Leonid Kurakin

The stability problem of a moving circular cylinder of radius $R$ and a system of $n$ identical point vortices uniformly distributed on a circle of radius $R_0^{}$, with $n\geqslant 2$, is considered. The center of the vortex polygon coincides with the center of the cylinder. The circulation around the cylinder is zero. There are three parameters in the problem: the number of point vortices $n$, the added mass of the cylinder $a$ and the parameter $q=\frac{R^2}{R_0^2}$.
The linearization matrix and the quadratic part of the Hamiltonian of the problem are studied. As a result, the parameter space of the problem is divided into the instability area and the area of linear stability where nonlinear analysis is required. In the case $n=2,\,3$ two domains of linear stability are found. In the case $n=4,\,5,\,6$ there is just one domain. In the case $n\geqslant 7$ the studied solution is unstable for any value of the problem parameters. The obtained results in the limiting case as $a\to\infty$ agree with the known results for the model of point vortices outside the circular domain.

The motion of the system of $N$ point vortices with identical intensity $\Gamma$ in the Alfven model of a two-fluid plasma is considered. The stability of the stationary rotation of $N$ identical vortices disposed uniformly on a circle with radius $R$ is studied for $N = 2,\ldots,5$. This problem has three parameters: the discrete parameter $N$ and two continuous parameters $R$ and $c$, where $c>0$ is the value characterizing the plasma. Two different definitions of the stability are used - the orbital stability and the stability of a three-dimensional invariant set founded by the orbits of a continuous family of stationary rotations. Instability is interpreted as instability of equilibrium of the reduced system. An analytical analysis of eigenvalues of the linearization matrix and the quadratic part of the Hamiltonian is given. As a result, the parameter space $(N,R,c)$ of this problem for two stability definitions considered is divided into three parts: the domain $\boldsymbol{A}$ of stability in an exact nonlinear problem setting, the linear stability domain $\boldsymbol{B}$, where the nonlinear analysis is needed, and the domain of exponential instability $\boldsymbol{C}$. The application of the stability theory of invariant sets for the systems with a few integrals for $N=2,3,4$ allows one to obtain new statements about the stability in the domains, where nonlinear analysis is needed in investigating the orbital stability.

The systems of differential equations with one cosymmetry are considered [1]. The ordinary object for such systems is a one-dimensional continuous family of equilibria. The stability spectrum changes along this family, but it necessarily contains zero. We consider the nondegeneracy condition, thus the boundary equilibria separate the family on linearly stable and instable areas. The stability of the boundary equilibria depends on nonlinear terms of the system.
The stability problem for the systems with one cosymmetry is studied in [2]. The general problem is to apply the stability criteria one needs to compute coefficients of the model system. It is especially difficult if the system has a large dimension, while a number of critical variables may be small. A method for calculating coefficients is proposed in [3].
In this work the expressions for the known stability criteria are proposed in a form convenient for calculation. The explicit formulas of the coefficients of the model system are given in semi-invariant form. They are expressed using the generalized eigenvectors of the linear matrix and its conjugate matrix.

In this paper the two-layer geostrophic model of the rotating fluid and the model of Bessel vortices are considered. Kirchhoff's model of vortices in a homogeneous fluid is the limiting case of both of these models. Part of the study is performed for an arbitrary Hamiltonian depending on the distances between point vortices.
The review of the stability problem of stationary rotation of regular Thomson's vortex $N$-gon of identical vortices is given for ${N\geqslant 2}$. The stability problem of the vortex tripole/quadrupole is also considered. This axisymmetric vortex structure consists of a~central vortex of an arbitrary intensity and two/three identical peripheral vortices. In the model of a two-layer fluid, peripheral vortices belong to one of the layers and the central vortex can belong to either another layer or the same.
The stability of the stationary rotation is interpreted as orbital stability (the stability of a one-parameter orbit of a stationary rotation of a vortex system). The instability of the stationary rotation is instability of equilibrium of the reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied.
The parameter space is divided into three parts: $\bf A$ is the domain of stability in an exact nonlinear setting, $\bf B$ is the linear stability domain, where the stability problem requires nonlinear analysis, and $\bf C$ is the instability domain.
In the stability problem of a vortex multipole, another definition of stability is used; it is the stability of an invariant three-parametric set of all trajectories of the families of stationary orbits. It is shown that in the case of non zero total intensity, the stability of the invariant set implies orbital stability.

The nonlinear stability analysis of a stationary rotation of a system of five identical point vortices lying uniform on a circle of radius $R_0$ outside a circular domain of radius $R$ is performed. The problem is reduced to the problem of equilibrium of Hamiltonian system with cyclic variable. The stability of stationary motion is interpreted as Routh stability. The conditions of stability, formal stability and instability are obtained subject to the parameter $q = R^2/R_0^2$.

We investigate the stability problem for a system of five stationary rotation 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 the author in paper (Doklady Physics, 2004, vol. 49, no 11, pp. 658–661).

The paper is devoted to stability of the stationary rotation of a system of $n$ equal point vortices located at vertices of a regular $n$-gon of radius $R_0$ inside a circular domain of radius $R$. T. H. Havelock stated (1931) that the corresponding linearized system has an exponentially growing solution for $n \geqslant 7$, and in the case $2 \leqslant n \leqslant 6$ — only if parameter $p=R_0^2/R^2$ is greater than a certain critical value: $p_{*n} < p < 1$. In the present paper the problem on stability is studied in exact nonlinear formulation for all other cases $0 < p \leqslant p_{*n}$, $n=2,\ldots,6$. We formulate the necessary and sufficient conditions for $n\neq5$. We give full proff only for the case of three vortices. A part of stability conditions is substantiated by the fact that the relative Hamiltonian of the system attains a minimum on the trajectory of a stationary motion of the vortex $n$-gon. The case when its sign is alternating, arising for $n=3$, did require a special study. This has been analyzed by the KAM theory methods. Besides, here are listed and investigated all resonances encountered up to forth order. In turned out that one of them lead to instability.