Lagrangian coherent structure

6747: 5320: 6742:{\displaystyle \nabla F_{t_{0}}^{t}(x_{0})\approx {\begin{pmatrix}{\frac {x^{1}(t;t_{0},x_{0}+\delta _{1})-x^{1}(t;t_{0},x_{0}-\delta _{1})}{\left|2\delta _{1}\right|}}&{\frac {x^{1}(t;t_{0},x_{0}+\delta _{2})-x^{1}(t;t_{0},x_{0}-\delta _{2})}{\left|2\delta _{2}\right|}}&{\frac {x^{1}(t;t_{0},x_{0}+\delta _{3})-x^{1}(t;t_{0},x_{0}-\delta _{3})}{\left|2\delta _{3}\right|}}\\{\frac {x^{2}(t;t_{0},x_{0}+\delta _{1})-x^{2}(t;t_{0},x_{0}-\delta _{1})}{\left|2\delta _{1}\right|}}&{\frac {x^{2}(t;t_{0},x_{0}+\delta _{2})-x^{2}(t;t_{0},x_{0}-\delta _{2})}{\left|2\delta _{2}\right|}}&{\frac {x^{2}(t;t_{0},x_{0}+\delta _{3})-x^{2}(t;t_{0},x_{0}-\delta _{3})}{\left|2\delta _{3}\right|}}\\{\frac {x^{3}(t;t_{0},x_{0}+\delta _{1})-x^{3}(t;t_{0},x_{0}-\delta _{1})}{\left|2\delta _{1}\right|}}&{\frac {x^{3}(t;t_{0},x_{0}+\delta _{2})-x^{3}(t;t_{0},x_{0}-\delta _{2})}{\left|2\delta _{2}\right|}}&{\frac {x^{3}(t;t_{0},x_{0}+\delta _{3})-x^{3}(t;t_{0},x_{0}-\delta _{3})}{\left|2\delta _{3}\right|}}\end{pmatrix}},} 16403: 14815:

boundary, without global transverse departure form the material vortex. (Exceptions are inviscid flows where such a global departure of LAVD level surfaces from a vortex is possible as fluid elements preserve their material rotation rate for all times). Remarkably, centers of rotationally coherent vortices (defined by local maxima of the LAVD field) can be proven to be the observed centers of attraction or repulsion for finite-size (inertial) particle motion in geophysical flows (see Fig. 11b). In three-dimensional flows, tubular level surfaces of the LAVD define initial positions of two-dimensional eddy boundary surfaces (see Fig. 11c) that remain rotationally coherent over a time intcenter|erval

4338: 12501: 3794: 9189: 1456: 12796:. In two-dimensions, therefore, (polar) elliptic LCSs are simply closed level curves of the PRA, which turn out to be objective. In three dimensions, (polar) elliptic LCSs are toroidal or cylindrical level surfaces of the PRA, which are, however, not objective and hence will generally change in rotating frames. Coherent Lagrangian vortex boundaries can be visualized as outermost members of nested families of elliptic LCSs. Two- and three-dimensional examples of elliptic LCS revealed by tubular level surfaces of the PRA are shown in Fig. 10a-b. 6843: 12090: 11800: 6752: 4333:{\displaystyle {\begin{aligned}\delta _{t_{0}}^{t_{1}}(x_{0})&=\lim _{\epsilon \to 0}{\frac {1}{\epsilon }}\max _{\left|\xi (t_{0})\right|=1}\left|\xi _{\epsilon }(t_{1};x_{0})\right|\\&=\max _{\left|\xi (t_{0})\right|=1}{\sqrt {\left\langle \nabla F_{t_{0}}^{t_{1}}(x_{0})\xi (t_{0}),\nabla F_{t_{0}}^{t_{1}}(x_{0})\xi (t_{0})\right\rangle }}\\&=\max _{\left|\xi (t_{0})\right|=1}{\sqrt {\left\langle \xi (t_{0}),C_{t_{0}}^{t_{1}}(x_{0})\xi (t_{0})\right\rangle }}\\\end{aligned}}} 11185: 16389: 13031: 17389:. In contrast to hyperbolic LCSs, however, parabolic LCSs satisfy more robust boundary conditions: they remain stationary curves of the material-line-averaged shear functional even under variations to their endpoints. This explains the high degree of robustness and observability that jet cores exhibit in mixing. This is to be contrasted with the highly sensitive and fading footprint of hyperbolic LCSs away from strongly hyperbolic regions in diffusive tracer patterns. 11416: 77: 25: 3086: 17523: 12496:{\displaystyle {\begin{aligned}\cos \theta _{t_{0}}^{t}&={\frac {1}{2}}\left(\sum _{i=1}^{3}{\frac {\left\langle \xi _{i},\nabla F_{t_{0}}^{t_{1}}\xi _{i}\right\rangle }{\sqrt {\lambda _{i}}}}-1\right),\\\sin \theta _{t_{0}}^{t}&={\frac {\left\langle \xi _{i},\nabla F_{t_{0}}^{t_{1}}\xi _{j}\right\rangle -\left\langle \xi _{j},\nabla F_{t_{0}}^{t_{1}}\xi _{i}\right\rangle }{2\epsilon _{ijk}e_{k}}},\qquad i\neq j,\end{aligned}}} 2832: 126: 11193: 14806: 19528: 15729: 13040: 16895: 15427: 11795:{\displaystyle {\begin{aligned}\cos \theta _{t_{0}}^{t_{1}}&={\frac {\langle \xi _{i},\nabla F_{t_{0}}^{t_{1}}\xi _{i}\rangle }{\sqrt {\lambda _{i}}}},\quad i=1\,\,or\,\,\,2,\\\sin \theta _{t_{0}}^{t_{1}}&=\left(-1\right)^{j}{\frac {\langle \xi _{i},\nabla F_{t_{0}}^{t_{1}}\xi _{j}\rangle }{\sqrt {\lambda _{j}}}},\qquad (i,j)=(1,2)\,\,or\,\,(2,1),\\\end{aligned}}} 14865: 982: 15432: 16583: 10388:

endpoints fixed. This is to be contrasted with parabolic LCSs (see below), which are also shearless LCSs but prevail as stationary curves to the shear functional even under arbitrary variations. As a consequence, individual trajectories are objective, and statements about the coherent structures they form should also be objective.

3278:. The heuristic element here is that instead of constructing a highly repelling material surface, one simply seeks points of large particle separation. Such a separation may well be due to strong shear along the set of points so identified; this set is not at all guaranteed to exert any normal repulsion on nearby trajectories. 15134: 4853: 16392:

Figure 11: An elliptic Lagrangian Coherent Structure (or LCS, in green, on the left) and its advected position under the flow map (on the right) of a chaotically forced ABC flow. Also shown in green is a circle of initial conditions placed around the LCS (on the left), advected for the same amount of

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Outermost complex tubular level curves of the LAVD define initial positions of rotationally coherent material vortex boundaries in two-dimensional unsteady flows (see Fig. 11a). By construction, these boundaries may exhibit transverse filamentation, but any developing filament keeps rotating with the

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Figure 11b: Materially advected rotationally coherent mesoscale eddy boundaries and eddy centers in the ocean, along with representative inertial particle trajectories initialised on the eddy boundaries. The eddy centers are obtained as local maxima of the LAVD field. As can be proven mathematically,

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As illustrated on the right, individual tracer trajectories forming coherent patterns are generally sensitive with respect to changes in their initial conditions and the system parameters. In contrast, the LCSs creating these trajectory patterns turn out to be robust and provide a simplified skeleton

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Individual trajectories in a model flow generally show vastly different behavior from trajectories starting from the same initial condition of the real flow. This is due to the inevitable accumulation of errors and uncertainties, as well as sensitive dependence on initial conditions, in any realistic

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As noted above under hyperbolic LCSs, a global variational approach has been developed in two dimensions to capture elliptic LCSs as closed stationary curves of the material-line-averaged Lagrangian strain functional. Such curves turn out to be closed null-geodesics of the generalized Green–Lagrange

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Figure 11c: A rotationally coherent mesoscale eddy (yellow) in the Southern Ocean State Estimate (SOSE) ocean model at t0 = May 15, 2006, computed as a tubular LAVD level surface over t1-t0=120 days. Also shown are nearby LAVD level surfaces to illustrate the rotational incoherence outside the eddy.

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In 3D flows, instead of solving the Frobenius PDE (see table above) for hyperbolic LCSs, an easier approach is to construct intersections of hyperbolic LCSs with select 2D planes, and fit a surface numerically to a large number of such intersection curves. Let us denote the unit normal of a 2D plane

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Figure 3: Instantaneous streamlines and the evolution of trajectories starting from the interior of one of them in a linear solution of the Navier–Stokes equation. This dynamical system is classified as elliptic by a number of frame-dependent coherence diagnostics, such as the Okubo–Weiss criterion.

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of possible particle positions is a material configuration space. In this space, LCSs are material surfaces, formed by trajectories. Whether or not a material trajectory is contained in an LCS is a property that is independent of the choice of coordinates, and hence cannot depend of the observer. As

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than any of the neighboring material surfaces. Such LCSs, defined as extrema (or more generally, stationary surfaces) for a finite-time coherence property, will indeed serve as observed centerpieces of trajectory patterns. Examples of attracting, repelling and shearing LCSs are in a direct numerical

13307:, respectively. Just as the classic polar decomposition, the DPD is valid in any finite dimension. Unlike the classic polar decomposition, however, the dynamic rotation and stretch tensors are obtained from solving linear differential equations, rather than from matrix manipulations. In particular, 10387:

eigenvalue field. This agrees with the conclusion of the local variational theory of LCSs. The geodesic approach, however, also sheds more light on the robustness of hyperbolic LCSs: hyperbolic LCSs only prevail as stationary curves of the averaged shear functional under variations that leave their

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FTLE ridges have proven to be a simple and efficient tool for the visualize hyperbolic LCSs in a number of physical problems, yielding intriguing images of initial positions of hyperbolic LCSs in different applications (see, e.g., Figs. 5a-b). However, FTLE ridges obtained over sliding time windows

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Fig. 2b: An attracting LCS is the locally most attracting material line (invariant manifold in the extended phase space of position and time), acting as the backbone curve of deforming tracer patterns over a finite time interval. In contrast, the unstable manifold of a saddle-type fixed point is an

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for fluid motion are well known to be frame-dependent, it might first seem counterintuitive to require frame-invariance for LCSs, which are composed of solutions of these frame-dependent equations. Recall, however, that the Newton and Navier–Stokes equations represent objective physical principles

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The level sets of the PRA are objective in two dimensions but not in three dimensions. An additional shortcoming of the polar rotation tensor is its dynamical inconsistency: polar rotations computed over adjacent sub-intervals of a total deformation do not sum up to the rotation computed for the

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For 3D flows, as in the case of hyperbolic LCSs, solving the Frobenius PDE can be avoided. Instead, one can construct intersections of a tubular elliptic LCS with select 2D planes, and fit a surface numerically to a large number of these intersection curves. As for hyperbolic LCSs above, let us

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that exert a major influence on nearby trajectories over a time interval of interest. The type of this influence may vary, but it invariably creates a coherent trajectory pattern for which the underlying LCS serves as a theoretical centerpiece. In observations of tracer patterns in nature, one

3057:-frame. While this process adds new terms (inertial forces) to the equations of motion, these inertial terms arise precisely to ensure the invariance of material trajectories. Fully composed of material trajectories, LCSs remain invariant in the transformed equation of motion defined in the 13034:

Figure 11a: Rotationally coherent mesoscale eddy boundaries in the ocean at time t0 = November 11, 2006, identified from satellite-based surface velocities, using the integration time t1-t0=90 days. The boundaries are identified as outermost closed contours of the LAVD with small convexity

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Elliptc LCSs are closed and nested material surfaces that act as building blocks of the Lagrangian equivalents of vortices, i.e., rotation-dominated regions of trajectories that generally traverse the phase space without substantial stretching or folding. They mimic the behavior of

15724:{\displaystyle n_{\pm }(x_{0})={\sqrt {\frac {\sqrt {\lambda _{1}(x_{0})}}{{\sqrt {\lambda _{1}(x_{0})}}+{\sqrt {\lambda _{3}(x_{0})}}}}}\xi _{1}(x_{0})\pm {\sqrt {\frac {\sqrt {\lambda _{3}(x_{0})}}{{\sqrt {\lambda _{1}(x_{0})}}+{\sqrt {\lambda _{3}(x_{0})}}}}}\xi _{3}(x_{0}),} 14734: 8014: 11041: 105:

of the overall dynamics of the system. The robustness of this skeleton makes LCSs ideal tools for model validation, model comparison and benchmarking. LCSs can also be used for now-casting and even short-term forecasting of pattern evolution in complex dynamical systems.

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are shearless material surfaces that delineate cores of jet-type sets of trajectories. Such LCSs are characterized by both low stretching (because they are inside a non-stretching structure), but also by low shearing (because material shearing is minimal in jet cores).

10124: 17519:. Fig. 14b shows an example of parabolic LCSs in Jupiter's atmosphere, located using this variational theory. The chevron-type shapes forming out of circular material blobs positioned along the jet core is characteristic of tracer deformation near parabolic LCSs. 16890:{\displaystyle \eta _{\lambda }^{\pm }(x_{0}):={\sqrt {\frac {\lambda _{2}(x_{0})-\lambda ^{2}}{\lambda _{2}(x_{0})-\lambda _{1}(x_{0})}}}\xi _{1}(x_{0})\pm {\sqrt {\frac {\lambda ^{2}-\lambda _{1}(x_{0})}{\lambda _{2}(x_{0})-\lambda _{1}(x_{0})}}}\xi _{2}(x_{0}),} 723: 2707: 2282: 12725: 2827:

referring to the Euclidean matrix norm. As seen in Fig. 3, however, trajectories grow exponentially along a rotating line and shrink exponentially along another rotating line. In material terms, therefore, the flow is hyperbolic (saddle-type) in any frame.

10651: 11135:, interpreted as the bulk solid-body rotation component of volume elements. In planar motions, this rotation is defined relative to the normal of the plane. In three dimensions, the rotation is defined relative to the axis defined by the eigenvector of 16344:. (Strictly speaking, the reduced shear ODE is not an ordinary differential equation, given that its right-hand side is not a vector field, but a direction field, which is generally not globally orientable). Intersections of tubular elliptic LCSs with 4848: 9185:. These starting points serve are initial positions of exceptional saddle-type trajectories in the flow. An example of the local variational computation of a repelling LCS is shown in FIg. 8. The computational algorithm is available in LCS Tool. 1840: 3760: 1447:

will generally be time-dependent, acting as the evolving skeletons of observed coherent trajectory patterns. Figure 2b shows the difference between an attracting LCS and a classic unstable manifold of a saddle point, for evolving times, in an

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Under variable endpoint boundary conditions, initial positions of parabolic LCSs turn out to be alternating chains of shrink lines and stretch lines that connect singularities of these line fields. These singularities occur at points where

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Physical phenomena governed by LCSs include floating debris, oil spills, surface drifters and chlorophyll patterns in the ocean; clouds of volcanic ash and spores in the atmosphere; and coherent crowd patterns formed by humans and animals.

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In two dimensions, parabolic LCSs are also solutions of the global shearless variational principle described above for hyperbolic LCSs. As such, parabolic LCSs are composed of shrink lines and stretch lines that represent geodesics of the

9559:. (Strictly speaking, this equation is not an ordinary differential equation, given that its right-hand side is not a vector field, but a direction field, which is generally not globally orientable). Intersections of hyperbolic LCSs with 13048:

An alternative to the classic polar decomposition provides a resolution to both the non-objectivity and the dynamic inconsistency issue. Specifically, the Dynamic Polar Decomposition (DPD) of the deformation gradient is also of the form

17954:

Olascoaga, M. J.; Beron-Vera, F. J.; Haller, G.; Triñanes, J.; Iskandarani, M.; Coelho, E. F.; Haus, B. K.; Huntley, H. S.; Jacobs, G.; Kirwan, A. D.; Lipphardt, B. L.; Özgökmen, T. M.; h. m. Reniers, A. J.; Valle-Levinson, A. (2013).

13188: 9087: 8871: 15422:{\displaystyle \eta ^{\pm }(x_{0}):={\sqrt {\frac {\lambda _{2}(x_{0})-1}{\lambda _{2}(x_{0})-\lambda _{1}(x_{0})}}}\xi _{1}(x_{0})\pm {\sqrt {\frac {1-\lambda _{1}(x_{0})}{\lambda _{2}(x_{0})-\lambda _{1}(x_{0})}}}\xi _{2}(x_{0}),} 11809: 8487:

Solving these local extremum principles for hyperbolic LCSs in two and three dimensions yields unit normal vector fields to which hyperbolic LCSs should everywhere be tangent. The existence of such normal surfaces also requires a

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Assume that the phase space of the underlying dynamical system is the material configuration space of a continuum, such as a fluid or a deformable body. For instance, for a dynamical system generated by an unsteady velocity field

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flow model. Yet an attracting LCS (such as the unstable manifold of a saddle point) is remarkably robust with respect to modelling errors and uncertainties. LCSs are, therefore, ideal tools for model validation and benchmarking

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planes, then fitting a surface to the limit cycle family yields a numerical approximation for 2D shear surface. A three-dimensional example of this local variational computation of an elliptic LCS is shown in Fig. 11.

