Skip to main content

Existence of Finite-Time Hyperbolic Trajectories for Planar Hamiltonian Flows

Abstract

We present a shadowing theorem on the existence of hyperbolic trajectories on finite time intervals based on the EPH partition by George Haller and apply it to an example which is inspired by the problem of two-dimensional symmetric vortex merger.

This is a preview of subscription content, access via your institution.

References

  1. 1

    Berger A., Doan T.S., Siegmund S.: Nonautonomous finite-time dynamics. Discret. Contin. Dyn. Syst. 9, 463–492 (2008)

    MATH  Article  Google Scholar 

  2. 2

    Berger A., Doan T.S., Siegmund S.: A remark on finite-time hyperbolicity. Proc. Appl. Math. Mech. 8, 10917–10918 (2008)

    Article  Google Scholar 

  3. 3

    Berger A., Doan T.S., Siegmund S.: A definition of spectrum for differential equations on finite time. J. Differ. Equ. 246, 1098–1118 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4

    Branicki M., Wiggins S.: An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems. Physica D 238, 1625–1657 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5

    Chorin A., Marsden J.: A Mathematical Introduction to Fluid Mechanics, 3rd edn. Texts in Applied Mathematics. Springer, New York (1993)

    Google Scholar 

  6. 6

    Duc, L.H.: Asymptotic and finite-time dynamics for nonautonomous dynamical systems. Doctoral thesis, J.W. Goethe University Frankfurt (2006)

  7. 7

    Duc L.H., Siegmund S.: Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals. Int. J. Bifur. Chaos Appl. Sci. Eng. 3, 641–674 (2008)

    MathSciNet  Article  Google Scholar 

  8. 8

    Hale J.: Ordinary Differential Equations. Krieger, Malabur (1980)

    MATH  Google Scholar 

  9. 9

    Haller G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids 13, 3365–3385 (2001)

    MathSciNet  Article  Google Scholar 

  10. 10

    Haller G.: An objective definition of a vortex. J. Fluid Mech. 525, 1–26 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11

    Haller G., Poje A.C.: Finite-time transport in aperiodic flows. Physica D 119, 352–380 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12

    Ide K., Small D., Wiggins S.: Distinguished hyperbolic trajectories in time-dependent fluid flows: analytical and computational approach for velocity fields defined as data sets. Nonlinear Process. Geophys. 9, 237–263 (2002)

    Article  Google Scholar 

  13. 13

    Malhotra N., Wiggins S.: Geometric structures, lobe dynamics, and Lagrangian transport in flows with aperiodic time-dependence with applications to Rossby wave flow. J. Nonlinear Sci. 8, 401–456 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14

    Okubo A.: Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences. Deep-Sea Res. 17, 445–454 (1970)

    Google Scholar 

  15. 15

    Poje A.C., Haller G., Mezic I.: The geometry and statistics of mixing in aperiodic flows. Phys. Fluids 11, 2963–2968 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16

    Trieling R.R., Velasco Fuentes O.U., van Heijst G.J.F.: Interaction of two unequal corotating vortices. Phys. Fluids 17, 087103-1–087103-17 (2005)

    MathSciNet  Article  Google Scholar 

  17. 17

    Velasco Fuentes O.U.: Chaotic advection by two interacting finite-area vortices. Phys. Fluids 27, 901–912 (2001)

    Article  Google Scholar 

  18. 18

    Velasco Fuentes O.U.: Vortex filmentation: its onset and its role on axisymmetrization and merger. Dyn Atmos Oceans 40, 23–42 (2005)

    Article  Google Scholar 

  19. 19

    Weiss J.: The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273–294 (1991)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Luu Hoang Duc.

Additional information

This paper is dedicated to Russell Johnson on the occasion of his birthday.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Duc, L.H., Siegmund, S. Existence of Finite-Time Hyperbolic Trajectories for Planar Hamiltonian Flows. J Dyn Diff Equat 23, 475–494 (2011). https://doi.org/10.1007/s10884-011-9211-8

Download citation

Keywords

  • EPH partition
  • Finite time dynamics
  • Vortex merger

Mathematics Subject Classification (2000)

  • Primary 37B55
  • 37D05
  • Secondary 37J20
  • 76F20