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A variational theory of hyperbolic Lagrangian coherent structures. (English) Zbl 1214.37056

This article deals with the development of a self-consistent theory of coherent trajectory patterns in dynamical systems defined over a finite time-interval. It is used the term Lagrangian coherent structures (LCSs) [G. Haller and G. Yuan, Physica D 147, No. 3–4, 352–370 (2000; Zbl 0970.76043)] to describe the core surfaces around which such trajectory patterns form. As proposed there repelling LCSs are the core structures generating stretching, attracting LCSs act as centerpieces of folding, and shear LCS delineate swirling and jet-type tracer patterns. In order to act as organizing center for Lagrangian patterns, LCSs are expected to have two key properties:

$\left(1\right)$ An LCs should be a material surface, i.e., a codimension-one invariant surface in the extended phase of a dynamical system. This is because (a) an LCS must have sufficiently high dimension to have visible impact and act as a transport barrier and (b) an LCS must move with the flow to act as observable core of evolving Lagrangian patterns.

$\left(2\right)$ An LCS should exhibit locally the strongest attraction, repulsion or shearing in the flow. This is essential to distinguish the LCS from all nearby material surfaces that will have the same stability type, as implied by the continuous dependence of the flow on initial conditions over finite times.

The authors develop a mathematical theory that clarifies the relationship between observable LCSs and invariants of the Cauchy-Green strain tensor field. Motivated by physical observations of trajectory patterns hyperbolic LCSs are defined as material surfaces (i.e., codimension-one invariant manifolds in the extendend phase space) that extremize an appropriate finite-time normal repulsion or attraction measure over all nearby material surfaces. Weak LCss (WLCSs) are defined as stationary solutions of the above variational problem. Solving these variational problems computable sufficient and necessary criteria are obtained for WLCSs and LCSs that link them rigorously to the Cauchy-Green strain tensor field. The notion of a Constrained LCS (CLCS) is introduced that extremizes normal repulsion or attractor under constraints. This construction allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node.

##### MSC:
 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology 76F25 Turbulent transport, mixing 37D35 Thermodynamic formalism, variational principles, equilibrium states
##### References:
 [1] Haller, G.; Yuan, G.: Lagrangian coherent structures and mixing in two-dimensional turbulence, Physica D 147, 352-370 (2000) · Zbl 0970.76043 · doi:10.1016/S0167-2789(00)00142-1 [2] Haller, G.: Distinguished material surfaces and coherent structures in three-dimensional fluid flows, Physica D 149, 248-277 (2001) · Zbl 1015.76077 · doi:10.1016/S0167-2789(00)00199-8 [3] Haller, G.: Lagrangian coherent structures from approximate velocity data, Phys. fluids A 14, 1851-1861 (2002) · Zbl 1185.76161 · doi:10.1063/1.1477449 [4] Shadden, S. C.; Lekien, F.; Marsden, J. E.: Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D 212, 271-304 (2005) · Zbl 1161.76487 · doi:10.1016/j.physd.2005.10.007 [5] Lekien, F.; Shadden, S. C.; Marsden, J. E.: Lagrangian coherent structures in n-dimensional systems, J. math. Phys. 48, 065404 (2007) · Zbl 1144.81374 · doi:10.1063/1.2740025 [6] Peacock, T.; Dabiri, J.: Introduction to focus issue: Lagrangian coherent structures, Chaos 20, 01750 (2010) [7] Lekien, F.; Coulliette, A.; Mariano, A. J.; Ryan, E. H.; Shay, L. K.; Haller, G.; Marsden, J.: Pollution release tied to invariant manifolds: a case study for the coast of florida, Physica D 210, 1-20 (2005) · Zbl 1149.86302 · doi:10.1016/j.physd.2005.06.023 [8] Tang, W.; Mathur, M.; Haller, G.; Hahn, D. C.; Ruggiero, F. H.: Lagrangian coherent structures near a subtropical jet stream, J. atmospheric sci. 67, 2307-2319 (2010) [9] Fenichel, N.: Persistence and smoothness of invariant manifolds for flows, Indiana univ. Math. J. 21, 193-225 (1971) · Zbl 0246.58015 · doi:10.1512/iumj.1971.21.21017 [10] Froyland, G.; Lloyd, S.; Santitissadeekorn, N.: Coherent sets for nonautonomous dynamical systems, Physica D 239, 1527-1541 (2010) · Zbl 1193.37032 · doi:10.1016/j.physd.2010.03.009 [11] Arnold, V. I.: Ordinary differential equations, (1978) [12] Haller, G.; Poje, A.: Finite-time transport in aperiodic flows, Physica D 119, 352-380 (1998) · Zbl 1194.76089 · doi:10.1016/S0167-2789(98)00091-8 [13] Haller, G.: Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos 10, 99-108 (2000) · Zbl 0979.37012 · doi:10.1063/1.166479 [14] Haller, G.: An objective definition of a vortex, J. fluid mech. 525, 1-26 (2005) · Zbl 1065.76031 · doi:10.1017/S0022112004002526 [15] Duc, L. H.; Siegmund, S.: Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals, Internat. J. Bifur. chaos appl. Sci. engrg. 18, 641-674 (2008) · Zbl 1147.37314 · doi:10.1142/S0218127408020562 [16] Berger, A.; Doan, T. S.; Siegmund, S.: Non-autonomous finite-time dynamics, Discrete contin. Dyn. syst. Ser. B 9, 463-492 (2008) · Zbl 1148.37010 · doi:10.3934/dcdsb.2008.9.463 [17] Berger, A.; Doan, T. S.; Siegmund, S.: A remark on finite-time hyperbolicity, PAMM proc. Appl. math. Mech. 8, 10917-10918 (2008) [18] A. Berger, On finite-time hyperbolicity, Discrete Contin. Dyn. Syst. Ser. S (2010) (in press). [19] Mathur, M.; Haller, G.; Peacock, T.; Ruppert-Felsot, J. E.; Swinney, H. L.: Uncovering the Lagrangian skeleton of turbulence, Phys. rev. Lett. 98, 144502 (2007) [20] Lin, K.; Young, L. -S.: Shear-induced chaos, Nonlinearity 21, 899-915 (2008) · Zbl 1153.37355 · doi:10.1088/0951-7715/21/5/002 [21] Beron-Vera, F. J.; Olascoaga, M. J.; Brown, M. G.; Koçak, H.; Rypina, I. I.: Invariant-tori-like Lagrangian coherent structures in geophysical flows, Chaos 20 (2010) [22] Haller, G.: Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Phys. fluids 13, 3365-3385 (2001) · Zbl 1184.76207 · doi:10.1063/1.1403336