Skip to main content
Log in

Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians

  • Published:
Journal of Nonlinear Science Aims and scope Submit manuscript

Abstract

We study the motion of N point vortices with N∈ℕ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N -ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5n or 6n with n∈ℕ, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N-ring. Using a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N-ring is shown to be chaotic.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1989)

    Book  Google Scholar 

  • Arnold, V.I.: Ordinary Differential Equations. Springer, New York (1992)

    Google Scholar 

  • Boatto, S., Cabral, H.E.: Nonlinear stability of a latitudinal ring of point-vortices on a nonrotating sphere. SIAM J. Appl. Math. 64, 216–230 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Boatto, S., Lasker, J.: Point-vortex cluster formation in the plane and on the sphere: An energy bifurcation condition. Chaos 13, 824–835 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Cabral, H.E., Meyer, K.R., Schmidt, D.S.: Stability and bifurcation of the N+1 vortex problem on sphere. Regul. Chaotic Dyn. 8, 259–282 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Champneys, A.R., Lord, G.J.: Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem. Physica D 102, 232–269 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Doedel, E., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.: AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont). Available by anonymous ftp from ftp.cs.concordia.ca, directory pub/doedel/auto (1997)

  • Dormand, J.R., Prince, P.J.: Practical Runge-Kutta processes. SIAM J. Sci. Stat. Comput. 10, 977–989 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  • Dritschel, D.G.: The stability and energetics of co-rotating uniform vortices. J. Fluid Mech. 358, 95–134 (1985)

    Article  MATH  Google Scholar 

  • Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. World Scientific, New Jersey (2001)

    Book  MATH  Google Scholar 

  • Grotta-Ragazzo, C.: Nonintegrability of some Hamiltonian systems, scattering and analytic continuation. Commun. Math. Phys. 166, 255–277 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    Book  MATH  Google Scholar 

  • Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  • Haller, G.: Chaos Near Resonance. Springer, New York (1999)

    Book  MATH  Google Scholar 

  • Hénon, M., Heiles, C.: The applicability of the third integral of motion: Some numerical experiments. Astron. J. 69, 73–79 (1964)

    Article  MathSciNet  Google Scholar 

  • Holmes, P.J., Marsden, J.E.: Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys. 23, 669–675 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  • Kidambi, R., Newton, P.K.: Motion of three point vortices on a sphere. Phys. D 116, 95–134 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Kidambi, R., Newton, P.K.: Collapse of three vortices on a sphere. Nouvo Cimento C 22, 779–791 (1999)

    Google Scholar 

  • Kurakin, L.G.: On nonlinear stability of the regular vortex systems on a sphere. Chaos 14, 592–602 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Lerman, L.M.: Hamiltonian systems with loops of a separatrix of a saddle-center. Sel. Math. Sov. 10, 297–306 (1991)

    MathSciNet  MATH  Google Scholar 

  • Lim, C.C.: Singular manifolds and quasi-periodic solutions of Hamiltonians for vortex lattices. Physica D 30, 343–362 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  • Lim, C.C.: On singular Hamiltonians: the existence of quasi-periodic solutions and nonlinear instability. Bull. Am. Math. Soc. 20(1), 35–40 (1989)

    Article  MATH  Google Scholar 

  • Lim, C.C.: Existence of KAM tori in the phase-space of lattice vortex systems. J. Appl. Math. Phys. (ZAMP) 41, 227–244 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • Lim, C.C.: A combinatorial perturbation method and Arnold whiskered tori in vortex dynamics. Physica D 64, 163–184 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Lim, C.C., Nebus, J.: Vorticity, Statistical Mechanics, and Monte Carlo Simulation. Springer Monographs in Mathematics. Springer, New York (2006)

    MATH  Google Scholar 

  • Lim, C.C., Montaldi, J., Roberts, M.: Relative equilibria of point vortices on the sphere. Physica D 148, 97–135 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Meyer, K.R., Hall, G.R.: Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Springer, New York (1992)

