Resonances in the Stability Problem of a Point Vortex Quadrupole on a Plane

A system of four point vortices on a plane is considered. Its motion is described by the Kirchhoff equations. Three vortices have unit intensity and one vortex has arbitrary intensity $\varkappa$. We study the stability problem for the stationary rotation of a vortex quadrupole consisting of three identical vortices located uniformly on a circle around a fourth vortex. It is known that for $ \varkappa> 1 $ the regime under study is unstable, and in the case of $ \varkappa <-3 $ and $ 0 <\varkappa <1 $ the orbital stability takes place. New results are obtained for $ -3 <\varkappa <0 $. It is found that, for all values of $ \varkappa $ in the stability problem, there is a resonance $1:1$ (diagonalizable case). Some other resonances through order four are found and investigated: double zero resonance (diagonalizable case), resonances $1:2$ and $1:3$, occurring with isolated values of $\varkappa $. The stability of the equilibrium of the system reduced by one degree of freedom with the involvement of the terms in the Hamiltonian through degree four is proved for all $ \varkappa \in (-3,0) $.