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    Leonid Kurakin

    8a, Milchakova st., Rostov-on-Don, 344090, Russia
    Department of Mechanics and Mathematics, Rostov University


    Kurakin L. G., Ostrovskaya I. V.
    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 $q = R^2/R_0^2$.
    Keywords: point vortices, stationary rotation, stability, resonance
    Citation: Kurakin L. G., Ostrovskaya I. V.,  The stability criterion of a regular vortex pentagon outside a circle, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 2, pp.  355-368
    Kurakin L. G.
    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).
    Keywords: point vortex, stationary motion, stability, resonance
    Citation: Kurakin L. G.,  On the Stability of Thomson’s Vortex Pentagon Inside a Circular Domain, Rus. J. Nonlin. Dyn., 2011, Vol. 7, No. 3, pp.  465-488
    Kurakin L. G.
    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.
    Keywords: point vortices, stationary motion, stability, resonance
    Citation: Kurakin L. G.,  The stability of Thomson's configurations of vortices in a circular domain, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 3, pp.  295-317

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