On the Stability of the Orbit and the Invariant Set of Thomson’s Vortex Polygon in a Two-Fluid Plasma

The motion of the system of $N$ point vortices with identical intensity $\Gamma$ in the Alfven model of a two-fluid plasma is considered. The stability of the stationary rotation of $N$ identical vortices disposed uniformly on a circle with radius $R$ is studied for $N = 2,\ldots,5$. This problem has three parameters: the discrete parameter $N$ and two continuous parameters $R$ and $c$, where $c>0$ is the value characterizing the plasma. Two different definitions of the stability are used - the orbital stability and the stability of a three-dimensional invariant set founded by the orbits of a continuous family of stationary rotations. Instability is interpreted as instability of equilibrium of the reduced system. An analytical analysis of eigenvalues of the linearization matrix and the quadratic part of the Hamiltonian is given. As a result, the parameter space $(N,R,c)$ of this problem for two stability definitions considered is divided into three parts: the domain $\boldsymbol{A}$ of stability in an exact nonlinear problem setting, the linear stability domain $\boldsymbol{B}$, where the nonlinear analysis is needed, and the domain of exponential instability $\boldsymbol{C}$. The application of the stability theory of invariant sets for the systems with a few integrals for $N=2,3,4$ allows one to obtain new statements about the stability in the domains, where nonlinear analysis is needed in investigating the orbital stability.