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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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