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This article is cited in 1 scientific paper (total in 1 paper)
Special Issue: 200th birthday of Hermann von Helmholtz
Resonances in the Stability Problem of a Point Vortex
Quadrupole on a Plane
Leonid G. Kurakinabc, Irina V. Ostrovskayaa a Institute for Mathematics, Mechanics and Computer Sciences, Southern Federal University,
ul. Milchakova 8a, 344090 Rostov-on-Don, Russia
b Southern Mathematical Institute, Vladikavkaz Scienific Center of RAS,
ul. Markusa 22, 362027 Vladikavkaz, Russia
c Water Problems Institute, RAS,
ul. Gubkina 3, 119333 Moscow, Russia
Abstract:
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) $.
Keywords:
$N+1$ vortex problem, point vortices, Hamiltonian equation, stability, resonances.
Received: 22.07.2021 Accepted: 20.08.2021
Citation:
Leonid G. Kurakin, Irina V. Ostrovskaya
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