Regular and Chaotic Dynamics

, Volume 22, Issue 8, pp 893–908 | Cite as

Dynamics of Two Point Vortices in an External Compressible Shear Flow

  • Evgeny V. VetchaninEmail author
  • Ivan S. Mamaev


This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincaré map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitch-fork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.


point vortices shear flow perturbation by an acoustic wave bifurcations reversible pitch-fork period doubling 

MSC2010 numbers



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Basic and Applied Problems of the Theory of Vortices, A. V.Borisov, I. S.Mamaev, M.A. Sokolovskiy (Eds.), Izhevsk: R&C Dynamics, Institute of Computer Science, 2003 (Russian).Google Scholar
  2. 2.
    Bogomolov, V.A., Interaction of Vortices in Plane-Parallel Flow, Izv. Akad. Nauk SSSR. Fiz. Atmos. Okeana, 1981, vol. 17, no. 2, pp. 199–201 (Russian).Google Scholar
  3. 3.
    Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557–571.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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. Math., 2005, vol. 71, no. 1, pp. 139–144; see also: Dokl. Ross. Akad. Nauk, 2005, vol. 400, no. 4, pp. 457–462.zbMATHGoogle Scholar
  5. 5.
    Borisov, A. V., Mamaev, I. S., and Kilin, A.A., Transition to Chaos in Dynamics of Four Point Vortices on a Plane, Dokl. Phys., 2006, vol. 51, no. 5, pp. 262–267; see also: Dokl. Akad. Nauk, 2006, vol. 408, no. 1, pp. 49–54.CrossRefzbMATHGoogle Scholar
  6. 6.
    Borisov, A. V. and Mamaev, I. S., The Dynamics of a Chaplygin Sleigh, J. Appl. Math. Mech., 2009, vol. 73, no. 2, pp. 156–161; see also: Prikl. Mat. Mekh., 2009, vol. 73, no. 2, pp. 219–225.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borisov, A.V. and Mamaev, I. S., Mathematical Methods in the Dynamics of Vortex Structures, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).zbMATHGoogle Scholar
  8. 8.
    Borisov, A.V., Kilin, A.A., and Mamaev, I. S., The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem, Regul. Chaotic Dyn., 2013, vol. 18, nos. 1–2, pp. 33–62.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Borisov, A. V., Kilin, A.A., Mamaev, I. S., and Tenenev, V.A., The Dynamics of Vortex Rings: Leapfrogging in an Ideal and Viscous Fluid, Fluid Dyn. Res., 2014, vol. 46, no. 3, 031415, 16 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Borisov, A. V. and Mamaev, I. S., An Integrability of the Problem on Motion of Cylinder and Vortex in the Ideal Fluid, Regul. Chaotic Dyn., 2003, vol. 8, no. 2, pp. 163–166.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Borisov, A. V., Mamaev, I. S., and Ramodanov, S. M., Motion of a Circular Cylinder and n Point Vortices in a Perfect Fluid, Regul. Chaotic Dyn., 2003, vol. 8, no. 4, pp. 449–462.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Dynamics of Nonholonomic Systems Consisting of a Spherical Shell with a Moving Rigid Body Inside, Regul. Chaotic Dyn., 2014, vol. 19, no. 2, pp. 198–213.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Dynamics of Three Vortex Sources, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 694–701.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Bizyaev, I.A., Borisov, A.V., and Mamaev, I. S., The Dynamics of Vortex Sources in a Deformation Flow, Regul. Chaotic Dyn., 2016, vol. 21, no. 3, pp. 367–376.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gallas, J.A.C., Structure of the Parameter Space of the Hénon Map, Phys. Rev. Lett., 1993, vol. 70, no. 18, pp. 2714–2717.CrossRefGoogle Scholar
  16. 16.
    Gonchar, V.Yu., Ostapchuk, P. N., Tur, A.V., and Yanovsky, V. V., Dynamics and Stochasticity in a Reversible System Describing Interaction of Point Vortices with a Potential Wave, Phys. Lett. A, 1991, vol. 152, nos. 5–6, pp. 287–292.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hairer, E., Nørsett, S.P., and Wanner, G., Solving Ordinary Differential Equations: Vol. 1. Nonstiff Problems, 2nd ed., Springer Ser. Comput. Math., vol. 8, New York: Springer, 1993.Google Scholar
  18. 18.
    von Helmholtz, H., Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. Reine Angew. Math., 1858, vol. 55, pp. 25–55; see also: Helmholtz, H., On Integrals of the Hydrodynamical Equations, which Express Vortex-Motion, Philos. Mag., 1867, vol. 33(226), no. 4, pp. 485–512.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Isakovich, M.A., General Acoustics, Moscow: Nauka, 1973 (Russian).Google Scholar
  20. 20.
    Kidambi, R. and Newton, P.K., Point Vortex Motion on a Sphere with Solid Boundaries, Phys. Fluids, 2000, vol. 12, no. 3, pp. 581–588.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kimura, Y., Motion of Two Point Vortices in a Circular Domain, J. Phys. Soc. Japan, 1988, vol. 57, no. 5, pp. 1641–1649.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kirchhoff, G., Vorlesungen áber mathematische Physik: Vol. 1. Mechanik, Leipzig: Teubner, 1876.zbMATHGoogle Scholar
  23. 23.
    Lamb, J. S.W. and Roberts, J.A.G., Time-Reversal Symmetry in Dynamical Systems: A Survey, Phys. D, 1998, vol. 112, nos. 1–2, pp. 1–39.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Middelkamp, S., Torres, P. J., Kevrekidis, P.G., Frantzeskakis, D. J., Carretero-González, R., Schmelcher, P., Freilich, D. V., and Hall, D. S., Guiding-Center Dynamics of Vortex Dipoles in Bose–Einstein Condensates, Phys. Rev. A, 2011, vol. 84, no. 1, 011605, 4 pp.CrossRefGoogle Scholar
  25. 25.
    Murray, A.V., Groszek, A. J., Kuopanportti, P., and Simula, T., Hamiltonian Dynamics of Two Same- Sign Point Vortices, Phys. Rev. A, 2016, vol. 93, no. 3, 033649, 8 pp.CrossRefGoogle Scholar
  26. 26.
    Politi, A., Oppo, G. L., and Badii, R., Coexistence of Conservative and Dissipative Behavior in Reversible Dynamical Systems, Phys. Rev. A, 1986, vol. 33, no. 6, pp. 4055–4060.CrossRefGoogle Scholar
  27. 27.
    Tronin, K.G., Absolute Choreographies of Point Vortices on a Sphere, Regul. Chaotic Dyn., 2006, vol. 11, no. 1, pp. 123–130.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Vetchanin, E.V. and Kazakov, A.O., Bifurcations and Chaos in the Dynamics of Two Point Vortices in an Acoustic Wave, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2016, vol. 26, no. 4, 1650063, 13 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Yates, J. E., Interaction with and Production of Sound by Vortex Flows, in Proc. of the 4th Aeroacoustics Conference (Atlanta,Ga., 1977), 82 pp.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Izhevsk State Technical UniversityIzhevskRussia

Personalised recommendations