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.
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Berger A., Doan T.S., Siegmund S.: Nonautonomous finite-time dynamics. Discret. Contin. Dyn. Syst. 9, 463–492 (2008)
Berger A., Doan T.S., Siegmund S.: A remark on finite-time hyperbolicity. Proc. Appl. Math. Mech. 8, 10917–10918 (2008)
Berger A., Doan T.S., Siegmund S.: A definition of spectrum for differential equations on finite time. J. Differ. Equ. 246, 1098–1118 (2009)
Branicki M., Wiggins S.: An adaptive method for computing invariant manifolds in non-autonomous, three-dimensional dynamical systems. Physica D 238, 1625–1657 (2009)
Chorin A., Marsden J.: A Mathematical Introduction to Fluid Mechanics, 3rd edn. Texts in Applied Mathematics. Springer, New York (1993)
Duc, L.H.: Asymptotic and finite-time dynamics for nonautonomous dynamical systems. Doctoral thesis, J.W. Goethe University Frankfurt (2006)
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)
Hale J.: Ordinary Differential Equations. Krieger, Malabur (1980)
Haller G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Phys. Fluids 13, 3365–3385 (2001)
Haller G.: An objective definition of a vortex. J. Fluid Mech. 525, 1–26 (2005)
Haller G., Poje A.C.: Finite-time transport in aperiodic flows. Physica D 119, 352–380 (1998)
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)
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)
Okubo A.: Horizontal dispersion of floatable particles in the vicinity of velocity singularities such as convergences. Deep-Sea Res. 17, 445–454 (1970)
Poje A.C., Haller G., Mezic I.: The geometry and statistics of mixing in aperiodic flows. Phys. Fluids 11, 2963–2968 (1999)
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)
Velasco Fuentes O.U.: Chaotic advection by two interacting finite-area vortices. Phys. Fluids 27, 901–912 (2001)
Velasco Fuentes O.U.: Vortex filmentation: its onset and its role on axisymmetrization and merger. Dyn Atmos Oceans 40, 23–42 (2005)
Weiss J.: The dynamics of enstrophy transfer in two-dimensional hydrodynamics. Physica D 48, 273–294 (1991)
This paper is dedicated to Russell Johnson on the occasion of his birthday.
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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
- EPH partition
- Finite time dynamics
- Vortex merger
Mathematics Subject Classification (2000)
- Primary 37B55
- Secondary 37J20