On the Stability of the System of Thomson’s Vortex $n$-Gon and a Moving Circular Cylinder

The linearization matrix and the quadratic part of the Hamiltonian of the problem are studied. As a result, the parameter space of the problem is divided into the instability area and the area of linear stability where nonlinear analysis is required. In the case $n=2,3$ two domains of linear stability are found. In the case $n=4,5,6$ there is just one domain. In the case $n\geqslant 7$ the studied solution is unstable for any value of the problem parameters. The obtained results in the limiting case as $a\to\infty$ agree with the known results for the model of point vortices outside the circular domain.