Stability Diagrams of a Regular System of Vortex Charges Outside the Circular Region in the Case of Arbitrary Circulation

The problem of the stability of the stationary rotation of a system of identical point vortex charges located at the vertices of a regular N-gon outside a circular region, in the case of arbitrary circulation of Γ around the boundary is considered. The interaction potential between the charges is inversely proportional to the distance between them. In this paper continue their research of the authors' articles, where cases of non-circulant flow around a boundary and arbitrary circulation for the logarithmic potential of interaction between point vortices were considered. This paper presents a general scheme for studying stability for an arbitrary N≥2, which is implemented for cases N=2,3. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are investigated analytically and numerically. There are constructed stability diagrams, which indicate the areas of orbital stability, instability, and areas of linear stability that require additional nonlinear analysis. All resonances up to and including the fourth order arising in this problem are listed and investigated numerically. An instability has been numerically detected on the resonance curve Γ03 corresponding to the two-fold zero eigenvalue of the linearization matrix at N=3. The results of the theoretical analysis are confirmed by numerical calculation of the trajectories of vortex charges.