## Resonances in the stability problem of a point vortex quadrupole on a plane.(English)Zbl 1479.76037

Summary: 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 $$\varkappa\in(-3,0)$$.

### MSC:

 76E07 Rotation in hydrodynamic stability 76B47 Vortex flows for incompressible inviscid fluids 76U05 General theory of rotating fluids
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