2853:. As long as correctly transformed from one frame to the other, these equations generate physically the same material trajectories in the new frame. In fact, we decide how to transform the equations of motion from an 1764:-dimensional vector representing time-dependent translations. As a consequence, any self-consistent LCS definition or criterion should be expressible in terms of quantities that are frame-invariant. For instance, the 857: 9893: 971: 13867: 13794: 4470: 1370:

Thus, attracting, repelling and shearing material surfaces are necessarily stacked on each other, i.e., occur in continuous families. This leads to the idea of seeking LCSs in finite-time dynamical systems as

1046:. Examples of such action are attraction, repulsion, or shear. In principle, any well-defined mathematical property qualifies that creates coherent patterns out of randomly selected nearby initial conditions. 307: 583: 16338: 9551: 2555: 1540: 17526:

Figure 14b: Parabolic LCSs delineating unsteady Lagrangian jet cores in the atmosphere of Jupiter. Also shown is the evolution of the elliptic LCS marking the boundary of the Great Red Spot. Video:Alireza

4690: 2550:, the ability of these quantities to capture material mixing, transport, and coherence is limited and a priori unknown in any given frame. As an example, consider the linear unsteady fluid particle motion 2062: 7287:

have been identified by some authors broadly with LCSs. In support of this identification, it is also often argued that the material flux over such sliding-window FTLE ridges should necessarily be small.

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Figure 6. FTLE ridges highlight both hyperbolic LCS and shearing material lines, such as the boundaries of a riverbed in a 3D model of the New River Inlet, Onslow, North Carolina (Image: Allen Sanderson).

16550: 14060: 6997: 973:, invariant manifolds and their associated material surfaces are abundant and generally undistinguished in the extended phase space. Only few of them will act as cores of coherent trajectory patterns. 16430:(GRS) of Jupiter. These LCSs were identified in a two-dimensional, unsteady velocity field reconstructed from a video footage of Jupiter. The color indicates the corresponding values of the parameter 17463: 13460: 8260: 8100: 10482: 5115:. A more expedient approach is to compute the FTLE field from a simple finite-difference approximation to the deformation gradient. For example, in a three-dimenisonal flow, we launch a trajectory 4964: 2515:, as well as the averages or eigenvalues of these quantities, are routinely used in heuristic LCS detection. While such quantities may effectively mark features of the instantaneous velocity field 8008: 15981: 12095: 11421: 10779: 7227: 3799: 3132: 766: 8451: 7933: 4701: 409: 14373: 13533: 13376: 13044:

heavy particles (cyan) converge to the centers of anti-cyclonic (clockwise) eddies. Light particles (black) converge to the centers of cyclonic (clockwise) eddies. (Movie: Alireza Hadjighasem)

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FTLE ridges mark hyperbolic LCS positions, but also highlight surfaces of high shear. A convoluted mixture of both types of surfaces often arises in applications (see Fig. 6 for an example).

5048: 15126: 15064: 14974: 13987: 11267: 7661: 7600: 16968: 13027:

are varied. For this reason, rotations predicted by the polar rotation tensor over varying time intervals divert from the experimentally observed mean material rotation of fluid elements.

12586: 14802:. This result applies both in two- and three dimensions, and enables the computation of a well-defined, objective and dynamically consistent material rotation angle along any trajectory. 17326: 13297: 9716:

seeks exceptionally coherent locations where this general trend fails, resulting in an order of magnitude smaller variability in shear or strain than what is normally expected across an

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Figure 2a: Hyperbolic LCS (attracting in red and repelling in blue) and elliptic LCS (boundaries of green regions) in a two-dimensional turbulence simulation. (Image: Mohammad Farazmand)

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Bozorgmagham, A. E.; Ross, S. D.; Schmale, D. G. (2013). "Real-time prediction of atmospheric Lagrangian coherent structures based on forecast data: An application and error analysis".

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requirement of continuum mechanics. The objectivity of LCSs requires them to be invariant with respect to all possible observer changes, i.e., linear coordinate changes of the form

15869: 15826: 15783: 8659: 8616: 8542: 7858: 7540: 5260: 3276: 3236: 3196: 1658: 1194: 1154: 927: 552: 3407: 2513: 12794: 10385: 10277: 9294: 9183: 9140: 3507: 512: 439: 17008: 13054: 3015: 17387: 17113: 15007: 14245: 14206: 12849: 12083: 11409: 11313: 11179: 11133: 11087: 10697: 9886: 8376: 8293: 8133: 1341: 1087: 1020: 5315: 5168: 3627: 3572: 3365: 16225: 15919: 12534: 10169: 9436: 8925: 8709: 214: 7402:

The local variational theory of hyperbolic LCSs builds on their original definition as strongest repelling or repelling material surfaces in the flow over the time interval

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Such null-geodesics can be proven to be tensorlines of the Cauchy–Green strain tensor, i.e., are tangent to the direction field formed by the strain eigenvector fields

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A sample application is shown in Fig. 9, where the sudden appearance of a hyperbolic core (strongest attracting part of a stretchline) within the oil spill caused the

9472: 9393: 9346: 2471: 7158: 14800: 2548: 2393: 2358: 2323: 1835: 1800: 17517: 17490: 17207: 17180: 13025: 12998: 12925: 12755: 11363: 11340: 8574: 8478: 7753: 7500: 7473: 6811: 5195: 4991: 4891: 4501: 3787: 3475: 16382: 16362: 16155: 16108: 14765: 9597: 9577: 9366: 9319: 6755:

Figure 5b. Attracting (blue) and repelling (red) LCSs extracted as FTLE ridges from a two-dimensional simulation of a von Karman vortex street (Image: Jens Kasten)

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Beron-Vera, F. J.; Olascoaga, M. A. J.; Brown, M. G.; KoçAk, H.; Rypina, I. I. (2010). "Invariant-tori-like Lagrangian coherent structures in geophysical flows".

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A general material surface experiences shear and strain in its deformation, both of which depend continuously on initial conditions by the continuity of the map

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of interest. A challenge in that in each material volume element, all individual material fibers (tangent vectors to trajectories) perform different rotations.

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can further be factorized into two deformation gradients: one for a spatially uniform (rigid-body) rotation, and one that deviates from this uniform rotation:

12971: 10468: 8343: 8179: 7446: 7354: 7285: 7125: 7067: 6912: 2052:{\displaystyle S(x,t)={\frac {1}{2}}\left(\nabla v(x,t)+(\nabla v(x,t))^{T}\right),\qquad W(x,t)={\frac {1}{2}}\left(\nabla v(x,t)-(\nabla v(x,t))^{T}\right),} 1423:

dynamical system. In contrast, LCSs are only required to be invariant in the extended phase space. This means that even if the underlying dynamical system is

16476:-line) bounding the core of the GRS, as well as the outermost elliptic LCS serving as the Lagrangian vortex boundary of the GRS. Image:Alireza Hadjighasem. 11089:

are fully captured by the singular values and singular vectors of the stretch tensors. The remaining factor in the deformation gradient is represented by

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Figure 10a. Elliptic LCSs revealed by closed level curves of the PRA distribution in a two-dimensional turbulence simulation. (Image: Mohammad Farazmand)

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is a positive parameter (Lagrange multiplier). The closed null-geodesics can be shown to coincide with limit cycles of the family of direction fields

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Figure 5a. Attracting (red) and repelling (blue) LCSs extracted as FTLE ridges from a two-dimensional turbulence experiment (Image: Manikandan Mathur)

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Fig. 11c Material advection of a rotationally coherent Lagrangian vortex and its core in the 3D SOSE model data set. (Animation: Alireza Hadjighasem)

9479: 1473: 18636:

Norgard, G.; Bremer, P. T. (2012). "Second derivative ridges are straight lines and the implications for computing Lagrangian Coherent Structures".

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Bozorgmagham, A. E.; Ross, S. D. (2015). "Atmospheric Lagrangian coherent structures considering unresolved turbulence and forecast uncertainty".

13304: 13300: 13232: 10786: 10782: 10700: 4506: 16402: 12030:{\displaystyle \theta _{t_{0}}^{t}=\left\pi +{\rm {sign\,}}\left(\sin \theta _{t_{0}}^{t}\right)\cos ^{-1}\left(\cos \theta _{t_{0}}^{t}\right).} 1460:

invariant curve in the phase space, acting as the asymptotic target for tracer patterns over infinite time intervals. Image: Mohammad Farazmand.

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While LCSs generally exist in any dynamical system, their role in creating coherent patterns is perhaps most readily observable in fluid flows.

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Lipinski, D.; Mohseni, K. (2010). "A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures".

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Haller, George; Hadjighasem, Alireza; Farazmand, Mohammad; Huhn, Florian (2016). "Defining Coherent Vortices Objectively from the Vorticity".

18489:(2005). "Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows". 17253:

Since both shearing and stretching are as low as possible along a parabolic LCS, one may seek initial positions of such material surfaces as

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The local variational theory of elliptic LCSs targets material surfaces that locally maximize material shear over the finite time interval

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corresponding to its unit eigenvalue. In higher-dimensional flows, the rotation tensor cannot be viewed as a rotation about a single axis.

10412:. There coherence can be approached either through their homogeneous material rotation or through their homogeneous stretching properties. 932: 13801: 4345: 14729:{\displaystyle \mathrm {LAVD} _{t_{0}}^{t_{1}}(x_{0}):=\int _{t_{0}}^{t_{1}}\left|\omega (x(s;x_{0}),s)-{\bar {\omega }}(s)\right|\,ds,} 14313:). The LAVD is defined as the trajectory-averaged magnitude of the deviation of the vorticity from its spatial mean. With the vorticity 9192:

Figure 8. A repelling LCS visualized as an FTLE ridge (left) and computed exactly as a shrink line (right), i.e., a solution of the ODE

219: 18671:

Schindler, B.; Peikert, R.; Fuchs, R.; Theisel, H. (2012). "Ridge Concepts for the Visualization of Lagrangian Coherent Structures".

11036:{\displaystyle C_{t_{0}}^{t_{1}}=^{T}\nabla F_{t_{0}}^{t_{1}}=U_{t_{0}}^{t_{1}}U_{t_{0}}^{t_{1}}=V_{t_{0}}^{t_{1}}V_{t_{0}}^{t_{1}},} 8489: 4473: 18747:"Erratum and addendum to "A variational theory of hyperbolic Lagrangian coherent structures" [Physica D 240 (2011) 574–598]" 18575: 13874: 4601: 3138:

can be defined as a locally strongest repelling material surface. Attracting and repelling LCSs together are usually referred to as

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Mathur, M.; Haller, G.; Peacock, T.; Ruppert-Felsot, J.; Swinney, H. (2007). "Uncovering the Lagrangian Skeleton of Turbulence".

17545: 16484: 13994: 10405: 6917: 19596:

Green, M. A.; Rowley, C. W.; Haller, G. (2007). "Detection of Lagrangian coherent structures in three-dimensional turbulence".

17328:. A geophysical example of a parabolic LCS (generalized jet core) revealed as a trench of the FTLE field is shown in Fig. 14a. 3143: 18956:

Farazmand, Mohammad; Haller, George (2016). "Polar rotation angle identifies elliptic islands in unsteady dynamical systems".

10119:{\displaystyle D_{t_{0}}^{t_{1}}(x_{0})={\frac {1}{2}}\left,\qquad \Omega ={\begin{pmatrix}0&-1\\1&0\\\end{pmatrix}}.} 19491: 19362: 18688: 18452: 18156: 17396: 8184: 8024: 18896:

Farazmand, M.; Blazevski, D.; Haller, G. (2014). "Shearless transport barriers in unsteady two-dimensional flows and maps".

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readily identifies coherent features, but it is often the underlying structure creating these features that is of interest.

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deficiency. Also shown in the background is the contour plot of the LAVD field for reference. (Image: Alireza Hadjighasem)

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Sanderson, A. R. (2014). "An Alternative Formulation of Lyapunov Exponents for Computing Lagrangian Coherent Structures".

7938: 15925: 10706: 7163: 3103: 737: 718:{\displaystyle {\mathcal {M}}=\{(x,t)\in {\mathcal {P}}\times {\mathcal {I}}\,\colon ^{-1}(x)\in {\mathcal {M}}(t_{0})\}} 18129:

Ali, S.; Shah, M. (2007). "A Lagrangian Particle Dynamics Approach for Crowd Flow Segmentation and Stability Analysis".

10424:

as a tubular material surface along which small material volumes complete the same net rotation over the time intervall

8381: 7863: 3077:-frame of reference. Consequently, any self-consistent LCS definition or detection method must also be frame-invariant. 2702:{\displaystyle {\dot {x}}=v(x,t)={\begin{pmatrix}\sin {4t}&2+\cos {4t}\\-2+\cos {4t}&-\sin {4t}\end{pmatrix}}x,} 349: 14316: 13467: 13310: 12543: 9828: 9599:

planes, then fitting a surface to the curve family so obtained yields a numerical approximation of a 2D repelling LCS.

2277:{\displaystyle {\tilde {S}}(y,t)=Q(t)^{T}S(x,t)Q(t),\qquad {\tilde {W}}(y,t)=Q(t)^{T}S(x,t)Q(t)-Q(t)^{T}{\dot {Q}}(t).} 1208:. They are not only special, but even locally unique in the phase space: no continuous family of attractors may exist. 45: 19411: 12720:{\displaystyle \left_{jk}=\left\langle \xi _{j},\nabla F_{t_{0}}^{t_{1}}\xi _{k}\right\rangle /{\sqrt {\lambda _{k}}}} 2715:. The (frame-dependent) Okubo-Weiss criterion classifies the whole domain in this flow as elliptic (vortical) because 10286: 10178: 9195: 8931: 8715: 7758: 7666: 4996: 63: 18840:

Blazevski, D.; Haller, G. (2014). "Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows".

15069: 15012: 14926: 11210: 7609: 7548: 3017:

to hold for all times. Temporal differentiation of this identity and substitution into the original equation in the

19475: 19179:

Hadjighasem, A.; Haller, G. (2016). "Geodesic Transport Barriers in Jupiter's Atmosphere: A Video-Based Analysis".

17522: 16928: 1572: 3198:

of repelling LCSs as set of initial conditions at which infinitesimal perturbations to trajectories starting from

17260: 13238: 9835: 7393:

There are several other types LCSs (elliptic and parabolic) beyond the hyperbolic LCSs highlighted by FTLE ridges

5060: 1449: 1424: 1420: 41: 18180:

Haller, G. (2001). "Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence".

18258:

Haller, G. (2001). "Distinguished material surfaces and coherent structures in three-dimensional fluid flows".

10646:{\displaystyle \nabla F_{t_{0}}^{t_{1}}=R_{t_{0}}^{t_{1}}U_{t_{0}}^{t_{1}}=V_{t_{0}}^{t_{1}}R_{t_{0}}^{t_{1}},} 9785: 9752: 9719: 9682: 9649: 7291:

The "FTLE ridge=LCS" identification, however, suffers form the following conceptual and mathematical problems:

34: 14253: 12854: 17998:"The impact of advective transport by the South Indian Ocean Countercurrent on the Madagascar plankton bloom" 17016: 312: 19543: 18786:

Farazmand, M.; Haller, G. (2012). "Computing Lagrangian coherent structures from their variational theory".

4598:. As this relative stretching tends to grow rapidly, it is more convenient to work with its growth exponent 3412: 2802: 17621: 15834: 15791: 15748: 10473:

To obtain a well-defined bulk rotation for each material element, one may employ the unique left and right

8624: 8581: 8507: 7823: 7505: 5200: 4843:{\displaystyle \mathrm {FTLE} _{t_{0}}^{t_{1}}(x_{0})={\frac {1}{2(t_{1}-t_{0})}}\log \lambda _{n}(x_{0}).} 3241: 3201: 3161: 1628: 1159: 1119: 892: 517: 3370: 3089:

Figure 4. Attracting and repelling LCSs in the extended phase space of a two-dimensional dynamical system.

2476: 19672: 17616: 12760: 10347: 10239: 9579:

are fastest contracting reduced shrink lines. Determining such shrink lines in a smooth family of nearby

9256: 9145: 9102: 7295:

Second-derivative FTLE ridges are necessarily straight lines and hence do not exist in physical problems.

