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 . 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 >1 the regime under study is unstable, and in the case of <-3 and 0<<1 the orbital stability takes place. New results are obtained for -3<<0 . It is found that, for all values of 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 . 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 ∈(-3,0) .