Irina Ostrovskaya
Mil’chakova 8a, RostovonDon, 344090, Russia
Southern Federal University, Faculty of Mathematics, Mechanics and Computer Sciences
Publications:
Kurakin L. G., Ostrovskaya I. V.
Resonances in the Stability Problem of a Point Vortex Quadrupole on a Plane
2021, vol. 26, no. 5, pp. 526542
Abstract
A system of four point vortices on a plane is considered. Its motion is described by
the Kirchhoff equations. Three vortices have unit intensity and one vortex has arbitrary intensity $\varkappa$. We study the stability problem for the stationary rotation of a vortex quadrupole consisting of three identical vortices located uniformly on a circle around a fourth vortex.
It is known that for $ \varkappa> 1 $ the regime under study is unstable,
and in the case of $ \varkappa <3 $ and $ 0 <\varkappa <1 $ the orbital stability takes place. New results are obtained for $ 3 <\varkappa <0 $. It is found that, for all values of $ \varkappa $ in the
stability problem, there is a resonance $1:1$ (diagonalizable case). Some other resonances
through order four are found and investigated: double zero resonance
(diagonalizable case), resonances 1:2 and 1:3, occurring with isolated values of $\varkappa $.
The stability of the equilibrium of the system reduced by one degree of freedom with the involvement of the
terms in the Hamiltonian through degree four is proved for all $ \varkappa \in (3,0) $.

Kurakin L. G., Ostrovskaya I. V., Sokolovskiy M. A.
On the Stability of Discrete Tripole, Quadrupole, Thomson’ Vortex Triangle and Square in a Twolayer/Homogeneous Rotating Fluid
2016, vol. 21, no. 3, pp. 291334
Abstract
A twolayer quasigeostrophic model is considered in the $f$plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case when the structure consists of a central vortex of arbitrary intensity $\Gamma$ and two/three identical peripheral vortices. The identical vortices, each having a unit intensity, are uniformly distributed over a circle of radius $R$ in a single layer. The central vortex lies either in the same or in another layer. The problem has three parameters $(R, \Gamma, \alpha)$, where $\alpha$ is the difference between layer thicknesses. A limiting case of a homogeneous fluid is also considered.
The theory of stability of steadystate motions of dynamic systems with a continuous symmetry group $\mathcal{G}$ is applied. The two definitions of stability used in the study are Routh stability and $\mathcal{G}$stability. The Routh stability is the stability of a oneparameter orbit of a steadystate rotation of a vortex multipole, and the $\mathcal{G}$stability is the stability of a threeparameter invariant set $O_\mathcal{G}$, formed by the orbits of a continuous family of steadystate rotations of a multipole. The problem of Routh stability is reduced to the problem of stability of a family of equilibria of a Hamiltonian system. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically.
The cases of zero total intensity of a tripole and a quadrupole are studied separately. Also, the Routh stability of a Thomson vortex triangle and square was proved at all possible values of problem parameters. The results of theoretical analysis are sustained by numerical calculations of vortex trajectories.

Kurakin L. G., Ostrovskaya I. V.
Nonlinear Stability Analysis of a Regular Vortex Pentagon Outside a Circle
2012, vol. 17, no. 5, pp. 385396
Abstract
A nonlinear stability analysis of the stationary rotation of a system of five identical point vortices lying uniformly on a circle of radius $R_0$ outside a circular domain of radius $R$ is performed. The problem is reduced to the problem of stability of an equilibrium position of a Hamiltonian system with a cyclic variable. The stability of stationary motion is interpreted as Routh stability. Conditions for stability, formal stability and instability are obtained depending on the values of the parameter $q = R^2/R_0^2$.
