Эта публикация цитируется в
2 статьях
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
Аннотация:
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) $.
Ключевые слова:
$N+1$ vortex problem, point vortices, Hamiltonian equation, stability, resonances.
MSC: 76B47,
76E20,
70K30,
70K45 Поступила в редакцию: 22.07.2021
Принята в печать: 20.08.2021
Язык публикации: английский
DOI:
10.1134/S1560354721050051