Advertisement

Regular and Chaotic Dynamics

, Volume 23, Issue 5, pp 595–612 | Cite as

Choreographies in the n-vortex Problem

  • Renato C. Calleja
  • Eusebius J. Doedel
  • Carlos García-Azpeitia
Article
  • 23 Downloads

Abstract

We consider the equations of motion of n vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, corresponds to choreographies of n vortices. We include numerical results for all cases, for various values of n, and we provide key details on the computational approach.

Keywords

n-vortex problem choreographies continuation methods 

MSC2010 numbers

34C25 37G40 47H11 54F45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aref, H., On the Equilibrium and Stability of a Row of Point Vortices, J. Fluid Mech., 1995, vol. 290, pp. 167–181.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., and Vainchtein, D., Vortex Crystals, in Advances in Applied Mechanics: Vol. 39, E. van derGiessen, H. Aref (Eds.), San Diego: Acad. Press, 2003, pp. 1–79.Google Scholar
  3. 3.
    Aref, H. and Pomphrey, N., Integrable and Chaotic Motions of Four Vortices: 1. The Case of Identical Vortices, Proc. Roy. Soc. London Ser. A, 1982, vol. 380, no. 1779, pp. 359–387.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bartsch, Th. and Dai, Q., Periodic Solutions of the N-Vortex Hamiltonian System in Planar Domains, J. Differential Equations, 2016, vol. 260, no. 3, pp. 2275–2295.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bartsch, Th. and Gebhard, B., Global Continua of Periodic Solutions of Singular First-Order Hamiltonian Systems of N-Vortex Type, Math. Ann., 2017, vol. 369, nos. 1–2, pp. 627–651.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Absolute and Relative Choreographies in the Problem of Point Vortices Moving on a Plane, Regul. Chaotic Dyn., 2004, vol. 9, no. 2, pp. 101–111.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., New Periodic Solutions for Three or Four Identical Vortices on a Plane and a Sphere, Discrete Contin. Dyn. Syst., 2005, suppl., 110–120.Google Scholar
  8. 8.
    Borisov, A. V. and Kilin, A.A., Stability of Thomson’s Configurations of Vortices on a Sphere, Regul. Chaotic Dyn., 2000, vol. 5, no. 2, pp. 189–200.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cabral, H.E. and Schmidt, D. S., Stability of Relative Equilibria in the Problem of N +1 Vortices, SIAM J. Math. Anal., 1999/2000, vol. 31, no. 2, pp. 231–250.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Calleja, R., Doedel, E., and García-Azpeitia, C., Symmetries and Choreographies in Families That Bifurcate from the Polygonal Relative Equilibrium of the N-Body Problem, Celest. Mech. Dyn. Astron., 2018, vol. 130, no. 7, Art. No. 48, 25 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Calleja, R., Doedel, E., García-Azpeitia, C., and Pando L., C. L., Choreographies in the Discrete Nonlinear Schrödinger Equations, Eur. Phys. J. Special Topics, 2018 (in press).Google Scholar
  12. 12.
    Carvalho, A.C. and Cabral, H.E., Lyapunov Orbits in the N-Vortex Problem, Regul. Chaotic Dyn., 2014, vol. 19, no. 3, pp. 348–362.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chenciner, A. and Féjoz, J., Unchained Polygons and the N-Body Problem, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 64–115.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chenciner, A. and Montgomery, R., A Remarkable Periodic Solution of the Three-Body Problem in the Case of Equal Masses, Ann. of Math. (2), 2000, vol. 152, no. 3, pp. 881–901.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Dai, Q., Gebhard, B., and Bartsch, Th., Periodic Solutions of N-Vortex Type Hamiltonian Systems near the Domain Boundary, SIAM J. Appl. Math., 2018, vol. 78, no. 2, pp. 977–995.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Doedel, E. J., Fairgrieve, Th. F., Sandstede, B., Champneys, A.R., Kuznetsov, Yu.A., and Wang, X., AUTO-07p: Continuation and Bifurcation Software for Ordinary Differential Equations, http://sourceforge.net/projects/auto-07p/files/auto07p/(2012).Google Scholar
