Loading [MathJax]/jax/output/SVG/config.js
Regular and Chaotic Dynamics
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Regul. Chaotic Dyn.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Regular and Chaotic Dynamics, 2007, Volume 12, Issue 2, Pages 117–126
DOI: https://doi.org/10.1134/S1560354707020013
(Mi rcd616)
 

This article is cited in 18 scientific papers (total in 18 papers)

Relative Equilibrium and Collapse Configurations of Four Point Vortices

K. A. O'Neil

Department of Mathematics, The University of Tulsa, 600 S. College Ave., Tulsa, Oklahoma 74104, USA
Citations (18)
Abstract: Relative equilibrium configurations of point vortices in the plane can be related to a system of polynomial equations in the vortex positions and circulations. For systems of four vortices the solution set to this system is proved to be finite, so long as a number of polynomial expressions in the vortex circulations are nonzero, and the number of relative equilibrium configurations is thereby shown to have an upper bound of 56. A sharper upper bound is found for the special case of vanishing total circulation. The polynomial system is simple enough to allow the complete set of relative equilibrium configurations to be found numerically when the circulations are chosen appropriately. Collapse configurations of four vortices are also considered; while finiteness is not proved, the approach provides an effective computational method that yields all configurations with a given ratio of velocity to position.
Keywords: point vortices, relative equilibrium.
Received: 19.01.2007
Accepted: 18.02.2007
Bibliographic databases:
Document Type: Article
MSC: 76B47, 70F10, 70H12
Language: English
Citation: K. A. O'Neil, “Relative Equilibrium and Collapse Configurations of Four Point Vortices”, Regul. Chaotic Dyn., 12:2 (2007), 117–126
Citation in format AMSBIB
\Bibitem{One07}
\by K.~A.~O'Neil
\paper Relative Equilibrium and Collapse Configurations of Four Point Vortices
\jour Regul. Chaotic Dyn.
\yr 2007
\vol 12
\issue 2
\pages 117--126
\mathnet{http://mi.mathnet.ru/rcd616}
\crossref{https://doi.org/10.1134/S1560354707020013}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2350301}
\zmath{https://zbmath.org/?q=an:1229.76024}
Linking options:
  • https://www.mathnet.ru/eng/rcd616
  • https://www.mathnet.ru/eng/rcd/v12/i2/p117
  • This publication is cited in the following 18 articles:
    1. Xiang Yu, “Finiteness of stationary configurations of the planar four-vortex problem”, Advances in Mathematics, 435 (2023), 109378  crossref
    2. Sreethin Sreedharan Kallyadan, Priyanka Shukla, “Self-similar vortex configurations: Collapse, expansion, and rigid-vortex motion”, Phys. Rev. Fluids, 7:11 (2022)  crossref
    3. Gotoda T., “Self-Similar Motions and Related Relative Equilibria in the N-Point Vortex System”, J. Dyn. Differ. Equ., 33:4 (2021), 1759–1777  crossref  isi  scopus
    4. Kudryashov N.A., “Generalized Hermite Polynomials For the Burgers Hierarchy and Point Vortices”, Chaos Solitons Fractals, 151 (2021), 111256  crossref  mathscinet  isi  scopus
    5. Wang Q., “Relative Periodic Solutions of the N-Vortex Problem Via the Variational Method”, Arch. Ration. Mech. Anal., 231:3 (2019), 1401–1425  crossref  mathscinet  zmath  isi  scopus
    6. Vikas S. Krishnamurthy, Mark A. Stremler, “Finite-time Collapse of Three Point Vortices in the Plane”, Regul. Chaotic Dyn., 23:5 (2018), 530–550  mathnet  crossref
    7. Dariya V. Safonova, Maria V. Demina, Nikolai A. Kudryashov, “Stationary Configurations of Point Vortices on a Cylinder”, Regul. Chaotic Dyn., 23:5 (2018), 569–579  mathnet  crossref
    8. M. V. Demina, N. A. Kudryashov, “Rotation, collapse, and scattering of point vortices”, Theor. Comput. Fluid Dyn., 28:3 (2014), 357  crossref
    9. Maria V. Demina, Nikolai A. Kudryashov, “Point Vortices and Classical Orthogonal Polynomials”, Regul. Chaotic Dyn., 17:5 (2012), 371–384  mathnet  crossref
    10. H. Aref, P. Beelen, M. Brøns, “Bilinear Relative Equilibria of Identical Point Vortices”, J Nonlinear Sci, 22:5 (2012), 849  crossref
    11. Maria V Demina, Nikolai A Kudryashov, “Vortices and polynomials: non-uniqueness of the Adler–Moser polynomials for the Tkachenko equation”, J. Phys. A: Math. Theor., 45:19 (2012), 195205  crossref
    12. Maria V. Demina, Nikolai A. Kudryashov, “Point Vortices and Polynomials of the Sawada–Kotera and Kaup–Kupershmidt Equations”, Regul. Chaotic Dyn., 16:6 (2011), 562–576  mathnet  crossref
    13. Kevin A. O'Neil, “Collapse and concentration of vortex sheets in two-dimensional flow”, Theor. Comput. Fluid Dyn., 24:1-4 (2010), 39  crossref
    14. Hassan Aref, “Self-similar motion of three point vortices”, Physics of Fluids, 22:5 (2010)  crossref
    15. Kevin A. O'Neil, Iutam Bookseries, 20, 150 Years of Vortex Dynamics, 2009, 55  crossref
    16. Alain Albouy, Yanning Fu, Shanzhong Sun, “Symmetry of planar four-body convex central configurations”, Proc. R. Soc. A., 464:2093 (2008), 1355  crossref
    17. Kevin A O'Neil, “Relative equilibria of point vortices that lie on a great circle of a sphere”, Nonlinearity, 21:9 (2008), 2043  crossref
    18. Kevin A. O'Neil, “Relative equilibrium and collapse configurations of heterogeneous vortex triple rings”, Physica D: Nonlinear Phenomena, 236:2 (2007), 123  crossref
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Statistics & downloads:
    Abstract page:73
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2025