Point vortices in a circular domain: stability, resonances, and instability of stationary rotation of a regular vortex polygon

Abstract

The paper is devoted to stability of the stationary rotation of a system of equal point vortices located at vertices of a regular n-gon of radius R0 inside a circular domain of radius R with a common center of symmetry. T. X. Havelock stated (1931) that the corresponding linearized system has an exponentially growing solution for n>=7, and in the case 2<=n <= 6 only if parameter p=R0^2}/R^2 is reater than a certain critical value: p(crit)n< p<1. In the present paper the problem on stability is studied in exact nonlinear formulation for all other cases 0<p<=p(crit)n,$ n=2,...,6. We formulate the necessary and sufficient conditions for n different from 5 . For the vortex pentagon it remains unclear the answer to the question about stability for a null set of parameter p. 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=3,5 , did require a special study. This has been analyzed by the KAM theory methods.