of strongly mixed trajectories of the system forms. As the amplitude of perturbation increases, the region grows in size due to the destruction and absorption of the nearest resonances. The order and multiplicity of the resonances are determined mainly by the relation ω/Ω, where ω is the perturbation frequency and Ω is the rotation frequency of the flow. The patterns of the resonances differ essentially whether this quantity is integer or fractional. The results of the numerical experiment are justified analytically. In the domain that is sufficiently far from the vortex, the Hamiltonian is represented in the angle-action variables. Based on the representation, the arrangement of the resonances on the phase plane is analyzed. In particular, a classification of the resonances, which is adequate to the numerical patterns, is proposed. The widths of the resonances are calculated. It is shown that, at large distances from the vortex, global chaotization of trajectories of the system is impossible.