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Choreographies in the n-vortex Problem

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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.

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References

  1. Aref, H., On the Equilibrium and Stability of a Row of Point Vortices, J. Fluid Mech., 1995, vol. 290, pp. 167–181.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. 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.

    Article  MathSciNet  MATH  Google Scholar 

  13. Chenciner, A. and Féjoz, J., Unchained Polygons and the N-Body Problem, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 64–115.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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. Doedel, E., AUTO: A Program for the Automatic Bifurcation Analysis of Autonomous Systems, Congr. Numer., 1981, vol. 30, pp. 265–284.

    MathSciNet  MATH  Google Scholar 

  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.

    Book  MATH  Google Scholar 

  19. García-Azpeitia, C., Relative Periodic Solutions of the n-Vortex Problem on the Sphere, arXiv:1805.10417 (2018).

    Google Scholar 

  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.

    Article  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MATH  Google Scholar 

  23. Ize, J. and Vignoli, A., Equivariant Degree Theory, de Gruyter Ser. Nonlinear Anal. Appl., vol. 8, Berlin: de Gruyter, 2003.

  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.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  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.

  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.

    Chapter  Google Scholar 

  28. Moore, Ch., Braids in Classical Gravity, Phys. Rev. Lett., 1993, vol. 70, no. 24, pp. 3675–3679.

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Article  MathSciNet  MATH  Google Scholar 

  30. Newton, P.K., The N-Vortex Problem: Analytical Techniques, Appl. Math. Sci., vol. 145, New York: Springer, 2001.

  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.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. IIMAS, Universidad Nacional Autónoma de México, Apdo. Postal 20–726, C.P., 01000, México, D.F., México

    Renato C. Calleja

  2. Concordia University, 1455 Boulevard De Maisonneuve West, Montreal, Quebec, H3G 1M8, Canada

    Eusebius J. Doedel

  3. Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior S/N, C.P. 04510, Ciudad Universitaria, CDMX, Ciudad, México

    Carlos García-Azpeitia

Authors
  1. Renato C. Calleja
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  2. Eusebius J. Doedel
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  3. Carlos García-Azpeitia
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Correspondence to Renato C. Calleja.

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Calleja, R.C., Doedel, E.J. & García-Azpeitia, C. Choreographies in the n-vortex Problem. Regul. Chaot. Dyn. 23, 595–612 (2018). https://doi.org/10.1134/S156035471805009X

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  • DOI: https://doi.org/10.1134/S156035471805009X

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