On the Stability of a System of Two Identical Point Vortices and a Cylinder

We consider the stability problem for a system of two identical point vortices and a circular cylinder located between them. The circulation around the cylinder is zero. There are two parameters in the problem: the added mass $a$ of the cylinder and $q=R^2/R_0^2$, where $R$ is the radius of the cylinder and $2R_0$ is the distance between vortices. We study the linearization matrix and the quadratic part of the Hamiltonian of the problem, find conditions of orbital stability and instability in nonlinear statement, and point out parameter domains in which linear stability holds and nonlinear analysis is required. The results for $a\to \infty$ are in agreement with the classical results for a fixed cylinder. We show that the mobility of the cylinder leads to the expansion of the stability region.