Contents Brief Overview Motivation The Finite Time Lyapunov Exponent How the FTLE is used LCS are ridges of the FTLE field LCS properties Examples Computation of FTLE MANGEN Additional Reading   

References

L. Barreira and Ya.B. Pesin 2002. Lyapunov Exponents and Smooth Ergodic Theory. University Lecture Series; 23, American Mathematical Society, Providence, R.I..

T. Bohr, M.H. Jensen, G. Paladin, and A. Vulpiani 1998. Dynamical Systems Approach to Turbulence. Cambridge University Press.

W. A. Coppel 1978. Dichotomies in Stability Theory. Springer Lecture Notes in Mathematics, volume 629, Springer-Verlag, New York.

C. Coulliette and S. Wiggins 2000. Intergyre transport in a wind-driven, quasigeostrophic double gyre: An application of lobe dynamics. Nonlinear Processes in Geophysics, 7:59-85.

C. Coulliette, F. Lekien, G. Haller, J. Paduan, and J.E. Marsden 2005. Optimal polution release in monterey bay based on nonlinear analysis of coastal radar data. Submitted to Environmental Science and Technology, submitted, 2004.

R. Doerner, B. Hübinger, W. Martienssen, A. Grossmann, S. Thomae 1999. Stable manifolds and predictability of dynamical systems. Chaos, Solitons & Fractals, 10(11):1759-1782.

J. D. Eldredge, T. Colonius, and A. Leonard 2002. A vortex particle method for two-dimensional compressible flow. J. Comput. Phys, 179:371-399.

J. D. Eldredge 2005. Efficient tools for the simulation of flapping wing flows, AIAA Paper 2005-0085.

J. Guckenheimer and P. Holmes 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, volume AMS 42. Springer-Verlag, New York.

E. Hairer, C. Lubich, and G. Wanner 2002. Geometric Numerical Integration. Springer, New York.

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

G. Haller 2000. Lagrangian coherent structures and mixing in two-dimensional turbulence. Chaos, 10(1):99-108.

G. Haller and G. Yuan 2000. Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D, 147:352-370.

G. Haller 2001. Distinguished material surfaces and coherent structures in 3d fluid flows. Physica D, 149:248-277.

G. Haller 2002. Lagrangian coherent structures from approximate velocity data. Phys. Fluids A, 14:1851-1861.

G. Haller 2004. Exact theory of unsteady separation for two-dimensional flows. J. Fluid Mech., 512:257-311.

P. Hartman 1973. Ordinary Differential Equations. John Wiley and Sons, Inc., Baltimore.

M. W. Hirsch and S. Smale 1974. Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, New York.

T. Inanc, S.C. Shadden, and J.E. Marsden 2005. Optimal trajectory generation in ocean flows. In Proc. of 24th American Control Conference, Portland, USA

F. Lekien and N. Leonard 2005. Dynamically consistent lagrangian coherent structures. In Experimental Chaos. American Institute of Physics, March 2005.

F. Lekien, C. Coulliette, A. J. Mariano, E. H. Ryan, L. K. Shay, G. Haller, and J.E. Marsden 2004. Pollution release tied to invariant manifolds: A case study for the coast of florida. Physica D, submitted, 2004.

F. Lekien and J. Marsden 2005. Tricubic interpolation in three dimensions. Int. Journal for Numerical Methods in Engineering, (to appear, 2005). 

A.M. Lyapunov 1992. The General Problem of the Stability of Motion. Taylor & Francis, Washington.

A.M. Mancho, S. Small, S. Wiggins, and K. Ide 2003. Computation of stable and unstable manifold of hyperbolic trajectories in two-dimensional aperiodically time-dependent vector fields. Physica D, 182:188-222.

J. E. Marsden and M. J. Hoffman 1993. Elementary Classical Analysis, 2nd Ed. W. H. Freeman and Company, New York.

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

B. O'Neill 1997. Elementary Differential Geometry, 2nd Ed. Academic Press, San Diego.

V. I. Oseledec 1968. A multiplicative ergodic theorem: Ljapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc., 19:197-231.

J.M. Ottino 1989. The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press, Cambridge.

H. Peters, L.K. Shay, A.J. Mariano, and T.M. Cook  2002. Current variability on a narrow shelf with large ambient vorticity. Journal of Geophysical Research-Oceans, 107(C8):art. no.-3087.

O. Perron 1930. Die Ordnungszahlen linearer Differentialgleichungssyteme. Math. Zs. 31:748-766.

R.T. Pierrehumbert 1991. Large-scale horizontal mixing in planetary atmospheres. Phys. Fluids A 3 (5):1250-1260.

R.T. Pierrehumbert and H. Yang  1993. Global chaotic mixing on isentropic surfaces. J Atmos. Sci. 50:2462-2480.

A. C. Poje and G. Haller  1999. Geometry of cross-stream mixing in a double- gyre ocean model. J. Phys. Oceanography, 29:1649-1665.

A. Provenzale 1999. Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech., 31:55-93.

A. Rogerson, P. D. Miller, L. J. Pratt, and C. K. R. T. J. Jones 1999. Lagrangian motion and fluid exchange in a barotropic meandering jet. Journal of Physical Oceanography, 29(10):2635-2655.

V. Rom-Kedar 1990. Transport rates of a class of two-dimensional maps and flows. Physica D, 43:229-268.

V. Rom-Kedar, A. Leonard, and S. Wiggins 1990. An analytical study of transport, mixing, and chaos in unsteady vortical flow. J. Fluid Mech., 214:347-394.

V. Rom-Kedar and S. Wiggins 1991. Transport in two-dimensional maps: Concepts, examples, and a comparison of the theory of Rom-Kedar and Wiggins with the markov model of Mackay, Meiss, Ott, and Percival. Physica D, 51:248-266.

S.C. Shadden, F. Lekien, and J.E. Marsden 2005. Definition and Properties of Lagrangian Coherent Structures. Submitted to Physica D.

L. K. Shay, T. M. Cook, H. Peters, A. J. Mariano, R. Weisberg, P. E. An, A. Soloviev, and M. Luther 2002. Very high frequency radar mapping of the surface currents. IEEE J. Oceanogr. Engin., 27:155-169.

H. Strogatz, S. H. 1994. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering. Reading, MA: Perseus Books, Cambridge MA.,

F. Verhulst 1996, Nonlinear Differential Equations and Dynamical Systems, 2nd Ed. Springer-Verlag, New York.

J. von Hardenberg, K. Fraedrich, F. Lunkeit and A. Provenzale 2000. Transient chaotic mixing during a baroclinic life cycle. Chaos, 10(1):122-134.

G.A. Voth, G. Haller, and J.P. Gollub 2002. Experimental measurements of stretching fields in fluid mixing. Phys. Rev. Letters, 88 (25):254501.1 - 254501.4.

K. C. Wang  1982. On the current controversy about unsteady separation. in Symposium On Numerical and Physical Aspects of Aerodynamic Flows. T. Cebecci.

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

S. Wiggins 1992. Chaotic Transport in Dynamical Systems. Springer-Verlag, New York.

G.C. Yuan, L.J. Pratt, and C.K.R.T Jones 2002. Barrier destruction and lagrangian predictability at depth in a meandering jet. Dynamics Of Atmospheres And Oceans, 35(1):41-61.