7023:

Lagrangian objects, such as hyperbolic LCSs. Indeed, a locally strongest repelling material surface over

3480: 2845: 2712: 487: 414: 19137:

Haller, G.; Beron-Vera, F. J. (2012). "Geodesic theory of transport barriers in two-dimensional flows".

17656:

Haller, G.; Yuan, G. (2000). "Lagrangian coherent structures and mixing in two-dimensional turbulence".

16973: 2949: 19652: 17346: 17072: 14979: 14211: 14172: 12808: 12042: 11368: 11272: 11138: 11092: 11046: 10656: 9845: 8348: 8265: 8105: 4476:. One then concludes that the maximum relative stretching experienced along a trajectory starting from 1313: 1059: 992: 17691:

Peacock, T.; Haller, G. (2013). "Lagrangian coherent structures: The hidden skeleton of fluid flows".

16364:

are limit cycles of the reduced shear ODE. Determining such limit cycles in a smooth family of nearby

5265: 5118: 3577: 3548: 3315: 19224:

Haller, G.; Beron-Vera, F. J. (2013). "Coherent Lagrangian vortices: The black holes of turbulence".

16187: 15881: 12506: 10131: 9398: 8887: 8671: 3755:{\displaystyle {\xi }_{\epsilon }(t_{1};x_{0})=\nabla F_{t_{0}}^{t_{1}}(x_{0})\epsilon {\xi }(t_{0})} 163: 19618: 18435: 18280: 1398: 1346: 1289: 1218: 1095: 1025: 862: 557: 139: 129:

Figure 1: An invariant manifold in the extended phase space, formed by an evolving material surface.

19449: 18139: 18074:"Lagrangian coherent structures are associated with fluctuations in airborne microbial populations" 17606: 16555: 14494:{\displaystyle {\bar {\omega }}(t)={\frac {\int _{U(t)}\omega (x,t)\,dV}{\mathrm {vol} \,(U(t))}},} 13669: 13195: 9610: 2401: 1250: 1244: 1050: 1022:

should exert a sustained and consistent action on nearby trajectories throughout the time interval

444: 19014:

Haller, George (2016). "Dynamic rotation and stretch tensors from a dynamic polar decomposition".

9749:

strip. Specifically, the geodesic theory searches for LCSs as special material lines around which

7229:. Nonetheless, evolving second-derivative FTLE ridges computed over sliding intervals of the form 4860:

Therefore, one expects hyperbolic LCSs to appear as codimension-one local maximizing surfaces (or

2896: 19657: 19505: 19465:"Development of an Efficient and Flexible Pipeline for Lagrangian Coherent Structure Computation" 14162:{\displaystyle {\dot {\beta }}=\Phi _{t}^{t_{0}}{\bar {W}}\left(t\right)\Phi _{t_{0}}^{t}\beta .} 6762: 3512: 16902: 16453: 11196:

Figure 10b. Elliptic LCSs revealed by closed level curves of the PRA distribution in the steady

8453:

is pointwise maximal (minimal) with respect to perturbations of the initial normal vector field

19613: 18430: 18275: 18134: 17957:"Drifter motion in the Gulf of Mexico constrained by altimetric Lagrangian coherent structures" 14872: 13798:

As a spatially independent rigid-body rotation, the proper orthogonal relative rotation tensor

12799: 9142:. Attracting LCSs are obtained as most attracting stretch lines, starting from local minima of 6816: 4866:) of the FTLE field. This expectation turns out to be justified in the majority of cases: time 3284: 2360:

is automatically frame-invariant. In contrast, an LCS approach depending on the eigenvalues of

1692: 17212: 17138: 17118: 17048: 16433: 16409: 16397: 16230: 16160: 16113: 9441: 9371: 9324: 2439: 19667: 19322:"Zonal Jets as Meridional Transport Barriers in the Subtropical and Polar Lower Stratosphere" 17915:"Surface coastal circulation patterns by in-situ detection of Lagrangian coherent structures" 13869:

is dynamically consistent, serving as the deformation gradient of the relative rotation flow

7130: 14770: 2518: 2363: 2328: 2293: 1805: 1770: 19605: 19568: 19333: 19286: 19243: 19146: 19103: 19033: 18975: 18911: 18855: 18795: 18758: 18716: 18645: 18602: 18545: 18498: 18397: 18358: 18312: 18267: 18224: 18189: 18085: 18046: 18009: 17968: 17926: 17867: 17828: 17793: 17758: 17700: 17665: 17611: 17495: 17468: 17185: 17158: 13183:{\displaystyle \nabla F_{t_{0}}^{t}=O_{t_{0}}^{t}M_{t_{0}}^{t}=N_{t_{0}}^{t}O_{t_{0}}^{t},} 13003: 12976: 12903: 12733: 11345: 11318: 8547: 8456: 7731: 7478: 7451: 6789: 5173: 4969: 4869: 4479: 3765: 3453: 1566: 18530: 17996:

Huhn, F.; von Kameke, A.; Pérez-Muñuzuri, V.; Olascoaga, M. J.; Beron-Vera, F. J. (2012).

16367: 16347: 16140: 16093: 14741: 9602: 9582: 9562: 9351: 9304: 9099:

Repelling LCSs are obtained as most repelling shrink lines, starting from local maxima of

9082:{\displaystyle \left\langle \nabla \times \xi _{3}(x_{0}),\xi _{3}(x_{0})\right\rangle =0} 8866:{\displaystyle \left\langle \nabla \times \xi _{1}(x_{0}),\xi _{1}(x_{0})\right\rangle =0} 2946:

precisely by upholding that trajectories are mapped into trajectories, i.e., by requiring

1718: 1663: 8: 19662: 18579: 10474: 3097:

is by requiring it to be a locally strongest attracting material surface in the extended

19609: 19572: 19533: 19337: 19290: 19247: 19150: 19107: 19037: 18979: 18915: 18859: 18799: 18762: 18720: 18707:

Haller, G. (2011). "A variational theory of hyperbolic Lagrangian Coherent Structures".

18649: 18606: 18549: 18502: 18401: 18362: 18316: 18271: 18228: 18193: 18089: 18050: 18013: 17972: 17930: 17871: 17832: 17797: 17770: 17762: 17704: 17669: 15733:

the criteria for two- and three-dimensional elliptic LCSs can be summarized as follows:

12757:

positions of elliptic LCSs are visualized as tubular level sets of the PRA distribution

19631: 19584: 19259: 19233: 19206: 19188: 19119: 19093: 19049: 19023: 18991: 18965: 18927: 18901: 18871: 18845: 18486: 18458: 18240: 18162: 17890: 17855: 12537: 10409: 7359: 7002: 3060: 3040: 3020: 2876: 2856: 1765: 1747: 1547: 1430: 1389: 1201: 728: 18289: 17677: 17248: 14880: 14818: 14504: 12930: 10427: 8302: 8138: 7405: 7301: 7232: 7072: 7026: 6859: 19497: 19487: 19302: 19263: 18811: 18684: 18618: 18448: 18328: 18152: 18111: 17895: 17537: 17338: 17331: 9839: 5054: 4693: 3147: 2287: 1212: 1197: 19588: 19210: 19123: 19053: 18995: 18931: 18875: 18349:

Haller, G. (2002). "Lagrangian coherent structures from approximate velocity data".

18244: 17465:, and hence no infinitesimal deformation takes place between the two time instances 6842: 19635: 19623: 19576: 19479: 19341: 19294: 19251: 19198: 19154: 19111: 19041: 18983: 18919: 18863: 18803: 18766: 18724: 18676: 18653: 18610: 18553: 18506: 18462: 18440: 18405: 18366: 18320: 18285: 18232: 18197: 18166: 18144: 18101: 18093: 18058: 18054: 18017: 17976: 17934: 17885: 17875: 17836: 17801: 17766: 17708: 17673: 3762:. Then the maximum relative stretching of infinitesimal perturbations at the point 3153: 1054: 885: 97: 19366: 18324: 18037:

Peng, J.; Peterson, R. (2012). "Attracting structures in volcanic ash transport".

12039:

For three-dimensional flows, the PRA can again be computed from the invariants of

8496:

Hyperbolic LCS conditions from local variational theory in dimensions n=2 and n=3

2325:, and hence an LCS approach depending only on the eigenvalues and eigenvectors of 19557:"Predicting transport by Lagrangian coherent structures with a high-order method" 19483: 18680: 15737:

Ellipitic LCS conditions from local variational theory in dimensions n=2 and n=3

13656:{\displaystyle {\dot {b}}=O_{t}^{t_{0}}S\left(x(t;x_{0}),t\right)O_{t_{0}}^{t}b.} 9188: 40:

It may require cleanup to comply with Knowledge's content policies, particularly

19158: 18987: 18923: 18867: 18771: 18746: 18728: 18657: 18510: 17840: 17805: 14767:

denoting the (possibly time-varying) domain of definition of the velocity field

19396: 19045: 18444: 17552: 17230: 16427: 16072:{\displaystyle \langle \nabla \times n_{\pm }(x_{0}),n_{\pm }(x_{0})\rangle =0} 8492:

in the three-dimensional case. All these results can be summarized as follows:

769: 481: 19627: 19580: 18384:

Kasten, J.; Petz, C.; Hotz, I.; Hege, H. C.; Noack, B. R.; Tadmor, G. (2010).

18236: 11043:

the local material straining described by the eigenvalues and eigenvectors of

2792:{\displaystyle q={\frac {1}{2}}({\vert S\vert }^{2}-{\vert W\vert }^{2})<0} 19646: 19501: 19470:. In Peer-Timo Bremer; Ingrid Hotz; Valerio Pascucci; Ronald Peikert (eds.). 18148: 17341: 7603: 7543: 7383:

In particular, a broadly referenced material flux formula for FTLE ridges is

6751: 4993:

positions of attracting LCSs are marked by ridges of the backward FTLE field

1455: 19415: 17880: 14873:

Stretching-based coherence from a local variational approach: Shear surfaces

12973:, these piecewise best fits do not form a family of rigid-body rotations as 12800:

Rotational coherence from the Lagrangian-averaged vorticity deviation (LAVD)

11365:. For two-dimensional flows, the PRA can be computed from the invariants of 4591:{\displaystyle \delta _{t_{0}}^{t_{1}}(x_{0})={\sqrt {\lambda _{n}(x_{0})}}} 3085: 19556: 19346: 19321: 19306: 18815: 18622: 18332: 18115: 17899: 17601: 16398:

Stretching-based coherence from a global variational approach: lambda-lines

13030: 10415: 93: 17065:-lines. Remarkably, they are initial positions of material lines that are 7397: 3238:

grow locally at the highest rate relative to trajectories starting off of

216:, consider a non-autonomous dynamical system defined through the flow map 19255: 19115: 18022: 17997: 17981: 17956: 17939: 17914: 17010:

for shearlines obtained above from the local variational theory of LCSs.

6850: 3142:, as they provide a finite-time generalization of the classic concept of 3098: 1393: 732: 134: 19320:

Beron-Vera, F. J.; Olascoaga, M. A. J.; Brown, M. G.; Koçak, H. (2012).

17583: 17559: 11184: 8480:. As earlier, we refer to repelling and attracting LCSs collectively as 18106: 17596: 17566: 12583:

is the eigenvector corresponding to the unit eigenvector of the matrix

9603:

Global variational approach: Shrink- and stretchlines as null-geodesics

19381: 19298: 19202: 18807: 18614: 18557: 18410: 18385: 18370: 18201: 18097: 17712: 17576: 9348:. The intersection curve of a 2D repelling LCS surface with the plane 8017:

Figure 7. Linearized flow geometry along an evolving material surface.

5057:

is solving a linear differential equation for the linearized flow map

4966:. By applying the same argument in backward time, we obtain that time 4852: 1204:

satisfying such an attraction property over infinite times are called

852:{\displaystyle {\mathcal {M}}(t)=F_{t_{0}}^{t}({\mathcal {M}}(t_{0}))} 17995: 17913:

Nencioli, F.; d'Ovidio, F.; Doglioli, A. M.; Petrenko, A. A. (2011).

8295:

strictly attracts nearby trajectories along its normal directions. A

7935:, the advected normal also develops a tangential component of length 5197:

of a grid of initial conditions. Using the coordinate representation

1205: 966:{\displaystyle {\mathcal {M}}\in {\mathcal {P}}\times {\mathcal {I}}} 18073: 17912: 8135:

strictly repels nearby trajectories by the end of the time interval

1715:

proper orthogonal matrix representing time-dependent rotations; and

19098: 19028: 18970: 17249:

Diagnostic approach: Finite-time Lyapunov exponents (FTLE) trenches

16388: 13862:{\displaystyle \Phi _{t_{0}}^{t}=\partial _{\alpha _{0}}\alpha (t)} 13789:{\displaystyle O_{t_{0}}^{t}=\Phi _{t_{0}}^{t}\Theta _{t_{0}}^{t}.} 11197: 4465:{\displaystyle C_{t_{0}}^{t_{1}}=\left^{T}\nabla F_{t_{0}}^{t_{1}}} 2831: 125: 19238: 19193: 18906: 18850: 17332:

Global variational approach: Heteroclinic chains of null-geodesics

17209:. As an example, Fig. 13 shows elliptic LCSs identified as closed 14805: 10420:

As a simplest approach to rotational coherence, one may define an

76: 18302: 17953: 16137:. Again, the intersection curves of elliptic LCSs with the plane 13039: 3093:

Motivated by the above discussion, the simplest way to define an

1156:

are carried by the flow into even smaller final perturbations to

302:{\displaystyle F_{t_{0}}^{t}\colon x_{0}\mapsto x(t,t_{0},x_{0})} 19555:

Salman, H.; Hesthaven, J. S.; Warburton, T.; Haller, G. (2006).

19430: 17856:"Forecasting sudden changes in environmental pollution patterns" 14247:

around its own axis of rotation is dynamically consistent. This

9831:

defined by the deformation field—hence the name of this theory.

6813:

coordinate direction. For two-dimensional flows, only the first

3154:

Diagnostic approach: Finite-time Lyapunov exponent (FTLE) ridges

1243:(i.e., locally unique) material surfaces. This follows from the 18131:

2007 IEEE Conference on Computer Vision and Pattern Recognition

17254: 16333:{\displaystyle x_{0}^{\prime }=n_{\pm }(x_{0})\times n_{\Pi },} 11804:

which yield a four-quadrant version of the PRA via the formula

9546:{\displaystyle x_{0}^{\prime }=\xi _{3}(x_{0})\times n_{\Pi },} 2059:

transform under Euclidean changes of frame into the quantities

1535:{\displaystyle v=v(x,t),\qquad x\in U\subset \mathbb {R} ^{3},} 19554: 19083: 11192: 4685:{\displaystyle (\log {\delta _{t_{0}}^{t_{1}}})/(t_{1}-t_{0})} 1367:, then so will any sufficiently close other material surface. 19319: 19276: 19174: 19172: 19170: 19168: 14864: 14305:

is also objective, and turns out to equal to one half of the

12805:

full-time interval of the same deformation. Therefore, while

8013: 4862: 1375:

material surfaces that exhibit a coherence-inducing property

18670: 17821:

Communications in Nonlinear Science and Numerical Simulation

7380:

Lagrangian and the flux through them is generally not small.

889:(see Fig. 1). Since any choice of the initial condition set 16545:{\displaystyle {\frac {1}{2}}(C_{t_{0}}^{t_{1}}-\lambda I)} 14055:{\displaystyle \Theta _{t_{0}}^{t}=D_{\beta _{0}}\beta (t)} 19472:

Topological Methods in Data Analysis and Visualization III

19165: 18215:

Haller, G. (2005). "An objective definition of a vortex".

13378:

is the deformation gradient of the purely rotational flow

9827:). Such LCSs turn out to be null-geodesics of appropriate 6992:{\displaystyle \mathrm {FTLE} _{t_{0}+T}^{t_{1}+T}(x_{0})} 989:

In order to create a coherent pattern, a material surface

19217: 19130: 18673:

Topological Methods in Data Analysis and Visualization II

18531:"Lagrangian coherent structures in n-dimensional systems" 17812: 17531: 14923:

of interest. This means that at initial point each point

13535:

is the deformation gradient of the purely straining flow

7860:. Therefore, in addition to a normal component of length 3574:. This perturbation generally grows along the trajectory 1383: 19463:

Ameli, Siavash; Desai, Yogin; Shadden, Shawn C. (2014).

