Regular and Chaotic Dynamics

, Volume 18, Issue 1–2, pp 33–62 | Cite as

The dynamics of vortex rings: Leapfrogging, choreographies and the stability problem

  • Alexey V. BorisovEmail author
  • Alexander A. Kilin
  • Ivan S. Mamaev


We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a method for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings.


ideal fluid vortex ring leapfrogging motion of vortex rings bifurcation complex, periodic solution integrability chaotic dynamics 

MSC2010 numbers



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alekseev, V.M., The Generalized Spatial Problem of Two Fixed Centers: The Classification of Motions, Bull. Inst. Teoret. Astr., 1965, vol. 10, no. 4(117), pp. 241–271 (Russian).Google Scholar
  2. 2.
    Akhmetov, D.G., Vortex Rings, Berlin: Springer, 2009.CrossRefGoogle Scholar
  3. 3.
    Bagrets, A. A. and Bagrets, D. A., Nonintegrability of Hamiltonian Systems in Vortex Dynamics, Regul. Chaotic Dyn., 1997, vol. 2, no. 1, pp. 36–43; no. 2, pp. 58–65 (Russian).zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bagrets, A. A. and Bagrets, D. A., Nonintegrability of Two Problems in Vortex Dynamics, Chaos, 1997, vol. 7, no 3, pp. 368–375.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Batchelor, G. K., An Introduction to Fluid Dynamics, 2nd ed., Cambridge: Cambridge Univ. Press, 1999.zbMATHGoogle Scholar
  6. 6.
    Blackmore, D., Brons, M., and Goullet, A., A Coaxial Vortex Ring Model for Vortex Breakdown, Phys. D, 2008, vol. 237, pp. 2817–2844.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Blackmore, D., Champanerkar, J., and Wang, Ch., A Generalized Poincaré-Birkhoff Theorem with Applications to Coaxial Vortex Ring Motion, Discrete Contin. Dyn. Syst. Ser. B, 2005, vol. 5, no. 1, pp. 15–33.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Blackmore, D. and Knio, O., Transition from Quasiperiodicity to Chaos for Three Coaxial Vortex Rings, ZAMM Z. Angew. Math. Mech., 2000, vol. 80, pp. 173–176.CrossRefGoogle Scholar
  9. 9.
    Blackmore, D. and Knio, O., KAM Theory Analysis of the Dynamics of Three Coaxial Vortex Rings, Phys. D, 2000, vol. 140, pp. 321–348.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Topology and Stability of Integrable Systems, Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71–132 [Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259–318].CrossRefMathSciNetGoogle Scholar
  11. 11.
    Bolsinov, A.V. and Fomenko, A.T., Integrable Hamiltonian Systems: Geometry, Topology and Classification, Boca Raton, FL: CRC Press, 2004.zbMATHGoogle Scholar
  12. 12.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Absolute and Relative Choreographies in the Problem of the Motion of Point Vortices in a Plane, Dokl. Ross. Akad. Nauk, 2005, vol. 400, no. 4, pp. 457–462 [Dokl. Math., 2005, vol. 71, no. 1, pp. 139–144].MathSciNetGoogle Scholar
  13. 13.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Reduction and Chaotic Behavior of Point Vortices on a Plane and a Sphere, Discrete Contin. Dyn. Syst., 2005, suppl., pp. 100–109.Google Scholar
  14. 14.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).zbMATHGoogle Scholar
  15. 15.
    Borisov, A.V. and Mamaev, I. S., Isomorphisms of Geodesic Flows on Quadrics, Regul. Chaotic Dyn., 2009, vol. 14, nos. 4–5, pp. 455–465.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Boyarintsev, V. I., Levchenko, E. S., and Savin, A. S., Motion of Two Vortex Rings, Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza, 1985, no. 5, pp. 176–177 [Fluid Dynam., 1985, vol. 20, no. 5, pp. 818–819].Google Scholar
  17. 17.
    Brutyan, M.A. and Krapivskii, P. L., The Motion of a System of Vortex Rings in an Incompressible Fluid, Prikl. Mat. Mekh., 1984, vol. 48, no. 3, pp. 503–506 [J. Appl. Math. Mech., 1984, vol. 48, no. 3, pp. 365–368].MathSciNetGoogle Scholar
  18. 18.
    Chaplygin S.A. Comments on Helmholtz’s Life and Works, in: H. Helmholtz, Dva issledovaniya po gidrodinamike (Two Studies in Hydrodynamics). Moscow, Palas, 1902, pp. 69–108 (Russian); reprinted edition: Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2002, pp. 57–73.Google Scholar
