The stability of Thomson's configurations of vortices in a circular domain

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 $n\ge 7$, and in the case $2\le n \le 6$ — 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 $0<p\le p_{*n},$ $n=2,\dots,6$. We formulate the necessary and sufficient conditions for $n\neq 5 $. We give full proof 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. It turned out that one of them lead to instability.