19279:

Chaos: An Interdisciplinary Journal of Nonlinear Science

18895: 18788:

Chaos: An Interdisciplinary Journal of Nonlinear Science

18595:

Chaos: An Interdisciplinary Journal of Nonlinear Science

18078:

Chaos: An Interdisciplinary Journal of Nonlinear Science

18065: 13991:

In contrast, the proper orthogonal mean rotation tensor

10416:

Rotational coherence from the polar rotation angle (PRA)

9815:

variability either in the material-line averaged shear (

2286:

A Euclidean frame change is, therefore, equivalent to a

1343:

attracts all nearby trajectories over the time interval

981: 976: 18071: 17847: 17783: 17777: 17458:{\displaystyle \lambda _{1}(x_{0})=\lambda _{2}(x_{0})} 16406:

Figure 13. Nested family of elliptic LCSs, obtained as

13455:{\displaystyle {\dot {a}}=W\left(x(t;x_{0}),t\right)a,} 11315:

for a volume element centered at the initial condition

11204:

In two and three dimensions, therefore, there exists a

10792:

Since the Cauchy–Green strain tensor can be written as

8255:{\displaystyle \rho _{t_{0}}^{t_{1}}(x_{0},n_{0})<1} 8095:{\displaystyle \rho _{t_{0}}^{t_{1}}(x_{0},n_{0})>1} 7542:(cf. Fig. 6). By the invariance of material lines, the 7398:

Local variational approach: Shrink and stretch surfaces

18072:

Tallapragada, P.; Ross, S. D.; Schmale, D. G. (2011).

14062:

is the deformation gradient of the mean-rotation flow

11269:

that characterises the material rotation generated by

10079: 6851:

Issues with inferring hyperbolic LCSs from FTLE ridges

5373: 4959:{\displaystyle \mathrm {FTLE} _{t_{0}}^{t_{1}}(x_{0})} 2600: 33:

A major contributor to this article appears to have a

18030: 17749:

Haller, G. (2015). "Lagrangian Coherent Structures".

17651: 17649: 17647: 17645: 17643: 17641: 17639: 17637: 17498: 17471: 17399: 17349: 17263: 17215: 17188: 17161: 17141: 17121: 17075: 17051: 17019: 16976: 16931: 16905: 16586: 16558: 16487: 16456: 16436: 16412: 16370: 16350: 16269: 16233: 16190: 16163: 16143: 16116: 16096: 15995: 15928: 15884: 15837: 15794: 15751: 15435: 15137: 15072: 15015: 14982: 14929: 14883: 14821: 14773: 14744: 14553: 14507: 14381: 14319: 14256: 14214: 14175: 14070: 13997: 13877: 13804: 13713: 13672: 13543: 13470: 13386: 13313: 13241: 13198: 13057: 13006: 12979: 12933: 12906: 12857: 12811: 12763: 12736: 12589: 12546: 12509: 12093: 12045: 11812: 11419: 11371: 11348: 11321: 11275: 11213: 11141: 11095: 11049: 10798: 10709: 10659: 10485: 10430: 10350: 10289: 10242: 10181: 10134: 9896: 9848: 9788: 9755: 9722: 9685: 9652: 9613: 9585: 9565: 9482: 9444: 9401: 9374: 9354: 9327: 9307: 9259: 9198: 9148: 9105: 9001: 8934: 8890: 8785: 8718: 8674: 8627: 8584: 8550: 8510: 8459: 8384: 8351: 8305: 8268: 8187: 8141: 8108: 8027: 7941: 7866: 7826: 7761: 7734: 7669: 7612: 7551: 7508: 7481: 7454: 7408: 7362: 7304: 7235: 7166: 7133: 7075: 7029: 7005: 6920: 6862: 6819: 6792: 6765: 5323: 5268: 5203: 5176: 5121: 5063: 4999: 4972: 4899: 4872: 4704: 4604: 4509: 4482: 4348: 3797: 3768: 3635: 3580: 3551: 3515: 3483: 3456: 3415: 3373: 3318: 3287: 3244: 3204: 3164: 3106: 3063: 3043: 3023: 2952: 2899: 2879: 2859: 2805: 2721: 2558: 2521: 2479: 2442: 2404: 2366: 2331: 2296: 2065: 1843: 1808: 1773: 1750: 1721: 1695: 1666: 1631: 1575: 1550: 1476: 1433: 1401: 1349: 1316: 1292: 1253: 1221: 1162: 1122: 1098: 1062: 1028: 995: 935: 895: 865: 778: 740: 586: 560: 520: 490: 447: 417: 352: 315: 222: 166: 142: 18528: 18524: 18522: 18520: 18484: 18480: 18478: 18476: 18474: 18472: 18386:"Lagrangian feature extraction of the cylinder wake" 17947: 16227:

of the LCS. As a consequence, an intersection curve

15429:

and the three-dimensional shear normal vector field

15066:

is the plane along which the local Lagrangian shear

9438:

of the LCS. As a consequence, an intersection curve

8003:{\displaystyle \sigma _{t_{0}}^{t_{1}}(x_{0},n_{0})} 7502:

denote a unit normal to an initial material surface

4893:

