Leonid Kurakin
8a, Milchakova st., RostovonDon, 344090, Russia
Department of Mechanics and Mathematics, Rostov University
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., Kurdoglyan A. V.
On the Isolation/Nonisolation of a Cosymmetric Equilibrium and Bifurcations in its Neighborhood
2021, vol. 26, no. 3, pp. 258270
Abstract
A dynamical system with a cosymmetry is considered. V. I. Yudovich showed that
a noncosymmetric equilibrium of such a system under the conditions of the general position
is a member of a oneparameter family. In this paper, it is assumed that the equilibrium is
cosymmetric, and the linearization matrix of the cosymmetry is nondegenerate. It is shown that,
in the case of an odddimensional dynamical system, the equilibrium is also nonisolated and
belongs to a oneparameter family of equilibria. In the evendimensional case, the cosymmetric
equilibrium is, generally speaking, isolated. The Lyapunov – Schmidt method is used to study
bifurcations in the neighborhood of the cosymmetric equilibrium when the linearization matrix
has a double kernel. The dynamical system and its cosymmetry depend on a real parameter.
We describe scenarios of branching for families of noncosymmetric equilibria.

Kurakin L. G., Ostrovskaya I. V.
On Stability of Thomson’s Vortex $N$gon in the Geostrophic Model of the Point Bessel Vortices
2017, vol. 22, no. 7, pp. 865879
Abstract
A stability analysis of the stationary rotation of a system of $N$ identical point Bessel vortices lying uniformly on a circle of radius $R$ is presented. The vortices have identical intensity $\Gamma$ and length scale $\gamma^{1}>0$. The stability of the stationary motion is interpreted as equilibrium stability of a reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied.
The cases for $N=2,\ldots,6$ are studied sequentially. The case of odd $N=2\ell+1\geqslant 7$ vortices and the case of even $N=2n\geqslant 8$ vortices are considered separately. It is shown that the $(2\ell+1)$gon is exponentially unstable for $0 < \gamma R < R_*(N)$. However, this $(2\ell+1)$gon is stable for $\gamma R\geqslant R_*(N)$ in the case of the linearized problem (the eigenvalues of the linearization matrix lie on the imaginary axis). The even $N=2n\geqslant 8$ vortex $2n$gon is exponentially unstable for $R>0$.

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 $q = R^2/R_0^2$.

Kurakin L. G.
On the Stability of Thomson’s Vortex Pentagon Inside a Circular Domain
2012, vol. 17, no. 2, pp. 150169
Abstract
We investigate the stability problem for stationary rotation of five identical point vortices located at the vertices of a regular pentagon inside a circular domain. The main result is the proof of theorems which have been announced by the author in Doklady Physics (2004, vol. 49, no. 11, pp. 658–661).

Kurakin L. G.
On the stability of Thomson’s vortex configurations inside a circular domain
2010, vol. 15, no. 1, pp. 4058
Abstract
The paper is devoted to the analysis of stability of the stationary rotation of a system of $n$ identical point vortices located at the vertices of a regular $n$gon of radius $R_0$ inside a circular domain of radius $R$. Havelock stated (1931) that the corresponding linearized system has exponentially growing solutions for $n \geqslant 7$ and in the case $2 \leqslant n \leqslant 6$ — only if the parameter $p = R^2_0/R^2$ is greater than a certain critical value: $p_{*n} < p < 1$. In the present paper the problem of nonlinear stability is studied for all other cases $0 < p \leqslant p_{*n}$, $n = 2, . . . ,6$. Necessary and sufficient conditions for stability and instability for $n \ne 5$ are formulated. A detailed proof for a vortex triangle is presented. A part of the stability conditions is substantiated by the fact that the relative Hamiltonian of the system attains a minimum on the trajectory of the stationary motion of the vortex triangle. The case where the sign of the Hamiltonian is alternating requires a special approach. The analysis uses results of KAM theory. All resonances up to and including the 4th order occurring here are enumerated and investigated. It has turned out that one of them leads to instability.