  19. 19.
    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, pp. 881–901.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Cheremnykh, O.K., On the Motion of Vortex Rings in an Incompressible Media, Rus. J. Nonlin. Dyn., 2008, vol. 4, no. 4, pp. 417–428 (Russian).Google Scholar
  21. 21.
    Dyson, F. W., The Potential of an Anchor Ring, Philos. Trans. Roy. Soc. London Ser. A, 1893, vol. 184, pp. 43–95.zbMATHCrossRefGoogle Scholar
  22. 22.
    Dyson, F. W., The Potential of an Anchor Ring: P. 2, Philos. Trans. Roy. Soc. London Ser. A, 1893, vol. 184, pp. 1041–1106.CrossRefGoogle Scholar
  23. 23.
    Fraenkel, L.E., On Steady Vortex Rings of Small Cross-Section in an Ideal Fluid, Proc. R. Soc. London Ser. A, 1970, vol. 316, pp. 29–62.zbMATHCrossRefGoogle Scholar
  24. 24.
    Fraenkel, L.E., Examples of Steady Vortex Rings of Small Cross-Section in an Ideal Fluid, J. Fluid Mech., 1972, vol. 51, pp. 119–135.zbMATHCrossRefGoogle Scholar
  25. 25.
    Fraenkel, L.E. and Berger, M. S., A Global Theory of Steady Vortex Rings in an Ideal Fluid, Acta Math., 1974, vol. 132, pp. 13–51.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Grinchenko, V.T., Meleshko, V.V., Gourzhii, A. A., van Heijst, G. J.F., and Eisenga, A. H. M., Two Approaches to the Analysis of the Coaxial Interaction of Vortex Rings, Appl. Hydromech., 2000, vol. 2, no. 3, pp. 40–52 (Russian).Google Scholar
  27. 27.
    Gröbli, W., Spezielle Probleme über die Bewegung geradliniger paralleler Wirbelfäden, Vierteljschr. Naturf. Ges. Zürich, 1877, vol. 22, pp. 37–82, 129–168.Google Scholar
  28. 28.
    Gurzhii, A.A., Konstantinov, M.Yu., and Meleshko, V.V., Interaction of Coaxial Vortex Rings in an Ideal Fluid, Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza, 1988, no. 2, pp. 78–84 [Fluid Dynam., 1988, vol. 23, no. 2, pp. 224–229].Google Scholar
  29. 29.
    Helmholtz, H., Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. Reine Angew. Math., 1858, vol. 55, pp. 25–55zbMATHCrossRefGoogle Scholar
  30. 30.
    Hicks, W.M., On the Steady Motion of a Hollow Vortex, Proc. R. Soc. London, 1883, vol. 35, pp. 304–308.zbMATHCrossRefGoogle Scholar
  31. 31.
    Hicks, W. M., Researches on the Theory of Vortex Rings: P. 2, Philos. Trans. Roy. Soc. London Ser. A, 1885, vol. 176, pp. 725–780.CrossRefGoogle Scholar
  32. 32.
    Hicks, W.M., On the Mutual Threading of Vortex Rings, Proc. R. Soc. London Ser. A, 1922, vol. 102, pp. 111–113.CrossRefGoogle Scholar
  33. 33.
    Hicks, W.M., Researches in Vortex Motion: P. 3. On Spiral or Gyrostatic Vortex Aggregates, Philos. Trans. Roy. Soc. London Ser. A, 1899, vol. 192, pp. 33–99.zbMATHCrossRefGoogle Scholar
  34. 34.
    Hill, M. J. M., On a Spherical Vortex, Philos. Trans. Roy. Soc. London Ser. A, 1894, vol. 185, pp. 213–245.zbMATHCrossRefGoogle Scholar
  35. 35.
    Konstantinov, M., Chaotic Phenomena in the Interaction of Vortex Rings, Phys. Fluids, 1994, vol. 6, no. 5, pp. 1752–1767.zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Lamb, H., Hydrodynamics, 6th ed. Cambridge: Cambridge Univ. Press, 1932.zbMATHGoogle Scholar
  37. 37.
    Levy, H. and Forsdyke, A.G., The Stability of an Infinite System of Circular Vortices, Proc. R. Soc. London Ser. A, 1927, vol. 114, pp. 594–604.zbMATHCrossRefGoogle Scholar
  38. 38.
    Levy, H. and Forsdyke, A.G., The Vibrations of an Infinite System of Vortex Rings, Proc. R. Soc. London Ser. A, 1927, vol. 116, pp. 352–379.zbMATHCrossRefGoogle Scholar
  39. 39.
    Llewellyn Smith, S.G. and Hattori, Y., Axisymmetric Magnetic Vortices with Swirl, Commun. Nonlinear Sci. Numer. Simul., 2012, vol. 17, no. 5, pp. 2101–2107.zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Maxwell, J. C., Letter to William Thomson, 6 October 1868, in The Scientific Letters and Papers of James Clerk Maxwell: Vol. 2, P.M. Harman (Ed.), Cambridge: Cambridge Univ. Press, 1990, pp. 446–448.Google Scholar
  41. 41.
    Maxwell, J.C., A Treatise on Electricity and Magnetism: In 2 Vols, Oxford: Clarendon, 1873.Google Scholar
  42. 42.
    Maxworthy, T., Some Experimental Studies of Vortex Rings, J. Fluid Mech., 1977, vol. 81, pp. 465–495.CrossRefGoogle Scholar