positions of repelling LCSs are marked by ridges of

18529:Lekien, F.; Shadden, S. C.; Marsden, J. E. (2007). 17906: 15976:{\displaystyle x_{0}^{\prime }=\eta ^{\pm }(x_{0})} 15131:Introducing the two-dimensional shear vector field 10774:{\displaystyle U_{t_{0}}^{t_{1}},V_{t_{0}}^{t_{1}}} 9712:variation within such a strip. The two-dimensional 7222:{\displaystyle \mathrm {FTLE} _{t_{0}+T}^{t_{1}+T}} 3127:{\displaystyle {\mathcal {P}}\times {\mathcal {I}}} 3037:-frame then yields the transformed equation in the 761:{\displaystyle {\mathcal {P}}\times {\mathcal {I}}} 18383: 17634: 17511: 17484: 17457: 17381: 17320: 17221: 17201: 17174: 17147: 17127: 17107: 17057: 17037: 17002: 16962: 16917: 16889: 16570: 16544: 16468: 16442: 16418: 16376: 16356: 16332: 16255: 16219: 16176: 16149: 16129: 16102: 16071: 15975: 15913: 15863: 15820: 15777: 15723: 15421: 15120: 15058: 15001: 14968: 14915: 14853: 14794: 14759: 14728: 14539: 14493: 14367: 14297: 14239: 14200: 14161: 14054: 13981: 13861: 13788: 13697: 13655: 13527: 13454: 13370: 13291: 13223: 13182: 13019: 12992: 12965: 12919: 12892: 12843: 12788: 12749: 12719: 12575: 12528: 12495: 12077: 12029: 11794: 11403: 11357: 11334: 11307: 11261: 11173: 11127: 11081: 11035: 10773: 10691: 10645: 10462: 10379: 10336: 10271: 10228: 10163: 10118: 9880: 9807: 9774: 9741: 9704: 9671: 9646:. The averaged strain and shear within a strip of 9638: 9591: 9571: 9545: 9466: 9430: 9387: 9360: 9340: 9313: 9288: 9245: 9177: 9134: 9081: 8981: 8919: 8865: 8765: 8703: 8653: 8610: 8568: 8536: 8472: 8446:{\displaystyle \rho _{t_{0}}^{t_{1}}(x_{0},n_{0})} 8445: 8370: 8337: 8287: 8254: 8173: 8127: 8094: 8002: 7928:{\displaystyle \rho _{t_{0}}^{t_{1}}(x_{0,}n_{0})} 7927: 7852: 7812: 7747: 7720: 7655: 7594: 7534: 7494: 7467: 7440: 7368: 7348: 7279: 7221: 7152: 7119: 7061: 7011: 6991: 6906: 6831: 6805: 6778: 6741: 5309: 5254: 5189: 5162: 5107: 5042: 4985: 4958: 4885: 4842: 4684: 4590: 4495: 4464: 4332: 3781: 3754: 3621: 3566: 3537: 3501: 3469: 3442: 3401: 3359: 3304: 3270: 3230: 3190: 3126: 3069: 3049: 3029: 3009: 2938: 2885: 2865: 2819: 2791: 2711:which is an exact solution of the two-dimensional 2701: 2542: 2507: 2465: 2428: 2387: 2352: 2317: 2276: 2051: 1829: 1794: 1756: 1736: 1707: 1681: 1652: 1617: 1556: 1534: 1439: 1411: 1359: 1335: 1302: 1278: 1231: 1188: 1148: 1108: 1081: 1038: 1014: 965: 921: 875: 851: 760: 717: 570: 546: 506: 472: 433: 404:{\displaystyle x(t,t_{0},x_{0})\in {\mathcal {P}}} 403: 338: 301: 208: 152: 19595: 19462: 19178: 18740: 18738: 18517: 18469: 14368:{\displaystyle \omega (x,t)=\nabla \times v(x,t)} 13528:{\displaystyle M_{t_{0}}^{t}=\nabla _{b_{0}}b(t)} 13371:{\displaystyle O_{t_{0}}^{t}=\nabla _{a_{0}}a(t)} 12576:{\displaystyle \mathbf {e} =\left\{e_{k}\right\}} 9679:-close material lines, therefore, typically show 5317:, we approximate the gradient of the flow map as 3450:be a small perturbation to the initial condition 1380:simulation of 2D turbulence are shown in Fig.2a. 19644: 19409: 19223: 19136: 18951: 18949: 18947: 18945: 18943: 18941: 18891: 18889: 18887: 18885: 18835: 18833: 18831: 18829: 18827: 18825: 17818: 4185: 3982: 3881: 3855: 2398:A number of frame-dependent quantities, such as 1049:Most such properties can be expressed by strict 18955: 18839: 18785: 18744: 18592: 18122: 17989: 17860:Proceedings of the National Academy of Sciences 17853: 10337:{\displaystyle x_{0}^{\prime }=\xi _{2}(x_{0})} 10229:{\displaystyle x_{0}^{\prime }=\xi _{1}(x_{0})} 9246:{\displaystyle x_{0}^{\prime }=\xi _{1}(x_{0})} 8982:{\displaystyle x_{0}^{\prime }=\xi _{1}(x_{0})} 8766:{\displaystyle x_{0}^{\prime }=\xi _{2}(x_{0})} 7813:{\displaystyle \nabla F_{t_{0}}^{t_{1}}(x_{0})} 7721:{\displaystyle \nabla F_{t_{0}}^{t_{1}}(x_{0})} 7298:FTLE ridges computed over sliding time windows 6914:do not form material surfaces. Thus, ridges of 5043:{\displaystyle \mathrm {FTLE} _{t_{1}}^{t_{0}}} 19313: 19270: 19079: 19077: 19075: 19073: 19071: 19069: 19067: 19065: 19063: 19016:Journal of the Mechanics and Physics of Solids 19009: 19007: 19005: 18735: 18675:. Mathematics and Visualization. p. 221. 18569: 18567: 16426:-lines, forming transport barriers around the 15121:{\displaystyle \sigma _{t_{0}}^{t_{1}}(x_{0})} 15059:{\displaystyle T_{x_{0}}{\mathcal {M}}(t_{0})} 14969:{\displaystyle x_{0}\in {\mathcal {M}}(t_{0})} 13982:{\displaystyle {\dot {\alpha }}=\left\alpha .} 11342:. This PRA is well-defined up to multiples of 11262:{\displaystyle \theta _{t_{0}}^{t_{1}}(x_{0})} 7656:{\displaystyle T_{x_{1}}{\mathcal {M}}(t_{1})} 7595:{\displaystyle T_{x_{0}}{\mathcal {M}}(t_{0})} 6839:minor matrix of the above matrix is relevant. 3158:Heuristically, one may seek initial positions 1660:is the vector of the transformed coordinates; 16:Distinguished surfaces of dynamic trajectories 18938: 18882: 18822: 18779: 18702: 18700: 18664: 18635: 18573: 18418: 18344: 18342: 17744: 17742: 17690: 16963:{\displaystyle \eta _{\lambda }^{\pm }(x_{0}} 16393:time (on the right). Image: Daniel Blazevski. 10703:and the symmetric, positive definite tensors 1565:a consequence, LCSs are subject to the basic 1116:if all small enough initial perturbations to 19561:Theoretical and Computational Fluid Dynamics 19360: 18586: 18036: 17740: 17738: 17736: 17734: 17732: 17730: 17728: 17726: 17724: 17722: 16060: 15996: 11705: 11647: 11527: 11469: 7728:. At the same time, the image of the normal 3281:The growth of an infinitesimal perturbation 2814: 2806: 2770: 2764: 2749: 2743: 712: 597: 19060: 19002: 18629: 18564: 18173: 17321:{\displaystyle FTLE_{t_{0}}^{t_{1}}(x_{0})} 13292:{\displaystyle M_{t_{0}}^{t},N_{t_{0}}^{t}} 9819:) or in the material-line averaged strain ( 7069:will generally not play the same role over 5108:{\displaystyle \nabla F_{t_{0}}^{t}(x_{0})} 1427:, the LCSs of the system over the interval 18697: 18377: 18339: 18296: 18208: 17655: 15735: 8494: 4692:, which is then precisely the finite-time 19617: 19345: 19237: 19192: 19097: 19027: 18969: 18905: 18849: 18770: 18434: 18427:2014 IEEE Pacific Visualization Symposium 18424: 18409: 18279: 18138: 18105: 18021: 17980: 17938: 17889: 17879: 17719: 14716: 14466: 14446: 11929: 11863: 11769: 11768: 11761: 11760: 11568: 11567: 11566: 11559: 11558: 9808:{\displaystyle {\mathcal {O}}(\epsilon )} 9775:{\displaystyle {\mathcal {O}}(\epsilon )} 9742:{\displaystyle {\mathcal {O}}(\epsilon )} 9705:{\displaystyle {\mathcal {O}}(\epsilon )} 9672:{\displaystyle {\mathcal {O}}(\epsilon )} 3554: 2813: 2809: 1640: 1567:objectivity (material frame-indifference) 1519: 635: 64:Learn how and when to remove this message 19410:Jimenez, Raymond; Vankerschaver, Joris. 17521: 16450:. Also shown is the perfectly coherent ( 16401: 16387: 14863: 14804: 14298:{\displaystyle \psi _{t_{0}}^{t}(x_{0})} 13038: 13029: 12893:{\displaystyle \nabla F_{t_{0}}^{t_{1}}} 11191: 11183: 9187: 8012: 7127:and hence its evolving position at time 6841: 6750: 4851: 3084: 2830: 1454: 980: 124: 75: 18128: 17038:{\displaystyle \eta _{\lambda }^{\pm }} 14307:Lagrangian-averaged vorticity deviation 3144:normally hyperbolic invariant manifolds 339:{\displaystyle x_{0}\in {\mathcal {P}}} 19645: 19013: 18706: 18348: 18257: 18214: 18179: 17748: 17532:Software packages for LCS computations 14208:implies that the total angle swept by 8490:Frobenius-type integrability condition 3443:{\displaystyle \epsilon {\xi }(t_{0})} 2820:{\displaystyle \vert \,\cdot \,\vert } 1464: 1384:LCSs vs. classical invariant manifolds 1310:. For instance, if a material surface 772:, we refer to the evolving time slice 115: 17854:Olascoaga, M. J.; Haller, G. (2012). 16090:denote the unit normal of a 2D plane 15864:{\displaystyle {\mathcal {M}}(t_{0})} 15821:{\displaystyle {\mathcal {M}}(t_{0})} 15778:{\displaystyle {\mathcal {M}}(t_{0})} 8654:{\displaystyle {\mathcal {M}}(t_{0})} 8611:{\displaystyle {\mathcal {M}}(t_{0})} 8537:{\displaystyle {\mathcal {M}}(t_{0})} 8345:can be defined as a material surface 8102:, then the evolving material surface 7853:{\displaystyle {\mathcal {M}}(t_{1})} 7535:{\displaystyle {\mathcal {M}}(t_{0})} 5255:{\displaystyle x=(x^{1},x^{2},x^{3})} 3545:denoting an arbitrary unit vector in 3367:is governed by the flow map gradient 3271:{\displaystyle {\mathcal {M}}(t_{0})} 3231:{\displaystyle {\mathcal {M}}(t_{0})} 3191:{\displaystyle {\mathcal {M}}(t_{0})} 1653:{\displaystyle y\in \mathbb {R} ^{3}} 1189:{\displaystyle {\mathcal {M}}(t_{1})} 1149:{\displaystyle {\mathcal {M}}(t_{0})} 977:LCSs as exceptional material surfaces 922:{\displaystyle {\mathcal {M}}(t_{0})} 547:{\displaystyle {\mathcal {M}}(t_{0})} 19361:Lekien, Francois; Coulliette, Chad. 17067:infinitesimally uniformly stretching 7820:generally does not remain normal to 3402:{\displaystyle \nabla F_{t_{0}}^{t}} 2508:{\displaystyle \nabla F_{t_{0}}^{t}} 1239:, strict inequalities do not define 1215:defined over a finite time interval 120: 18: 19443: 19394: 19326:Journal of the Atmospheric Sciences 17771:10.1146/annurev-fluid-010313-141322 16970:coincides with the direction field 13192:where the proper orthogonal tensor 12789:{\displaystyle \theta _{t_{0}}^{t}} 10653:where the proper orthogonal tensor 10380:{\displaystyle \lambda _{1}(x_{0})} 10272:{\displaystyle \lambda _{2}(x_{0})} 9289:{\displaystyle \lambda _{2}(x_{0})} 9178:{\displaystyle \lambda _{1}(x_{0})} 9135:{\displaystyle \lambda _{2}(x_{0})} 3502:{\displaystyle 0<\epsilon \ll 1} 2893:-frame through a coordinate change 507:{\displaystyle t\in {\mathcal {I}}} 434:{\displaystyle t\in {\mathcal {I}}} 13: 19519: 19379: 18745:Farazmand, M.; Haller, G. (2012). 17135:-line is stretched by a factor of 17003:{\displaystyle \eta ^{\pm }(x_{0}} 16371: 16351: 16340:whose trajectories we refer to as 16322: 16280: 16169: 16144: 16122: 16097: 15999: 15939: 15840: 15797: 15754: 15035: 14985: 14945: 14565: 14562: 14559: 14556: 14462: 14459: 14456: 14341: 14216: 14177: 14132: 14087: 13999: 13831: 13806: 13762: 13740: 13497: 13340: 13058: 12858: 12851:is the closest rotation tensor to 12651: 12392: 12324: 12189: 11926: 11923: 11920: 11917: 11860: 11857: 11854: 11851: 11663: 11485: 10876: 10834: 10486: 10406:Kolmogorov–Arnold–Moser (KAM) tori 10344:starting from local minims of the 10300: 10236:starting from local maxima of the 10192: 10068: 10011: 10005: 9791: 9758: 9725: 9688: 9655: 9586: 9566: 9555:whose trajectories we refer to as 9535: 9493: 9380: 9355: 9333: 9308: 9253:starting from a global maximum of 9209: 9007: 8945: 8791: 8729: 8630: 8587: 8513: 8354: 8271: 8111: 7829: 7762: 7670: 7632: 7571: 7511: 7178: 7175: 7172: 7169: 6932: 6929: 6926: 6923: 5324: 5064: 5011: 5008: 5005: 5002: 4911: 4908: 4905: 4902: 4716: 4713: 4710: 4707: 4430: 4387: 4100: 4030: 3680: 3374: 3247: 3207: 3167: 3119: 3109: 3080: 3010:{\displaystyle x(t)=Q(t)y(t)+b(t)} 2480: 2405: 2395:is generally not frame-invariant. 2010: 1983: 1907: 1880: 1404: 1352: 1319: 1295: 1224: 1165: 1125: 1101: 1065: 1031: 998: 958: 948: 938: 898: 868: 825: 781: 753: 743: 691: 630: 620: 589: 563: 523: 499: 426: 396: 331: 169: 145: 14: 19684: 19474:. Mathematics and Visualization. 19428: 17382:{\displaystyle D_{t_{0}}^{t_{1}}} 17236: 17108:{\displaystyle F_{t_{0}}^{t_{1}}} 15002:{\displaystyle {\mathcal {M}}(t)} 14240:{\displaystyle \Phi _{t_{0}}^{t}} 14201:{\displaystyle \Phi _{t_{0}}^{t}} 12844:{\displaystyle R_{t_{0}}^{t_{1}}} 12078:{\displaystyle C_{t_{0}}^{t_{1}}} 11404:{\displaystyle C_{t_{0}}^{t_{1}}} 11308:{\displaystyle R_{t_{0}}^{t_{1}}} 11174:{\displaystyle R_{t_{0}}^{t_{1}}} 11128:{\displaystyle R_{t_{0}}^{t_{1}}} 11082:{\displaystyle C_{t_{0}}^{t_{1}}} 10692:{\displaystyle R_{t_{0}}^{t_{1}}} 10477:of the flow gradient in the form 9881:{\displaystyle D_{t_{0}}^{t_{1}}} 8371:{\displaystyle {\mathcal {M}}(t)} 8288:{\displaystyle {\mathcal {M}}(t)} 8128:{\displaystyle {\mathcal {M}}(t)} 1336:{\displaystyle {\mathcal {M}}(t)} 1082:{\displaystyle {\mathcal {M}}(t)} 1015:{\displaystyle {\mathcal {M}}(t)} 19526: 17751:Annual Review of Fluid Mechanics 17115:. Specifically, any subset of a 16263:satisfies the reduced shear ODE 12927:norm over a fixed time interval 12548: 10398: 5310:{\displaystyle x(t;t_{0},x_{0})} 5163:{\displaystyle x(t;t_{0},x_{0})} 4474:right Cauchy–Green strain tensor 3622:{\displaystyle x(t,t_{0},x_{0})} 3567:{\displaystyle \mathbb {R} ^{n}} 3360:{\displaystyle x(t,t_{0},x_{0})} 92:) are distinguished surfaces of 44:. Please discuss further on the 23: 19456: 19437: 19422: 19403: 19388: 19373: 19354: 18538:Journal of Mathematical Physics 18251: 16220:{\displaystyle n_{\pm }(x_{0})} 15914:{\displaystyle n_{\pm }(x_{0})} 12529:{\displaystyle \epsilon _{ijk}} 12476: 11726: 11548: 10395:in the shape of the oil spill. 