  17. 17.
    Doedel, E., AUTO: A Program for the Automatic Bifurcation Analysis of Autonomous Systems, Congr. Numer., 1981, vol. 30, pp. 265–284.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Zermelo, E., Collected Works/Gesammelte Werke: Vol. 2. Calculus of Variations, Applied Mathematics, and Physics/Variationsrechnung, Angewandte Mathematik und Physik, Berlin: Springer, 2013, pp. 300–463.CrossRefzbMATHGoogle Scholar
  19. 19.
    García-Azpeitia, C., Relative Periodic Solutions of the n-Vortex Problem on the Sphere, arXiv:1805.10417 (2018).Google Scholar
  20. 20.
    García-Azpeitia, C. and Ize, J., Global Bifurcation of Polygonal Relative Equilibria for Masses, Vortices and dNLS Oscillators, J. Differential Equations, 2011, vol. 251, no. 11, pp. 3202–3227.CrossRefzbMATHGoogle Scholar
  21. 21.
    García-Azpeitia, C. and Ize, J., Bifurcation of Periodic Solutions from a Ring Configuration in the Vortex and Filament Problems, J. Differential Equations, 2012, vol. 252, no. 10, pp. 5662–5678.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Havelock, T.H., The Stability of Motion of Rectilinear Vortices in Ring Formation, Philos. Mag. (7), 1931, vol. 11, no. 70, pp. 617–633.CrossRefzbMATHGoogle Scholar
  23. 23.
    Ize, J. and Vignoli, A., Equivariant Degree Theory, de Gruyter Ser. Nonlinear Anal. Appl., vol. 8, Berlin: de Gruyter, 2003.Google Scholar
  24. 24.
    Lim, C., Montaldi, J., and Roberts, M., Relative Equilibria of Point Vortices on the Sphere, Phys. D, 2001, vol. 148, nos. 1–2, pp. 97–135.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lin, C. C., On the Motion of Vortices in Two Dimensions: 1. Existence of the Kirchhoff–Routh Function, Proc. Natl. Acad. Sci. USA, 1941, vol. 27, no. 12, pp. 570–575.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kurakin, L., Point Vortices in a Circular Domain: Stability, Resonances, and Instability of Stationary Rotation of a Regular Vortex Polygon, in 18ème Congrès Français de Mécanique (Grenoble, 2007), 6 pp.Google Scholar
  27. 27.
    Montaldi, J. and Tokieda, T., Deformation of Geometry and Bifurcations of Vortex Rings, in Recent Trends in Dynamical Systems, A. Johann, H.P. Kruse, F. Rupp, S. Schmitz (Eds.), Springer Proc. Math. Stat., vol. 35, Basel: Springer, 2013, pp. 335–370.CrossRefGoogle Scholar
  28. 28.
    Moore, Ch., Braids in Classical Gravity, Phys. Rev. Lett., 1993, vol. 70, no. 24, pp. 3675–3679.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Mu˜noz-Almaraz, F. J., Freire, E., Galán, J., Doedel, E., and Vanderbauwhede, A., Continuation of Periodic Orbits in Conservative and Hamiltonian Systems, Phys. D, 2003, vol. 181, nos. 1–2, pp. 1–38.MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Newton, P.K., The N-Vortex Problem: Analytical Techniques, Appl. Math. Sci., vol. 145, New York: Springer, 2001.Google Scholar
  31. 31.
    Simó, C., New Families of Solutions in N-Body Problems, in European Congress of Mathematics (Barcelona, 2000): Vol. 1, C. Casacuberta, R. M. Miró-Roig, J. Verdera, S. Xambó-Descamps (Eds.), Progr. Math., vol. 201, Basel: Birkhäuser, 2001, pp. 101–115.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Renato C. Calleja
    • 1
  • Eusebius J. Doedel
    • 2
  • Carlos García-Azpeitia
    • 3
  1. 1.IIMASUniversidad Nacional Autónoma de MéxicoMéxicoMéxico
  2. 2.Concordia UniversityMontreal, QuebecCanada
  3. 3.Facultad de CienciasUniversidad Nacional Autónoma de MéxicoCiudadMéxico

Personalised recommendations