A. V. Borisov, I. S. Mamaev, A. A. Kilin, “Absolute and relative choreographies in the problem of point vortices moving on a plane”, Regul. Chaotic Dyn., 9:2 (2004), 101

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Regular and Chaotic Dynamics, 2004, Volume 9, Issue 2, Pages 101–111
DOI: https://doi.org/10.1070/RD2004v009n02ABEH000269 (Mi rcd735)

This article is cited in 28 scientific papers (total in 28 papers)

Absolute and relative choreographies in the problem of point vortices moving on a plane

A. V. Borisov, I. S. Mamaev, A. A. Kilin

Institute of Computer Science, 1, Universitetskaya str. 426034, Izhevsk, Russia

Citations (28)

DOI: https://doi.org/10.1070/RD2004v009n02ABEH000269

Abstract: We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the case of four vortices, the order is reduced to two degrees of freedom, and the system is not integrable. We present relative and absolute choreographies of three and four vortices of the same intensity which are periodic motions of vortices in some rotating and fixed frame of reference, where all the vortices move along the same closed curve. Similar choreographies have been recently obtained by C. Moore, A. Chenciner, and C. Simo for the

$n$ -body problem in celestial mechanics [6, 7, 17]. Nevertheless, the choreographies that appear in vortex dynamics have a number of distinct features.

Received: 05.04.2004

Bibliographic databases:

Document Type: Article

MSC: 76B47, 37J35, 70E40

Language: English

Citation: A. V. Borisov, I. S. Mamaev, A. A. Kilin, “Absolute and relative choreographies in the problem of point vortices moving on a plane”, Regul. Chaotic Dyn., 9:2 (2004), 101–111

Citation in format AMSBIB

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\by A. V. Borisov, I. S. Mamaev, A. A. Kilin

\paper Absolute and relative choreographies in the problem of point vortices moving on a plane

\jour Regul. Chaotic Dyn.

\yr 2004

\vol 9

\issue 2

\pages 101--111

\mathnet{http://mi.mathnet.ru/rcd735}

\crossref{https://doi.org/10.1070/RD2004v009n02ABEH000269}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2081550}

\zmath{https://zbmath.org/?q=an:1079.76020}

Linking options:

https://www.mathnet.ru/eng/rcd735

https://www.mathnet.ru/eng/rcd/v9/i2/p101

This publication is cited in the following 28 articles:

E. M. Artemova, “Dinamika dvukh vikhrei na konechnom ploskom tsilindre”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 33:4 (2023), 642–658
Artemova E. Kilin A., “Nonlinear Stability of Regular Vortex Polygons in a Bose-Einstein Condensate”, Phys. Fluids, 33:12 (2021), 127105
Elizaveta M. Artemova, Alexander A. Kilin, 2021 International Conference “Nonlinearity, Information and Robotics” (NIR), 2021, 1
Sokolovskiy M.A., Koshel K.V., Dritschel D.G., Reinaud J.N., “N-Symmetric Interaction of N Hetons. i. Analysis of the Case N=2”, Phys. Fluids, 32:9 (2020), 096601
Ivan S. Mamaev, Ivan A. Bizyaev, 2020 International Conference Nonlinearity, Information and Robotics (NIR), 2020, 1
P. E. Ryabov, S. V. Sokolov, “Phase Topology of Two Vortices of Identical Intensities in a Bose – Einstein Condensate”, Rus. J. Nonlin. Dyn., 15:1 (2019), 59–66
Alexander A. Kilin, Lizaveta M. Artemova, “Integrability and Chaos in Vortex Lattice Dynamics”, Regul. Chaotic Dyn., 24:1 (2019), 101–113
Pavel E. Ryabov, Artemiy A. Shadrin, “Bifurcation Diagram of One Generalized Integrable Model of Vortex Dynamics”, Regul. Chaotic Dyn., 24:4 (2019), 418–431
Garcia-Azpeitia C., “Relative Periodic Solutions of the N-Vortex Problem on the Sphere”, J. Geom. Mech., 11:3 (2019), 427–438
P. E. Ryabov, “Bifurcations of Liouville tori in a system of two vortices of positive intensity in a Bose–Einstein condensate”, Dokl. Math., 99:2 (2019), 225–229
Wang Q., “Relative Periodic Solutions of the N-Vortex Problem Via the Variational Method”, Arch. Ration. Mech. Anal., 231:3 (2019), 1401–1425
Renato C. Calleja, Eusebius J. Doedel, Carlos García-Azpeitia, “Choreographies in the n-vortex Problem”, Regul. Chaotic Dyn., 23:5 (2018), 595–612
Renato Calleja, Eusebius Doedel, Carlos García-Azpeitia, Carlos L. Pando L., “Choreographies in the discrete nonlinear Schrödinger equations”, Eur. Phys. J. Spec. Top., 227:5-6 (2018), 615
Antonio Hernández-Garduño, Banavara N Shashikanth, “Reconstruction phases in the planar three- and four-vortex problems”, Nonlinearity, 31:3 (2018), 783
Adecarlos C. Carvalho, Hildeberto E. Cabral, “Lyapunov Orbits in the $n$-Vortex Problem on the Sphere”, Regul. Chaotic Dyn., 20:3 (2015), 234–246
Adecarlos C. Carvalho, Hildeberto E. Cabral, “Lyapunov Orbits in the $n$-Vortex Problem”, Regul. Chaotic Dyn., 19:3 (2014), 348–362
E. V. Vetchanin, A. O. Kazakov, “Bifurkatsii i khaos v zadache o dvizhenii dvukh tochechnykh vikhrei v akusticheskoi volne”, Nelineinaya dinam., 10:3 (2014), 329–343
Joris Vankerschaver, Melvin Leok, “A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects”, J Nonlinear Sci, 24:1 (2014), 1
Mikhail A. Sokolovskiy, Jacques Verron, Atmospheric and Oceanographic Sciences Library, 47, Dynamics of Vortex Structures in a Stratified Rotating Fluid, 2014, 317
Mikhail A. Sokolovskiy, Jacques Verron, Atmospheric and Oceanographic Sciences Library, 47, Dynamics of Vortex Structures in a Stratified Rotating Fluid, 2014, 179

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