10164:{\displaystyle \xi _{i}(x_{0})} 10067: 9834:Shearless LCSs are found to be 9431:{\displaystyle \xi _{3}(x_{0})} 8920:{\displaystyle \xi _{n}(x_{0})} 8704:{\displaystyle \xi _{1}(x_{0})} 7387:, even for straight FTLE ridges 2148: 1946: 1504: 209:{\displaystyle {\mathcal {I}}=} 19139:Physica D: Nonlinear Phenomena 18958:Physica D: Nonlinear Phenomena 18898:Physica D: Nonlinear Phenomena 18842:Physica D: Nonlinear Phenomena 18751:Physica D: Nonlinear Phenomena 18709:Physica D: Nonlinear Phenomena 18638:Physica D: Nonlinear Phenomena 18491:Physica D: Nonlinear Phenomena 18260:Physica D: Nonlinear Phenomena 18059:10.1016/j.atmosenv.2011.05.053 17786:Physica D: Nonlinear Phenomena 17684: 17658:Physica D: Nonlinear Phenomena 17452: 17439: 17423: 17410: 17315: 17302: 16987: 16947: 16881: 16868: 16851: 16838: 16822: 16809: 16794: 16781: 16748: 16735: 16718: 16705: 16689: 16676: 16648: 16635: 16615: 16602: 16539: 16498: 16311: 16298: 16250: 16244: 16214: 16201: 16057: 16044: 16028: 16015: 15970: 15957: 15908: 15895: 15858: 15845: 15815: 15802: 15772: 15759: 15715: 15702: 15683: 15670: 15650: 15637: 15620: 15607: 15587: 15574: 15555: 15542: 15522: 15509: 15492: 15479: 15459: 15446: 15413: 15400: 15383: 15370: 15354: 15341: 15326: 15313: 15287: 15274: 15257: 15244: 15228: 15215: 15194: 15181: 15161: 15148: 15115: 15102: 15053: 15040: 14996: 14990: 14963: 14950: 14910: 14884: 14848: 14822: 14789: 14777: 14754: 14748: 14708: 14702: 14696: 14684: 14675: 14656: 14650: 14607: 14594: 14534: 14508: 14501:the LAVD over a time interval 14482: 14479: 14473: 14467: 14443: 14431: 14423: 14417: 14400: 14394: 14388: 14362: 14350: 14335: 14323: 14292: 14279: 14114: 14049: 14043: 13951: 13928: 13909: 13856: 13850: 13611: 13592: 13522: 13516: 13432: 13413: 13365: 13359: 12960: 12934: 11782: 11770: 11757: 11745: 11739: 11727: 11256: 11243: 10867: 10831: 10457: 10431: 10408:that form elliptic regions in 10393:notable Tiger-Tail instability 10374: 10361: 10331: 10318: 10266: 10253: 10223: 10210: 10158: 10145: 10056: 10043: 10002: 9989: 9939: 9926: 9802: 9796: 9769: 9763: 9736: 9730: 9699: 9693: 9666: 9660: 9524: 9511: 9461: 9455: 9425: 9412: 9283: 9270: 9240: 9227: 9172: 9159: 9129: 9116: 9065: 9052: 9036: 9023: 8976: 8963: 8914: 8901: 8849: 8836: 8820: 8807: 8760: 8747: 8698: 8685: 8648: 8635: 8605: 8592: 8531: 8518: 8440: 8414: 8365: 8359: 8332: 8306: 8282: 8276: 8243: 8217: 8168: 8142: 8122: 8116: 8083: 8057: 7997: 7971: 7922: 7896: 7847: 7834: 7807: 7794: 7715: 7702: 7650: 7637: 7589: 7576: 7529: 7516: 7435: 7409: 7343: 7305: 7274: 7236: 7114: 7076: 7056: 7030: 6986: 6973: 6901: 6863: 6699: 6654: 6638: 6593: 6549: 6504: 6488: 6443: 6399: 6354: 6338: 6293: 6247: 6202: 6186: 6141: 6097: 6052: 6036: 5991: 5947: 5902: 5886: 5841: 5795: 5750: 5734: 5689: 5645: 5600: 5584: 5539: 5495: 5450: 5434: 5389: 5362: 5349: 5304: 5272: 5249: 5210: 5157: 5125: 5102: 5089: 4953: 4940: 4834: 4821: 4799: 4773: 4758: 4745: 4679: 4653: 4645: 4605: 4583: 4570: 4552: 4539: 4316: 4303: 4297: 4284: 4249: 4236: 4210: 4197: 4164: 4151: 4145: 4132: 4094: 4081: 4075: 4062: 4007: 3994: 3963: 3937: 3906: 3893: 3862: 3844: 3831: 3749: 3736: 3725: 3712: 3674: 3648: 3616: 3584: 3532: 3519: 3437: 3424: 3354: 3322: 3299: 3293: 3265: 3252: 3225: 3212: 3185: 3172: 3004: 2998: 2989: 2983: 2977: 2971: 2962: 2956: 2933: 2927: 2915: 2909: 2851:material particle trajectories 2780: 2738: 2589: 2577: 2537: 2525: 2460: 2448: 2423: 2411: 2382: 2370: 2347: 2335: 2312: 2300: 2268: 2262: 2241: 2234: 2225: 2219: 2213: 2201: 2189: 2182: 2173: 2161: 2155: 2142: 2136: 2130: 2118: 2106: 2099: 2090: 2078: 2072: 2032: 2028: 2016: 2007: 2001: 1989: 1962: 1950: 1929: 1925: 1913: 1904: 1898: 1886: 1859: 1847: 1824: 1812: 1789: 1777: 1731: 1725: 1676: 1670: 1609: 1603: 1591: 1585: 1498: 1486: 1412:{\displaystyle {\mathcal {P}}} 1360:{\displaystyle {\mathcal {I}}} 1330: 1324: 1303:{\displaystyle {\mathcal {I}}} 1232:{\displaystyle {\mathcal {I}}} 1183: 1170: 1143: 1130: 1109:{\displaystyle {\mathcal {I}}} 1076: 1070: 1039:{\displaystyle {\mathcal {I}}} 1009: 1003: 916: 903: 876:{\displaystyle {\mathcal {M}}} 846: 843: 830: 820: 792: 786: 709: 696: 683: 677: 665: 639: 612: 600: 571:{\displaystyle {\mathcal {P}}} 541: 528: 388: 356: 296: 264: 258: 203: 177: 153:{\displaystyle {\mathcal {P}}} 86:Lagrangian coherent structures 1: 19544:Lagrangian coherent structure 18325:10.1103/PhysRevLett.98.144502 18290:10.1016/S0167-2789(00)00199-8 17678:10.1016/S0167-2789(00)00142-1 17627: 17538:Finite-Time Lyapunov Exponent 16571:{\displaystyle \lambda >0} 14810:(Image: Alireza Hadjighasem) 13698:{\displaystyle O_{t_{0}}^{t}} 13235:and the non-singular tensors 13224:{\displaystyle O_{t_{0}}^{t}} 11200:. (Image: Mohammad Farazmand) 10279:eigenvalue field. Similarly, 9639:{\displaystyle F_{t_{0}}^{t}} 9296:. (Image: Mohammad Farazmand) 5053:The classic way of computing 3629:into the perturbation vector 3134:(see. Fig. 4) . Similarly, a 2836:(Image: Francisco Beron-Vera) 2429:{\displaystyle \nabla v(x,t)} 1618:{\displaystyle x=Q(t)y+b(t),} 1279:{\displaystyle F_{t_{0}}^{t}} 929:yields an invariant manifold 768:. Borrowing terminology from 473:{\displaystyle F_{t_{0}}^{t}} 309:, mapping initial conditions 19484:10.1007/978-3-319-04099-8_13 18681:10.1007/978-3-642-23175-9_15 18485:Shadden, S. C.; Lekien, F.; 18002:Geophysical Research Letters 17961:Geophysical Research Letters 17919:Geophysical Research Letters 17622:Coherent turbulent structure 13666:The dynamic rotation tensor 13305:right dynamic stretch tensor 5262:for the evolving trajectory 2939:{\displaystyle x=Q(t)y+b(t)} 2844:for particle motion and the 7: 19159:10.1016/j.physd.2012.06.012 18988:10.1016/j.physd.2015.09.007 18924:10.1016/j.physd.2014.03.008 18868:10.1016/j.physd.2014.01.007 18772:10.1016/j.physd.2011.09.013 18729:10.1016/j.physd.2010.11.010 18658:10.1016/j.physd.2012.05.006 18511:10.1016/j.physd.2005.10.007 17841:10.1016/j.cnsns.2014.07.011 17806:10.1016/j.physd.2013.05.003 17617:Eulerian coherent structure 17590: 14169:The dynamic consistency of 13301:left dynamic stretch tensor 7663:by the linearized flow map 6779:{\displaystyle \delta _{i}} 3538:{\displaystyle \xi (t_{0})} 10: 19689: 19598:Journal of Fluid Mechanics 19226:Journal of Fluid Mechanics 19086:Journal of Fluid Mechanics 19046:10.1016/j.jmps.2015.10.002 18445:10.1109/PacificVis.2014.27 18217:Journal of Fluid Mechanics 16918:{\displaystyle \lambda =1} 16469:{\displaystyle \lambda =1} 11206:polar rotation angle (PRA) 8297:repelling (attracting) LCS 1392:are invariant sets in the 1053:. For instance, we call a 514:, then for any smooth set 19628:10.1017/S0022112006003648 19581:10.1007/s00162-006-0031-0 18237:10.1017/S0022112004002526 15875: 15830: 15787: 15744: 15741: 14547:therefore takes the form 8881: 8665: 8620: 8577: 8503: 8500: 6832:{\displaystyle 2\times 2} 3305:{\displaystyle {\xi }(t)} 1708:{\displaystyle 3\times 3} 554:of initial conditions in 160:and over a time interval 18149:10.1109/CVPR.2007.382977 17607:Dynamical systems theory 17222:{\displaystyle \lambda } 17148:{\displaystyle \lambda } 17128:{\displaystyle \lambda } 17058:{\displaystyle \lambda } 16443:{\displaystyle \lambda } 16419:{\displaystyle \lambda } 16256:{\displaystyle x_{0}(s)} 16177:{\displaystyle n_{\Pi }} 16130:{\displaystyle n_{\Pi }} 15128:is maximal (cf. Fig 7). 14249:intrinsic rotation angle 9782:material strips show no 9467:{\displaystyle x_{0}(s)} 9388:{\displaystyle n_{\Pi }} 9341:{\displaystyle n_{\Pi }} 7160:will not be a ridge for 2466:{\displaystyle {W}(y,t)} 18305:Physical Review Letters 18039:Atmospheric Environment 17881:10.1073/pnas.1118574109 17536:Particle advection and 16184:and to the unit normal 15831:Frobenius-type PDE for 15745:Normal vector field of 13233:dynamic rotation tensor 9714:geodesic theory of LCSs 9395:and to the unit normal 8621:Frobenius-type PDE for 8504:Normal vector field of 7153:{\displaystyle t_{0}+T} 2846:Navier–Stokes equations 2713:Navier–Stokes equations 19347:10.1175/JAS-D-11-084.1 18574:Shadden, S.C. (2005). 18396:(9): 091108–091108–1. 17528: 17513: 17486: 17459: 17383: 17322: 17223: 17203: 17176: 17149: 17129: 17109: 17059: 17039: 17004: 16964: 16925:, the direction field 16919: 16891: 16572: 16546: 16477: 16470: 16444: 16420: 16394: 16378: 16358: 16334: 16257: 16221: 16178: 16151: 16131: 16104: 16073: 15977: 15915: 15865: 15822: 15779: 15725: 15423: 15122: 15060: 15003: 14970: 14917: 14869: 14855: 14811: 14796: 14795:{\displaystyle v(x,t)} 14761: 14730: 14541: 14495: 14369: 14299: 14241: 14202: 14163: 14056: 13983: 13863: 13790: 13699: 13657: 13529: 13456: 13372: 13293: 13225: 13184: 13045: 13036: 13021: 12994: 12967: 12921: 12894: 12845: 12790: 12751: 12721: 12577: 12530: 12497: 12168: 12079: 12031: 11796: 11405: 11359: 11336: 11309: 11263: 11201: 11189: 11175: 11129: 11083: 11037: 10775: 10693: 10647: 10464: 10381: 10338: 10273: 10230: 10165: 10120: 9882: 9809: 9776: 9743: 9706: 9673: 9640: 9593: 9573: 9547: 9468: 9432: 9389: 9362: 9342: 9315: 9297: 9290: 9247: 9179: 9136: 9083: 8983: 8921: 8867: 8767: 8705: 8655: 8612: 8570: 8538: 8474: 8447: 8372: 8339: 8289: 8256: 8175: 8129: 8096: 8018: 8004: 7929: 7854: 7814: 7749: 7722: 7657: 7596: 7536: 7496: 7469: 7448:. At an initial point 7442: 7370: 7350: 7281: 7223: 7154: 7121: 7063: 7013: 6993: 6908: 6847: 6833: 6807: 6780: 6756: 6743: 5311: 5256: 5191: 5164: 5109: 5044: 4987: 4960: 4887: 4857: 4844: 4686: 4592: 4497: 4466: 4334: 3783: 3756: 3623: 3568: 3539: 3503: 3471: 3444: 3403: 3361: 3306: 3272: 3232: 3192: 3128: 3090: 3071: 3051: 3031: 3011: 2940: 2887: 2867: 2837: 2821: 2793: 2703: 2544: 2543:{\displaystyle v(x,t)} 2509: 2467: 2430: 2389: 2388:{\displaystyle W(x,t)} 2354: 2353:{\displaystyle S(x,t)} 2319: 2318:{\displaystyle S(x,t)} 2278: 2053: 1831: 1830:{\displaystyle W(x,t)} 1796: 1795:{\displaystyle S(x,t)} 1758: 1738: 1709: 1683: 1654: 1619: 1558: 1536: 1461: 1441: 1413: 1361: 1337: 1304: 1280: 1233: 1190: 1150: 1110: 1083: 1040: 1016: 986: 967: 923: 877: 853: 762: 719: 572: 548: 508: 474: 435: 405: 340: 303: 210: 154: 130: 82: 17525: 17514: 17512:{\displaystyle t_{1}} 17487: 17485:{\displaystyle t_{0}} 17460: 17384: 17323: 17224: 17204: 17202:{\displaystyle t_{1}} 17177: 17175:{\displaystyle t_{0}} 17150: 17130: 17110: 17060: 17040: 17005: 16965: 16920: 16892: 16573: 16547: 16481:strain tensor family 16471: 16445: 16421: 16405: 16391: 16379: 16359: 16335: 16258: 16222: 16179: 16152: 16132: 16105: 16074: 15978: 15916: 15866: 15823: 15780: 15726: 15424: 15123: 15061: 15004: 14971: 14918: 14867: 14856: 14808: 14797: 14762: 14731: 14542: 14496: 14375:and its spatial mean 14370: 14300: 14242: 14203: 14164: 14057: 13984: 13864: 13791: 13700: 13658: 13530: 13457: 13373: 13294: 13226: 13185: 13042: 13033: 13022: 13020:{\displaystyle t_{1}} 12995: 12993:{\displaystyle t_{0}} 12968: 12922: 12920:{\displaystyle L^{2}} 12895: 12846: 12791: 12752: 12750:{\displaystyle t_{0}} 12722: 12578: 12531: 12498: 12148: 12080: 12032: 11797: 11406: 11360: 11358:{\displaystyle 2\pi } 11337: 11335:{\displaystyle x_{0}} 11310: 11264: 11195: 11187: 11176: 11130: 11084: 11038: 10776: 10694: 10648: 10465: 10382: 10339: 10274: 10231: 10166: 10121: 9883: 9810: 9777: 9744: 9707: 9674: 9641: 9594: 9574: 9548: 9469: 9433: 9390: 9363: 9343: 9316: 9291: 9248: 9191: 9180: 9137: 9084: 8984: 8922: 8868: 8768: 8706: 8656: 8613: 8571: 8569:{\displaystyle n=2,3} 8539: 8475: 8473:{\displaystyle n_{0}} 8448: 8373: 8340: 8290: 8257: 8176: 8130: 8097: 8016: 8005: 7930: 7855: 7815: 7750: 7748:{\displaystyle n_{0}} 7723: 7658: 7597: 7537: 7497: 7495:{\displaystyle n_{0}} 7470: 7468:{\displaystyle x_{0}} 7443: 7371: 7351: 7282: 7224: 7155: 7122: 7064: 7014: 6994: 6909: 6845: 6834: 6808: 6806:{\displaystyle x^{i}} 6781: 6754: 6744: 5312: 5257: 5192: 5190:{\displaystyle x_{0}} 5165: 5110: 5045: 4988: 4986:{\displaystyle t_{1}} 4961: 4888: 4886:{\displaystyle t_{0}} 4855: 4845: 4687: 4593: 4498: 4496:{\displaystyle x_{0}} 4467: 4335: 3784: 3782:{\displaystyle x_{0}} 3757: 3624: 3569: 3540: 3504: 3472: 3470:{\displaystyle x_{0}} 3445: 3404: 3362: 3307: 3273: 3233: 3193: 3129: 3088: 3072: 3052: 3032: 3012: 2941: 2888: 2868: 2834: 2822: 2794: 2704: 2545: 2510: 2468: 2431: 2390: 2355: 2320: 2279: 2054: 1832: 1797: 1759: 1739: 1710: 1684: 1655: 1620: 1559: 1537: 1458: 1442: 1414: 1362: 1338: 1305: 1281: 1234: 1191: 1151: 1111: 1084: 1041: 1017: 984: 968: 924: 878: 854: 763: 720: 573: 549: 509: 475: 436: 406: 341: 304: 211: 155: 128: 79: 42:neutral point of view 19478:. pp. 201–215. 19256:10.1017/jfm.2013.391 19116:10.1017/jfm.2016.151 18429:. pp. 277–280. 18023:10.1029/2012GL051246 17982:10.1002/2013GL058624 17940:10.1029/2011GL048815 17612:Spectral submanifold 17496: 17469: 17397: 17347: 17261: 17213: 17186: 17159: 17139: 17119: 17073: 17049: 17017: 16974: 16929: 16903: 16584: 16556: 16485: 16454: 16434: 16410: 16377:{\displaystyle \Pi } 16368: 16357:{\displaystyle \Pi } 16348: 16267: 16231: 16188: 16161: 16150:{\displaystyle \Pi } 16141: 16114: 16103:{\displaystyle \Pi } 16094: 15993: 15926: 15882: 15835: 15792: 15749: 15433: 15135: 15070: 15013: 15009:, the tangent space 14980: 14927: 14881: 14819: 14771: 14760:{\displaystyle U(t)} 14742: 14551: 14505: 14379: 14317: 14254: 14212: 14173: 14068: 13995: 13875: 13802: 13711: 13670: 13541: 13468: 13384: 13311: 13239: 13196: 13055: 13004: 12977: 12931: 12904: 12855: 12809: 12761: 12734: 12587: 12544: 12507: 12091: 12043: 11810: 11417: 11369: 11346: 11319: 11273: 11211: 11139: 11093: 11047: 10796: 10787:right stretch tensor 10707: 10657: 10483: 10475:polar decompositions 10428: 10348: 10287: 10283:are trajectories of 10240: 10179: 10175:are trajectories of 10132: 9894: 9846: 9786: 9753: 9720: 9683: 9650: 9611: 9592:{\displaystyle \Pi } 9583: 9572:{\displaystyle \Pi } 9563: 9557:reduced shrink lines 9480: 9442: 9399: 9372: 9361:{\displaystyle \Pi } 9352: 9325: 9314:{\displaystyle \Pi } 9305: 9257: 9196: 9146: 9103: 8999: 8932: 8888: 8783: 8716: 8672: 8625: 8582: 8548: 8508: 8457: 8382: 8378:whose net repulsion 8349: 8303: 8266: 8185: 8139: 8106: 8025: 7939: 7864: 7824: 7759: 7732: 7667: 7610: 7549: 7506: 7479: 7452: 7406: 7360: 7302: 7233: 7164: 7131: 7073: 7027: 7003: 6918: 6860: 6817: 6790: 6763: 6759:with a small vector 5321: 5266: 5201: 5174: 5119: 5061: 4997: 4970: 4897: 4870: 4702: 4602: 4507: 4480: 4346: 3795: 3766: 3633: 3578: 3549: 3513: 3481: 3454: 3413: 3371: 3316: 3285: 3242: 3202: 3162: 3104: 3061: 3041: 3021: 2950: 2897: 2877: 2857: 2803: 2719: 2556: 2519: 2477: 2440: 2402: 2364: 2329: 2294: 2288:similarity transform 2063: 1841: 1806: 1802:and the spin tensor 1771: 1748: 1737:{\displaystyle b(t)} 1719: 1693: 1682:{\displaystyle Q(t)} 1664: 1629: 1573: 1548: 1474: 1431: 1399: 1347: 1314: 1290: 1251: 1219: 1160: 1120: 1096: 1060: 1026: 993: 933: 893: 863: 776: 738: 584: 558: 518: 488: 445: 415: 350: 346:into their position 313: 220: 164: 140: 19610:2007JFM...572..111G 19573:2007ThCFD..21...39S 19444:Du Toit, Philip C. 19338:2012JAtS...69..753B 19291:2010Chaos..20a7514B 19248:2013JFM...731R...4H 19151:2012PhyD..241.1680H 19108:2016JFM...795..136H 19038:2016JMPSo..86...70H 18980:2016PhyD..315....1F 18916:2014PhyD..278...44F 18860:2014PhyD..273...46B 18800:2012Chaos..22a3128F 18763:2012PhyD..241..439F 18721:2011PhyD..240..574H 18650:2012PhyD..241.1475N 18607:2010Chaos..20a7504L 18550:2007JMP....48f5404L 18503:2005PhyD..212..271S 18402:2010PhFl...22i1108K 18363:2002PhFl...14.1851H 18317:2007PhRvL..98n4502M 18272:2001PhyD..149..248H 18229:2005JFM...525....1H 18194:2001PhFl...13.3365H 18090:2011Chaos..21c3122T 18051:2012AtmEn..48..230P 18014:2012GeoRL..39.6602H 17973:2013GeoRL..40.6171O 17931:2011GeoRL..3817604N 17872:2012PNAS..109.4738O 17833:2015CNSNS..22..964B 17798:2013PhyD..258...47B 17763:2015AnRFM..47..137H 17705:2013PhT....66b..41P 17670:2000PhyD..147..352H 17378: 17301: 17104: 17069:under the flow map 17045:are referred to as 17034: 16946: 16601: 16529: 16342:reduced shear lines 16284: 16157:are normal to both 15943: 15738: 15101: 14976:of an elliptic LCS 14641: 14593: 14278: 14236: 14197: 14152: 14107: 14019: 13826: 13782: 13760: 13735: 13694: 13646: 13580: 13492: 13335: 13288: 13263: 13220: 13176: 13154: 13129: 13107: 13082: 12889: 12840: 12785: 12682: 12616: 12423: 12355: 12295: 12220: 12125: 12074: 12018: 11964: 11898: 11834: 11694: 11613: 11516: 11458: 11411:using the formulas 11400: 11304: 11242: 11170: 11124: 11078: 11029: 11000: 10968: 10939: 10907: 10865: 10827: 10783:left stretch tensor 10770: 10738: 10688: 10639: 10610: 10578: 10549: 10517: 10410:Hamiltonian systems 10304: 10196: 10042: 9988: 9925: 9877: 9635: 9497: 9213: 8949: 8733: 8497: 8413: 8216: 8056: 7970: 7895: 7793: 7701: 7602:is mapped into the 7218: 6972: 5348: 5088: 5039: 4939: 4744: 4643: 4538: 4461: 4418: 4377: 4283: 4131: 4061: 3830: 3789:can be computed as 3711: 3398: 3312:along a trajectory 2504: 1465:Objectivity of LCSs 1390:invariant manifolds 1275: 1202:invariant manifolds 819: 663: 469: 244: 116:General definitions 19673:Flow visualization 19395:Shadden, Shawn C. 18900:. 