  43. 43.
    Meleshko, V.V., Coaxial Axisymmetric Vortex Rings: 150 Years after Helmholtz, Theor. Comput. Fluid Dynam., 2010, vol. 24, pp. 403–431.zbMATHCrossRefGoogle Scholar
  44. 44.
    Moore, C., Braids in Classical Gravity, Phys. Rev. Lett., 1993, vol. 70, pp. 3675–3679.zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Moore, D. W. and Saffman, P. G., A Note on the Stability of a Vortex Ring of Small Cross-Section, Proc. R. Soc. London Ser. A, 1974, vol. 338, no. 1615, pp. 535–537.CrossRefGoogle Scholar
  46. 46.
    Novikov, E. A., Generalized Dynamics of Three-Dimensional Vortical Singularities (Vortons), Zh. Eksp. Teor. Fiz., 1983, vol. 84, no. 3, pp. 975–981 [J. Exp. Theor. Phys., 1983, vol. 57, no. 3, pp. 566–569].Google Scholar
  47. 47.
    Novikov, E.A., Hamiltonian Description of Axisymmetric Vortex Flows and the System of Vortex Rings, Phys. Fluids, 1985, vol. 28, no. 9, pp. 2921–2922.zbMATHCrossRefGoogle Scholar
  48. 48.
    Pocklington, H.C., The Complete System of the Periods of a Hollow Vortex Ring, Philos. Trans. Roy. Soc. London Ser. A, 1895, vol. 186, pp. 603–619.zbMATHCrossRefGoogle Scholar
  49. 49.
    Roberts, P.H. and Donnelly, R. J., Dynamics of Vortex Rings, Phys. Lett. A, 1970, vol. 31, pp. 137–138.CrossRefGoogle Scholar
  50. 50.
    Saffman, P.G., The Velocity of Viscous Vortex Rings, Stud. Appl. Math., 1970, vol. 49, pp. 371–380.zbMATHGoogle Scholar
  51. 51.
    Saffman, P.G., Vortex Dynamics, Cambridge: Cambridge Univ. Press, 1992.zbMATHGoogle Scholar
  52. 52.
    Shariff, K., Vortex Rings, Annu. Rev. Fluid Mech., 1992, vol. 24, pp. 235–279.CrossRefMathSciNetGoogle Scholar
  53. 53.
    Sharlier, C. L., Die Mechanik des Himmels, Berlin: Walter de Gryter, 1927.Google Scholar
  54. 54.
    Shashikanth, B.N., Marsden, J. E., Leapfrogging Vortex Rings: Hamiltonian Structure, Geometric Phases and Discrete Reduction, Fluid Dyn. Research, 2003, vol. 33, pp. 333–356.zbMATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Shashikanth, B.N., Symmetric Pairs of Point Vortices Interacting with a Neutrally Buoyant 2-dimensional Circular Cylinder, Phys. Fluids 2006, vol. 18, 127103 (17 p.).Google Scholar
  56. 56.
    Shashikanth, B.N., Sheshmani, A., Kelly, S.D., Marsden, J.E. Hamiltonian Structure for a Neutrally Buoyant Rigid Body Interacting with N Vortex Rings of Arbitrary Shape: The Case of Arbitrary Smooth Body Shape, Theor. Comput. Fluid Dyn., 2008, vol. 22, no. 1, pp. 37–64zbMATHCrossRefGoogle Scholar
  57. 57.
    Simó, C., Dynamical Properties of the Figure Eight Solution of the Three-Body Problem, in Celestial Mechanics (Evanston, IL, 1999), Contemp. Math., vol. 292, Providence, RI: AMS, 2002, pp. 209–228.CrossRefGoogle Scholar
  58. 58.
    Simó, C. and Stuchi, T. J., Central Stable/Unstable Manifolds and the Destruction of KAM Tori in the Planar Hill Problem, Phys. D, 2000, vol. 140, nos. 1–2, pp. 1–32.zbMATHCrossRefMathSciNetGoogle Scholar
  59. 59.
    Thomson, J. J., On the Vibrations of a Vortex Ring, and the Action upon Each Other of Two Vortices in a Perfect Fluid, Philos. Trans. Roy. Soc. London, 1882, vol. 173, pp. 493–521.zbMATHCrossRefGoogle Scholar
  60. 60.
    Thomson, W., On Vortex Atoms, Philos. Mag. Ser. 4, 1867, vol. 34, pp. 15–24.Google Scholar
  61. 61.
    Vasilev, N. S., On the Motion of an Infinite Row of Coaxial Circular Vortex Rings with the Same Initial Radii, Zap. Fiz.-Mat. Fak. Imp. Novoross. Univ., 1914, vol. 10, pp. 1–44 (Russian).Google Scholar
  62. 62.
    Vasilev, N. S., Reduction of the Equations of Motion of Coaxial Vortex Rings to Canonical Form, Zap. Fiz.-Mat. Fak. Imp. Novoross. Univ., 1913, vol. 21, pp. 1–12 (Russian).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
    • 3
    Email author
  • Alexander A. Kilin
    • 1
    • 2
    • 3
  • Ivan S. Mamaev
    • 1
    • 2
    • 3
  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia
  2. 2.A.A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  3. 3.Institute of Mathematics and Mechanics of the Ural Branch of RASEkaterinburgRussia

Personalised recommendations