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Rus. J. Nonlin. Dyn., 2020, Volume 16, Number 1, Pages 3–11 (Mi nd690)  

This article is cited in 2 scientific papers (total in 2 papers)

Nonlinear physics and mechanics

On the Stability of the Orbit and the Invariant Set of Thomsonís Vortex Polygon in a Two-Fluid Plasma

L. G. Kurakinab, I. A. Lysenkoc

a Water Problems Institute of RAS, ul. Gubkina 3, Moscow, 119333 Russia
b Southern Mathematical Institute, Vladikavkaz Scientific Center of RAS, ul. Markusa 22, Vladikavkaz, 362027 Russia
c Southern Federal University, ul. Milchakova 8a, Rostov-on-Don, 344090 Russia

Abstract: 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.

Keywords: point vortex, two-fluid plasma, stability, stationary rotation, Hamiltonian system, invariant set

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation VnGr/2020-04-IM
Research was financially supported by Southern Federal University, 2020 (Ministry of Science and Higher Education of the Russian Federation), VnGr/2020-04-IM.


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MSC: 93B18, 93B52
Received: 26.06.2019

Citation: L. G. Kurakin, I. A. Lysenko, “On the Stability of the Orbit and the Invariant Set of Thomsonís Vortex Polygon in a Two-Fluid Plasma”, Rus. J. Nonlin. Dyn., 16:1 (2020), 3–11

Citation in format AMSBIB
\by L. G. Kurakin, I. A. Lysenko
\paper On the Stability of the Orbit and the Invariant Set of Thomsonís Vortex Polygon in a Two-Fluid Plasma
\jour Rus. J. Nonlin. Dyn.
\yr 2020
\vol 16
\issue 1
\pages 3--11

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    This publication is cited in the following articles:
    1. Kurakin L., Ostrovskaya I., “on the Effects of Circulation Around a Circle on the Stability of a Thomson Vortexn-Gon”, Mathematics, 8:6 (2020), 1033  crossref  isi  scopus
    2. Kilin A.A., Artemova E.M., “Stability of Regular Vortex Polygons in Bose-Einstein Condensate”, Izv. Inst. Mat. Inform., 56 (2020), 20–29  crossref  mathscinet  zmath  isi  scopus
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