278–279: 44–57. 18844:. 273–274: 46–62. 17529: 17509: 17482: 17455: 17379: 17350: 17318: 17273: 17257:of the FTLE field 17229:-lines within the 17219: 17199: 17172: 17155:between the times 17145: 17125: 17105: 17076: 17055: 17035: 17020: 17000: 16960: 16932: 16915: 16887: 16587: 16568: 16542: 16501: 16478: 16466: 16440: 16416: 16395: 16374: 16354: 16330: 16270: 16253: 16217: 16174: 16147: 16127: 16100: 16069: 15973: 15929: 15911: 15861: 15818: 15775: 15736: 15721: 15419: 15118: 15073: 15056: 14999: 14966: 14913: 14870: 14851: 14812: 14792: 14757: 14726: 14613: 14554: 14537: 14491: 14365: 14295: 14257: 14237: 14215: 14198: 14176: 14159: 14131: 14086: 14052: 13998: 13979: 13859: 13805: 13786: 13761: 13739: 13714: 13695: 13673: 13653: 13625: 13559: 13525: 13471: 13452: 13368: 13314: 13289: 13267: 13242: 13221: 13199: 13180: 13155: 13133: 13108: 13086: 13061: 13046: 13037: 13017: 12990: 12963: 12917: 12890: 12861: 12841: 12812: 12786: 12764: 12747: 12717: 12654: 12595: 12573: 12538:Levi-Civita symbol 12526: 12493: 12491: 12395: 12327: 12274: 12192: 12104: 12085:from the formulas 12075: 12046: 12027: 11997: 11943: 11877: 11813: 11792: 11790: 11666: 11585: 11488: 11430: 11401: 11372: 11355: 11332: 11305: 11276: 11259: 11214: 11202: 11190: 11171: 11142: 11125: 11096: 11079: 11050: 11033: 11001: 10972: 10940: 10911: 10879: 10837: 10799: 10771: 10742: 10710: 10689: 10660: 10643: 10611: 10582: 10550: 10521: 10489: 10460: 10377: 10334: 10290: 10269: 10226: 10182: 10161: 10116: 10107: 10014: 9960: 9897: 9878: 9849: 9805: 9772: 9739: 9702: 9669: 9636: 9614: 9589: 9569: 9543: 9483: 9474:satisfies the ODE 9464: 9428: 9385: 9368:is normal to both 9358: 9338: 9311: 9298: 9286: 9243: 9199: 9175: 9132: 9079: 8979: 8935: 8917: 8863: 8763: 8719: 8701: 8651: 8608: 8566: 8534: 8495: 8470: 8443: 8385: 8368: 8335: 8299:over the interval 8285: 8252: 8188: 8171: 8125: 8092: 8028: 8019: 8000: 7942: 7925: 7867: 7850: 7810: 7765: 7745: 7718: 7673: 7653: 7592: 7532: 7492: 7465: 7438: 7366: 7346: 7277: 7219: 7167: 7150: 7117: 7059: 7019:cannot be used to 7009: 6989: 6921: 6904: 6848: 6829: 6803: 6776: 6757: 6739: 6730: 5327: 5307: 5252: 5187: 5160: 5105: 5067: 5055:Lyapunov exponents 5040: 5000: 4983: 4956: 4900: 4883: 4858: 4840: 4705: 4682: 4615: 4588: 4510: 4493: 4462: 4433: 4390: 4349: 4330: 4328: 4255: 4225: 4103: 4033: 4022: 3921: 3869: 3802: 3779: 3752: 3683: 3619: 3564: 3535: 3499: 3467: 3440: 3399: 3377: 3357: 3302: 3268: 3228: 3188: 3124: 3091: 3067: 3047: 3027: 3007: 2936: 2883: 2863: 2838: 2817: 2789: 2699: 2687: 2540: 2505: 2483: 2463: 2426: 2385: 2350: 2315: 2274: 2049: 1827: 1792: 1754: 1734: 1705: 1679: 1650: 1615: 1554: 1532: 1462: 1452:dynamical system. 1437: 1409: 1357: 1333: 1300: 1276: 1254: 1229: 1186: 1146: 1106: 1092:over the interval 1079: 1036: 1012: 987: 963: 919: 873: 849: 798: 758: 729:invariant manifold 715: 642: 568: 544: 504: 484:for any choice of 470: 448: 441:. If the flow map 431: 401: 336: 299: 223: 206: 150: 131: 83: 19653:Dynamical systems 19493:978-3-319-04099-8 19299:10.1063/1.3271342 19203:10.1137/140983665 18808:10.1063/1.3690153 18690:978-3-642-23174-2 18615:10.1063/1.3270049 18558:10.1063/1.2740025 18454:978-1-4799-2873-6 18411:10.1063/1.3483220 18390:Physics of Fluids 18371:10.1063/1.1477449 18351:Physics of Fluids 18202:10.1063/1.1403336 18188:(11): 3365–3385. 18182:Physics of Fluids 18158:978-1-4244-1179-5 18098:10.1063/1.3624930 17866:(13): 4738–4743. 17713:10.1063/PT.3.1886 16856: 16855: 16723: 16722: 16496: 16087: 16086: 15690: 15689: 15686: 15653: 15623: 15562: 15561: 15558: 15525: 15495: 15388: 15387: 15262: 15261: 14699: 14486: 14391: 14117: 14080: 13954: 13887: 13553: 13396: 12715: 12471: 12248: 12247: 12141: 11721: 11720: 11543: 11542: 9953: 9840:Lorentzian metric 9097: 9096: 7369:{\displaystyle T} 7012:{\displaystyle T} 6726: 6576: 6426: 6274: 6124: 5974: 5822: 5672: 5522: 5170:from any element 4803: 4694:Lyapunov exponent 4586: 4324: 4184: 4172: 3981: 3880: 3878: 3854: 3148:dynamical systems 3070:{\displaystyle y} 3050:{\displaystyle y} 3030:{\displaystyle x} 2886:{\displaystyle y} 2866:{\displaystyle x} 2842:Newton’s equation 2736: 2568: 2259: 2158: 2075: 1976: 1873: 1757:{\displaystyle 3} 1557:{\displaystyle U} 1440:{\displaystyle I} 1213:dynamical systems 1198:dynamical systems 121:Material surfaces 74: 73: 66: 37:with its subject. 19680: 19639: 19621: 19592: 19530: 19529: 19513: 19512: 19510: 19504:. Archived from 19469: 19460: 19454: 19453: 19448:. Archived from 19441: 19435: 19434: 19426: 19420: 19419: 19414:. Archived from 19407: 19401: 19400: 19392: 19386: 19385: 19382:"LCS MATLAB Kit" 19380:Dabiri, John O. 19377: 19371: 19370: 19365:. Archived from 19358: 19352: 19351: 19349: 19317: 19311: 19310: 19274: 19268: 19267: 19241: 19221: 19215: 19214: 19196: 19176: 19163: 19162: 19134: 19128: 19127: 19101: 19081: 19058: 19057: 19031: 19011: 19000: 18999: 18973: 18953: 18936: 18935: 18909: 18893: 18880: 18879: 18853: 18837: 18820: 18819: 18783: 18777: 18776: 18774: 18742: 18733: 18732: 18704: 18695: 18694: 18668: 18662: 18661: 18633: 18627: 18626: 18590: 18584: 18583: 18578:. Archived from 18571: 18562: 18561: 18535: 18526: 18515: 18514: 18497:(3–4): 271–304. 18482: 18467: 18466: 18438: 18422: 18416: 18415: 18413: 18381: 18375: 18374: 18357:(6): 1851–1861. 18346: 18337: 18336: 18300: 18294: 18293: 18283: 18255: 18249: 18248: 18212: 18206: 18205: 18177: 18171: 18170: 18142: 18126: 18120: 18119: 18109: 18069: 18063: 18062: 18034: 18028: 18027: 18025: 17993: 17987: 17986: 17984: 17951: 17945: 17944: 17942: 17910: 17904: 17903: 17893: 17883: 17851: 17845: 17844: 17827:(1–3): 964–979. 17816: 17810: 17809: 17781: 17775: 17774: 17746: 17717: 17716: 17688: 17682: 17681: 17653: 17551:LCS MATLAB Kit ( 17518: 17516: 17515: 17510: 17508: 17507: 17491: 17489: 17488: 17483: 17481: 17480: 17464: 17462: 17461: 17456: 17451: 17450: 17438: 17437: 17422: 17421: 17409: 17408: 17388: 17386: 17385: 17380: 17377: 17376: 17375: 17365: 17364: 17363: 17327: 17325: 17324: 17319: 17314: 17313: 17300: 17299: 17298: 17288: 17287: 17286: 17228: 17226: 17225: 17220: 17208: 17206: 17205: 17200: 17198: 17197: 17181: 17179: 17178: 17173: 17171: 17170: 17154: 17152: 17151: 17146: 17134: 17132: 17131: 17126: 17114: 17112: 17111: 17106: 17103: 17102: 17101: 17091: 17090: 17089: 17064: 17062: 17061: 17056: 17044: 17042: 17041: 17036: 17033: 17028: 17013:Trajectories of 17009: 17007: 17006: 17001: 16999: 16998: 16986: 16985: 16969: 16967: 16966: 16961: 16959: 16958: 16945: 16940: 16924: 16922: 16921: 16916: 16896: 16894: 16893: 16888: 16880: 16879: 16867: 16866: 16857: 16854: 16850: 16849: 16837: 16836: 16821: 16820: 16808: 16807: 16797: 16793: 16792: 16780: 16779: 16767: 16766: 16756: 16755: 16747: 16746: 16734: 16733: 16724: 16721: 16717: 16716: 16704: 16703: 16688: 16687: 16675: 16674: 16664: 16663: 16662: 16647: 16646: 16634: 16633: 16623: 16622: 16614: 16613: 16600: 16595: 16577: 16575: 16574: 16569: 16551: 16549: 16548: 16543: 16528: 16527: 16526: 16516: 16515: 16514: 16497: 16489: 16475: 16473: 16472: 16467: 16449: 16447: 16446: 16441: 16425: 16423: 16422: 16417: 16383: 16381: 16380: 16375: 16363: 16361: 16360: 16355: 16339: 16337: 16336: 16331: 16326: 16325: 16310: 16309: 16297: 16296: 16283: 16278: 16262: 16260: 16259: 16254: 16243: 16242: 16226: 16224: 16223: 16218: 16213: 16212: 16200: 16199: 16183: 16181: 16180: 16175: 16173: 16172: 16156: 16154: 16153: 16148: 16136: 16134: 16133: 16128: 16126: 16125: 16109: 16107: 16106: 16101: 16078: 16076: 16075: 16070: 16056: 16055: 16043: 16042: 16027: 16026: 16014: 16013: 15982: 15980: 15979: 15974: 15969: 15968: 15956: 15955: 15942: 15937: 15920: 15918: 15917: 15912: 15907: 15906: 15894: 15893: 15870: 15868: 15867: 15862: 15857: 15856: 15844: 15843: 15827: 15825: 15824: 15819: 15814: 15813: 15801: 15800: 15784: 15782: 15781: 15776: 15771: 15770: 15758: 15757: 15739: 15730: 15728: 15727: 15722: 15714: 15713: 15701: 15700: 15691: 15688: 15687: 15682: 15681: 15669: 15668: 15659: 15654: 15649: 15648: 15636: 15635: 15626: 15619: 15618: 15606: 15605: 15596: 15595: 15594: 15586: 15585: 15573: 15572: 15563: 15560: 15559: 15554: 15553: 15541: 15540: 15531: 15526: 15521: 15520: 15508: 15507: 15498: 15491: 15490: 15478: 15477: 15468: 15467: 15466: 15458: 15457: 15445: 15444: 15428: 15426: 15425: 15420: 15412: 15411: 15399: 15398: 15389: 15386: 15382: 15381: 15369: 15368: 15353: 15352: 15340: 15339: 15329: 15325: 15324: 15312: 15311: 15295: 15294: 15286: 15285: 15273: 15272: 15263: 15260: 15256: 15255: 15243: 15242: 15227: 15226: 15214: 15213: 15203: 15193: 15192: 15180: 15179: 15169: 15168: 15160: 15159: 15147: 15146: 15127: 15125: 15124: 15119: 15114: 15113: 15100: 15099: 15098: 15088: 15087: 15086: 15065: 15063: 15062: 15057: 15052: 15051: 15039: 15038: 15032: 15031: 15030: 15029: 15008: 15006: 15005: 15000: 14989: 14988: 14975: 14973: 14972: 14967: 14962: 14961: 14949: 14948: 14939: 14938: 14922: 14920: 14919: 14916:{\displaystyle } 14914: 14909: 14908: 14896: 14895: 14861:(see Fig. 11d). 14860: 14858: 14857: 14854:{\displaystyle } 14852: 14847: 14846: 14834: 14833: 14801: 14799: 14798: 14793: 14766: 14764: 14763: 14758: 14735: 14733: 14732: 14727: 14715: 14711: 14701: 14700: 14692: 14674: 14673: 14640: 14639: 14638: 14628: 14627: 14626: 14606: 14605: 14592: 14591: 14590: 14580: 14579: 14578: 14568: 14546: 14544: 14543: 14540:{\displaystyle } 14538: 14533: 14532: 14520: 14519: 14500: 14498: 14497: 14492: 14487: 14485: 14465: 14453: 14427: 14426: 14407: 14393: 14392: 14384: 14374: 14372: 14371: 14366: 14304: 14302: 14301: 14296: 14291: 14290: 14277: 14272: 14271: 14270: 14246: 14244: 14243: 14238: 14235: 14230: 14229: 14228: 14207: 14205: 14204: 14199: 14196: 14191: 14190: 14189: 14168: 14166: 14165: 14160: 14151: 14146: 14145: 14144: 14130: 14119: 14118: 14110: 14106: 14105: 14104: 14094: 14082: 14081: 14073: 14061: 14059: 14058: 14053: 14039: 14038: 14037: 14036: 14018: 14013: 14012: 14011: 13988: 13986: 13985: 13980: 13972: 13968: 13967: 13956: 13955: 13947: 13941: 13937: 13927: 13926: 13889: 13888: 13880: 13868: 13866: 13865: 13860: 13846: 13845: 13844: 13843: 13825: 13820: 13819: 13818: 13795: 13793: 13792: 13787: 13781: 13776: 13775: 13774: 13759: 13754: 13753: 13752: 13734: 13729: 13728: 13727: 13704: 13702: 13701: 13696: 13693: 13688: 13687: 13686: 13662: 13660: 13659: 13654: 13645: 13640: 13639: 13638: 13624: 13620: 13610: 13609: 13579: 13578: 13577: 13567: 13555: 13554: 13546: 13534: 13532: 13531: 13526: 13512: 13511: 13510: 13509: 13491: 13486: 13485: 13484: 13461: 13459: 13458: 13453: 13445: 13441: 13431: 13430: 13398: 13397: 13389: 13377: 13375: 13374: 13369: 13355: 13354: 13353: 13352: 13334: 13329: 13328: 13327: 13298: 13296: 13295: 13290: 13287: 13282: 13281: 13280: 13262: 13257: 13256: 13255: 13230: 13228: 13227: 13222: 13219: 13214: 13213: 13212: 13189: 13187: 13186: 13181: 13175: 13170: 13169: 13168: 13153: 13148: 13147: 13146: 13128: 13123: 13122: 13121: 13106: 13101: 13100: 13099: 13081: 13076: 13075: 13074: 13026: 13024: 13023: 13018: 13016: 13015: 12999: 12997: 12996: 12991: 12989: 12988: 12972: 12970: 12969: 12966:{\displaystyle } 12964: 12959: 12958: 12946: 12945: 12926: 12924: 12923: 12918: 12916: 12915: 12899: 12897: 12896: 12891: 12888: 12887: 12886: 12876: 12875: 12874: 12850: 12848: 12847: 12842: 12839: 12838: 12837: 12827: 12826: 12825: 12795: 12793: 12792: 12787: 12784: 12779: 12778: 12777: 12756: 12754: 12753: 12748: 12746: 12745: 12726: 12724: 12723: 12718: 12716: 12714: 12713: 12704: 12702: 12697: 12693: 12692: 12691: 12681: 12680: 12679: 12669: 12668: 12667: 12647: 12646: 12629: 12628: 12620: 12615: 12610: 12609: 12608: 12582: 12580: 12579: 12574: 12572: 12568: 12567: 12551: 12535: 12533: 12532: 12527: 12525: 12524: 12502: 12500: 12499: 12494: 12492: 12472: 12470: 12469: 12468: 12459: 12458: 12439: 12438: 12434: 12433: 12432: 12422: 12421: 12420: 12410: 12409: 12408: 12388: 12387: 12370: 12366: 12365: 12364: 12354: 12353: 12352: 12342: 12341: 12340: 12320: 12319: 12304: 12294: 12289: 12288: 12287: 12260: 12256: 12249: 12246: 12245: 12236: 12235: 12231: 12230: 12229: 12219: 12218: 12217: 12207: 12206: 12205: 12185: 12184: 12170: 12167: 12162: 12142: 12134: 12124: 12119: 12118: 12117: 12084: 12082: 12081: 12076: 12073: 12072: 12071: 12061: 12060: 12059: 12036: 12034: 12033: 12028: 12023: 12019: 12017: 12012: 12011: 12010: 11982: 11981: 11969: 11965: 11963: 11958: 11957: 11956: 11931: 11930: 11908: 11904: 11903: 11899: 11897: 11892: 11891: 11890: 11865: 11864: 11833: 11828: 11827: 11826: 11801: 11799: 11798: 11793: 11791: 11722: 11719: 11718: 11709: 11708: 11704: 11703: 11693: 11692: 11691: 11681: 11680: 11679: 11659: 11658: 11645: 11643: 11642: 11637: 11633: 11612: 11611: 11610: 11600: 11599: 11598: 11544: 11541: 11540: 11531: 11530: 11526: 11525: 11515: 11514: 11513: 11503: 11502: 11501: 11481: 11480: 11467: 11457: 11456: 11455: 11445: 11444: 11443: 11410: 11408: 11407: 11402: 11399: 11398: 11397: 11387: 11386: 11385: 11364: 11362: 11361: 11356: 11341: 11339: 11338: 11333: 11331: 11330: 11314: 11312: 11311: 11306: 11303: 11302: 11301: 11291: 11290: 11289: 11268: 11266: 11265: 11260: 11255: 11254: 11241: 11240: 11239: 11229: 11228: 11227: 11180: 11178: 11177: 11172: 11169: 11168: 11167: 11157: 11156: 11155: 11134: 11132: 11131: 11126: 11123: 11122: 11121: 11111: 11110: 11109: 11088: 11086: 11085: 11080: 11077: 11076: 11075: 11065: 11064: 11063: 11042: 11040: 11039: 11034: 11028: 11027: 11026: 11016: 11015: 11014: 10999: 10998: 10997: 10987: 10986: 10985: 10967: 10966: 10965: 10955: 10954: 10953: 10938: 10937: 10936: 10926: 10925: 10924: 10906: 10905: 10904: 10894: 10893: 10892: 10875: 10874: 10864: 10863: 10862: 10852: 10851: 10850: 10826: 10825: 10824: 10814: 10813: 10812: 10789:, respectively. 