    Book  MATH  Google Scholar 

  • Mielke, A., Holmes, P.J., O’Reilly, O.: Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center. J. Dyn. Differ. Equ. 4, 95–126 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Newton, P.K.: The N-Vortex Problem, Analytical Techniques. Springer, New York (2001)

    Book  MATH  Google Scholar 

  • Newton, P.K.: The N-vortex problem on a rotating sphere: III. Ring configurations coupled to a background field. Proc. Roy. Soc. Lond. Ser. A 463, 961–977 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Nusse, H.E., Yorke, J.A.: Dynamics: Numerical Explorations, 2nd edn. Springer, New York (1998)

    Book  MATH  Google Scholar 

  • Polvani, L.M., Dritschel, D.G.: Wave and vortex dynamics on the surface of a sphere. J. Fluid Mech. 255, 35–64 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Sakajo, T.: The motion of three point vortices on a sphere. Jpn. J. Indust. Appl. Math. 16, 321–347 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  • Sakajo, T.: Motion of a vortex sheet on a sphere with pole vortices. Phys. Fluids 16, 717–727 (2004a)

    Article  MathSciNet  MATH  Google Scholar 

  • Sakajo, T.: Transition of global dynamics of a polygonal vortex ring on a sphere with pole vortices. Physica D 196, 243–264 (2004b)

    Article  MathSciNet  MATH  Google Scholar 

  • Sakajo, T.: High-dimensional heteroclinic and homoclinic connections in odd point-vortex ring on a sphere. Nonlinearity 19, 75–93 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Sakajo, T.: Invariant dynamical systems embedded in the N-vortex problem on a sphere with pole vortices. Physica D 217, 142–152 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Sakajo, T.: Erratum: Invariant dynamical systems embedded in the N-vortex problem on a sphere with pole vortices. Physica D 225, 235–236 (2007a)

    Article  MathSciNet  Google Scholar 

  • Sakajo, T.: Integrable four-vortex motion on sphere with zero moment of vorticity. Phys. Fluids 19(1), 017019 (2007b)

    Article  MathSciNet  MATH  Google Scholar 

  • Sakajo, T., Yagasaki, K.: Chaotic motion of the N-vortex problem on a sphere: II. Saddle centers in three-degree-of-freedom Hamiltonians. Physica D (2008) DOI: 10.1016/j.physd.2008.02.001

  • Wiggins, S.: Global Bifurcations and Chaos. Springer, New York (1988)

    Book  MATH  Google Scholar 

  • Wiggins, S.: Global Dynamics, Phase Space Transport, Orbits Homoclinic to Resonances, and Applications. American Mathematical Society, Providence (1993)

    Book  MATH  Google Scholar 

  • Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  • Yagasaki, K.: Horseshoes in two-degree-of-freedom Hamiltonian systems with saddle-centers. Arch. Rat. Mech. Anal. 154, 275–296 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  • Yagasaki, K.: Homoclinic and heteroclinic orbits to invariant tori in multi-degree-of-freedom Hamiltonian systems with saddle-centers. Nonlinearity 18, 1331–1350 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Yagasaki, K.: Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers (2008a, submitted for publication)

  • Yagasaki, K.: Numerical analysis for global bifurcations of periodic orbits in autonomous differential equations (2008b, in preparation)

  • Yagasaki, K.: Numerical computation of homoclinic and heteroclinic orbits to nonhyperbolic trajectories (2008c, in preparation)

  • Yagasaki, K.: Numerical computation of stable and unstable manifolds of normally hyperbolic invariant manifolds (2008d, in preparation)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Takashi Sakajo.

Additional information

Communicated by P.K. Newton.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sakajo, T., Yagasaki, K. Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians. J Nonlinear Sci 18, 485–525 (2008). https://doi.org/10.1007/s00332-008-9019-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00332-008-9019-9

Keywords

PACS

Navigation