10780: 10778: 10777: 10772: 10769: 10768: 10767: 10757: 10756: 10755: 10737: 10736: 10735: 10725: 10724: 10723: 10698: 10696: 10695: 10690: 10687: 10686: 10685: 10675: 10674: 10673: 10652: 10650: 10649: 10644: 10638: 10637: 10636: 10626: 10625: 10624: 10609: 10608: 10607: 10597: 10596: 10595: 10577: 10576: 10575: 10565: 10564: 10563: 10548: 10547: 10546: 10536: 10535: 10534: 10516: 10515: 10514: 10504: 10503: 10502: 10469: 10467: 10466: 10463:{\displaystyle } 10461: 10456: 10455: 10443: 10442: 10386: 10384: 10383: 10378: 10373: 10372: 10360: 10359: 10343: 10341: 10340: 10335: 10330: 10329: 10317: 10316: 10303: 10298: 10278: 10276: 10275: 10270: 10265: 10264: 10252: 10251: 10235: 10233: 10232: 10227: 10222: 10221: 10209: 10208: 10195: 10190: 10171:. Specifically, 10170: 10168: 10167: 10162: 10157: 10156: 10144: 10143: 10125: 10123: 10122: 10117: 10112: 10111: 10063: 10059: 10055: 10054: 10041: 10040: 10039: 10029: 10028: 10027: 10001: 10000: 9987: 9986: 9985: 9975: 9974: 9973: 9954: 9946: 9938: 9937: 9924: 9923: 9922: 9912: 9911: 9910: 9887: 9885: 9884: 9879: 9876: 9875: 9874: 9864: 9863: 9862: 9814: 9812: 9811: 9806: 9795: 9794: 9781: 9779: 9778: 9773: 9762: 9761: 9748: 9746: 9745: 9740: 9729: 9728: 9711: 9709: 9708: 9703: 9692: 9691: 9678: 9676: 9675: 9670: 9659: 9658: 9645: 9643: 9642: 9637: 9634: 9629: 9628: 9627: 9598: 9596: 9595: 9590: 9578: 9576: 9575: 9570: 9552: 9550: 9549: 9544: 9539: 9538: 9523: 9522: 9510: 9509: 9496: 9491: 9473: 9471: 9470: 9465: 9454: 9453: 9437: 9435: 9434: 9429: 9424: 9423: 9411: 9410: 9394: 9392: 9391: 9386: 9384: 9383: 9367: 9365: 9364: 9359: 9347: 9345: 9344: 9339: 9337: 9336: 9320: 9318: 9317: 9312: 9295: 9293: 9292: 9287: 9282: 9281: 9269: 9268: 9252: 9250: 9249: 9244: 9239: 9238: 9226: 9225: 9212: 9207: 9184: 9182: 9181: 9176: 9171: 9170: 9158: 9157: 9141: 9139: 9138: 9133: 9128: 9127: 9115: 9114: 9088: 9086: 9085: 9080: 9072: 9068: 9064: 9063: 9051: 9050: 9035: 9034: 9022: 9021: 8988: 8986: 8985: 8980: 8975: 8974: 8962: 8961: 8948: 8943: 8926: 8924: 8923: 8918: 8913: 8912: 8900: 8899: 8875:stretch surfaces 8872: 8870: 8869: 8864: 8856: 8852: 8848: 8847: 8835: 8834: 8819: 8818: 8806: 8805: 8772: 8770: 8769: 8764: 8759: 8758: 8746: 8745: 8732: 8727: 8710: 8708: 8707: 8702: 8697: 8696: 8684: 8683: 8660: 8658: 8657: 8652: 8647: 8646: 8634: 8633: 8617: 8615: 8614: 8609: 8604: 8603: 8591: 8590: 8575: 8573: 8572: 8567: 8543: 8541: 8540: 8535: 8530: 8529: 8517: 8516: 8498: 8479: 8477: 8476: 8471: 8469: 8468: 8452: 8450: 8449: 8444: 8439: 8438: 8426: 8425: 8412: 8411: 8410: 8400: 8399: 8398: 8377: 8375: 8374: 8369: 8358: 8357: 8344: 8342: 8341: 8338:{\displaystyle } 8336: 8331: 8330: 8318: 8317: 8294: 8292: 8291: 8286: 8275: 8274: 8261: 8259: 8258: 8253: 8242: 8241: 8229: 8228: 8215: 8214: 8213: 8203: 8202: 8201: 8180: 8178: 8177: 8174:{\displaystyle } 8172: 8167: 8166: 8154: 8153: 8134: 8132: 8131: 8126: 8115: 8114: 8101: 8099: 8098: 8093: 8082: 8081: 8069: 8068: 8055: 8054: 8053: 8043: 8042: 8041: 8009: 8007: 8006: 8001: 7996: 7995: 7983: 7982: 7969: 7968: 7967: 7957: 7956: 7955: 7934: 7932: 7931: 7926: 7921: 7920: 7911: 7910: 7894: 7893: 7892: 7882: 7881: 7880: 7859: 7857: 7856: 7851: 7846: 7845: 7833: 7832: 7819: 7817: 7816: 7811: 7806: 7805: 7792: 7791: 7790: 7780: 7779: 7778: 7754: 7752: 7751: 7746: 7744: 7743: 7727: 7725: 7724: 7719: 7714: 7713: 7700: 7699: 7698: 7688: 7687: 7686: 7662: 7660: 7659: 7654: 7649: 7648: 7636: 7635: 7629: 7628: 7627: 7626: 7601: 7599: 7598: 7593: 7588: 7587: 7575: 7574: 7568: 7567: 7566: 7565: 7541: 7539: 7538: 7533: 7528: 7527: 7515: 7514: 7501: 7499: 7498: 7493: 7491: 7490: 7474: 7472: 7471: 7466: 7464: 7463: 7447: 7445: 7444: 7441:{\displaystyle } 7439: 7434: 7433: 7421: 7420: 7375: 7373: 7372: 7367: 7355: 7353: 7352: 7349:{\displaystyle } 7347: 7336: 7335: 7317: 7316: 7286: 7284: 7283: 7280:{\displaystyle } 7278: 7267: 7266: 7248: 7247: 7228: 7226: 7225: 7220: 7217: 7210: 7209: 7199: 7192: 7191: 7181: 7159: 7157: 7156: 7151: 7143: 7142: 7126: 7124: 7123: 7120:{\displaystyle } 7118: 7107: 7106: 7088: 7087: 7068: 7066: 7065: 7062:{\displaystyle } 7060: 7055: 7054: 7042: 7041: 7018: 7016: 7015: 7010: 6998: 6996: 6995: 6990: 6985: 6984: 6971: 6964: 6963: 6953: 6946: 6945: 6935: 6913: 6911: 6910: 6907:{\displaystyle } 6905: 6894: 6893: 6875: 6874: 6838: 6836: 6835: 6830: 6812: 6810: 6809: 6804: 6802: 6801: 6786:pointing in the 6785: 6783: 6782: 6777: 6775: 6774: 6748: 6746: 6745: 6740: 6735: 6734: 6727: 6725: 6721: 6720: 6719: 6702: 6698: 6697: 6685: 6684: 6672: 6671: 6653: 6652: 6637: 6636: 6624: 6623: 6611: 6610: 6592: 6591: 6581: 6577: 6575: 6571: 6570: 6569: 6552: 6548: 6547: 6535: 6534: 6522: 6521: 6503: 6502: 6487: 6486: 6474: 6473: 6461: 6460: 6442: 6441: 6431: 6427: 6425: 6421: 6420: 6419: 6402: 6398: 6397: 6385: 6384: 6372: 6371: 6353: 6352: 6337: 6336: 6324: 6323: 6311: 6310: 6292: 6291: 6281: 6275: 6273: 6269: 6268: 6267: 6250: 6246: 6245: 6233: 6232: 6220: 6219: 6201: 6200: 6185: 6184: 6172: 6171: 6159: 6158: 6140: 6139: 6129: 6125: 6123: 6119: 6118: 6117: 6100: 6096: 6095: 6083: 6082: 6070: 6069: 6051: 6050: 6035: 6034: 6022: 6021: 6009: 6008: 5990: 5989: 5979: 5975: 5973: 5969: 5968: 5967: 5950: 5946: 5945: 5933: 5932: 5920: 5919: 5901: 5900: 5885: 5884: 5872: 5871: 5859: 5858: 5840: 5839: 5829: 5823: 5821: 5817: 5816: 5815: 5798: 5794: 5793: 5781: 5780: 5768: 5767: 5749: 5748: 5733: 5732: 5720: 5719: 5707: 5706: 5688: 5687: 5677: 5673: 5671: 5667: 5666: 5665: 5648: 5644: 5643: 5631: 5630: 5618: 5617: 5599: 5598: 5583: 5582: 5570: 5569: 5557: 5556: 5538: 5537: 5527: 5523: 5521: 5517: 5516: 5515: 5498: 5494: 5493: 5481: 5480: 5468: 5467: 5449: 5448: 5433: 5432: 5420: 5419: 5407: 5406: 5388: 5387: 5377: 5361: 5360: 5347: 5342: 5341: 5340: 5316: 5314: 5313: 5308: 5303: 5302: 5290: 5289: 5261: 5259: 5258: 5253: 5248: 5247: 5235: 5234: 5222: 5221: 5196: 5194: 5193: 5188: 5186: 5185: 5169: 5167: 5166: 5161: 5156: 5155: 5143: 5142: 5114: 5112: 5111: 5106: 5101: 5100: 5087: 5082: 5081: 5080: 5049: 5047: 5046: 5041: 5038: 5037: 5036: 5026: 5025: 5024: 5014: 4992: 4990: 4989: 4984: 4982: 4981: 4965: 4963: 4962: 4957: 4952: 4951: 4938: 4937: 4936: 4926: 4925: 4924: 4914: 4892: 4890: 4889: 4884: 4882: 4881: 4849: 4847: 4846: 4841: 4833: 4832: 4820: 4819: 4804: 4802: 4798: 4797: 4785: 4784: 4765: 4757: 4756: 4743: 4742: 4741: 4731: 4730: 4729: 4719: 4691: 4689: 4688: 4683: 4678: 4677: 4665: 4664: 4652: 4644: 4642: 4641: 4640: 4630: 4629: 4628: 4597: 4595: 4594: 4589: 4587: 4582: 4581: 4569: 4568: 4559: 4551: 4550: 4537: 4536: 4535: 4525: 4524: 4523: 4502: 4500: 4499: 4494: 4492: 4491: 4471: 4469: 4468: 4463: 4460: 4459: 4458: 4448: 4447: 4446: 4429: 4428: 4423: 4419: 4417: 4416: 4415: 4405: 4404: 4403: 4376: 4375: 4374: 4364: 4363: 4362: 4339: 4337: 4336: 4331: 4329: 4325: 4323: 4319: 4315: 4314: 4296: 4295: 4282: 4281: 4280: 4270: 4269: 4268: 4248: 4247: 4227: 4224: 4217: 4213: 4209: 4208: 4177: 4173: 4171: 4167: 4163: 4162: 4144: 4143: 4130: 4129: 4128: 4118: 4117: 4116: 4093: 4092: 4074: 4073: 4060: 4059: 4058: 4048: 4047: 4046: 4024: 4021: 4014: 4010: 4006: 4005: 3974: 3970: 3966: 3962: 3961: 3949: 3948: 3936: 3935: 3920: 3913: 3909: 3905: 3904: 3879: 3871: 3868: 3843: 3842: 3829: 3828: 3827: 3817: 3816: 3815: 3788: 3786: 3785: 3780: 3778: 3777: 3761: 3759: 3758: 3753: 3748: 3747: 3735: 3724: 3723: 3710: 3709: 3708: 3698: 3697: 3696: 3673: 3672: 3660: 3659: 3647: 3646: 3641: 3628: 3626: 3625: 3620: 3615: 3614: 3602: 3601: 3573: 3571: 3570: 3565: 3563: 3562: 3557: 3544: 3542: 3541: 3536: 3531: 3530: 3508: 3506: 3505: 3500: 3476: 3474: 3473: 3468: 3466: 3465: 3449: 3447: 3446: 3441: 3436: 3435: 3423: 3408: 3406: 3405: 3400: 3397: 3392: 3391: 3390: 3366: 3364: 3363: 3358: 3353: 3352: 3340: 3339: 3311: 3309: 3308: 3303: 3292: 3277: 3275: 3274: 3269: 3264: 3263: 3251: 3250: 3237: 3235: 3234: 3229: 3224: 3223: 3211: 3210: 3197: 3195: 3194: 3189: 3184: 3183: 3171: 3170: 3133: 3131: 3130: 3125: 3123: 3122: 3113: 3112: 3076: 3074: 3073: 3068: 3056: 3054: 3053: 3048: 3036: 3034: 3033: 3028: 3016: 3014: 3013: 3008: 2945: 2943: 2942: 2937: 2892: 2890: 2889: 2884: 2872: 2870: 2869: 2864: 2826: 2824: 2823: 2818: 2798: 2796: 2795: 2790: 2779: 2778: 2773: 2758: 2757: 2752: 2737: 2729: 2708: 2706: 2705: 2700: 2692: 2691: 2684: 2665: 2638: 2616: 2570: 2569: 2561: 2549: 2547: 2546: 2541: 2514: 2512: 2511: 2506: 2503: 2498: 2497: 2496: 2472: 2470: 2469: 2464: 2447: 2435: 2433: 2432: 2427: 2394: 2392: 2391: 2386: 2359: 2357: 2356: 2351: 2324: 2322: 2321: 2316: 2283: 2281: 2280: 2275: 2261: 2260: 2252: 2249: 2248: 2197: 2196: 2160: 2159: 2151: 2114: 2113: 2077: 2076: 2068: 2058: 2056: 2055: 2050: 2045: 2041: 2040: 2039: 1977: 1969: 1942: 1938: 1937: 1936: 1874: 1866: 1836: 1834: 1833: 1828: 1801: 1799: 1798: 1793: 1763: 1761: 1760: 1755: 1744:is an arbitrary 1743: 1741: 1740: 1735: 1714: 1712: 1711: 1706: 1689:is an arbitrary 1688: 1686: 1685: 1680: 1659: 1657: 1656: 1651: 1649: 1648: 1643: 1624: 1622: 1621: 1616: 1563: 1561: 1560: 1555: 1541: 1539: 1538: 1533: 1528: 1527: 1522: 1446: 1444: 1443: 1438: 1418: 1416: 1415: 1410: 1408: 1407: 1366: 1364: 1363: 1358: 1356: 1355: 1342: 1340: 1339: 1334: 1323: 1322: 1309: 1307: 1306: 1301: 1299: 1298: 1285: 1283: 1282: 1277: 1274: 1269: 1268: 1267: 1247:of the flow map 1238: 1236: 1235: 1230: 1228: 1227: 1211:In contrast, in 1195: 1193: 1192: 1187: 1182: 1181: 1169: 1168: 1155: 1153: 1152: 1147: 1142: 1141: 1129: 1128: 1115: 1113: 1112: 1107: 1105: 1104: 1088: 1086: 1085: 1080: 1069: 1068: 1055:material surface 1045: 1043: 1042: 1037: 1035: 1034: 1021: 1019: 1018: 1013: 1002: 1001: 972: 970: 969: 964: 962: 961: 952: 951: 942: 941: 928: 926: 925: 920: 915: 914: 902: 901: 886:material surface 882: 880: 879: 874: 872: 871: 859:of the manifold 858: 856: 855: 850: 842: 841: 829: 828: 818: 813: 812: 811: 785: 784: 767: 765: 764: 759: 757: 756: 747: 746: 731:in the extended 724: 722: 721: 716: 708: 707: 695: 694: 676: 675: 662: 657: 656: 655: 634: 633: 624: 623: 593: 592: 577: 575: 574: 569: 567: 566: 553: 551: 550: 545: 540: 539: 527: 526: 513: 511: 510: 505: 503: 502: 479: 477: 476: 471: 468: 463: 462: 461: 440: 438: 437: 432: 430: 429: 410: 408: 407: 402: 400: 399: 387: 386: 374: 373: 345: 343: 342: 337: 335: 334: 325: 324: 308: 306: 305: 300: 295: 294: 282: 281: 257: 256: 243: 238: 237: 236: 215: 213: 212: 207: 202: 201: 189: 188: 173: 172: 159: 157: 156: 151: 149: 148: 98:dynamical system 69: 62: 58: 55: 49: 35:close connection 27: 26: 19: 19688: 19687: 19683: 19682: 19681: 19679: 19678: 19677: 19643: 19642: 19619:10.1.1.506.7756 19551: 19550: 19549: 19531: 19527: 19522: 19520:Further reading 19517: 19516: 19508: 19494: 19467: 19461: 19457: 19442: 19438: 19427: 19423: 19408: 19404: 19393: 19389: 19378: 19374: 19359: 19355: 19318: 19314: 19275: 19271: 19222: 19218: 19177: 19166: 19135: 19131: 19082: 19061: 19012: 19003: 18954: 18939: 18894: 18883: 18838: 18823: 18784: 18780: 18743: 18736: 18705: 18698: 18691: 18669: 18665: 18634: 18630: 18591: 18587: 18572: 18565: 18533: 18527: 18518: 18483: 18470: 18455: 18436:10.1.1.657.3742 18423: 18419: 18382: 18378: 18347: 18340: 18301: 18297: 18281:10.1.1.331.6383 18256: 18252: 18213: 18209: 18178: 18174: 18159: 18127: 18123: 18070: 18066: 18035: 18031: 17994: 17990: 17952: 17948: 17911: 17907: 17852: 17848: 17817: 17813: 17782: 17778: 17747: 17720: 17689: 17685: 17654: 17635: 17630: 17593: 17534: 17503: 17499: 17497: 17494: 17493: 17476: 17472: 17470: 17467: 17466: 17446: 17442: 17433: 17429: 17417: 17413: 17404: 17400: 17398: 17395: 17394: 17371: 17367: 17366: 17359: 17355: 17354: 17348: 17345: 17344: 17334: 17309: 17305: 17294: 17290: 17289: 17282: 17278: 17277: 17262: 17259: 17258: 17251: 17239: 17214: 17211: 17210: 17193: 17189: 17187: 17184: 17183: 17166: 17162: 17160: 17157: 17156: 17140: 17137: 17136: 17120: 17117: 17116: 17097: 17093: 17092: 17085: 17081: 17080: 17074: 17071: 17070: 17050: 17047: 17046: 17029: 17024: 17018: 17015: 17014: 16994: 16990: 16981: 16977: 16975: 16972: 16971: 16954: 16950: 16941: 16936: 16930: 16927: 16926: 16904: 16901: 16900: 16875: 16871: 16862: 16858: 16845: 16841: 16832: 16828: 16816: 16812: 16803: 16799: 16798: 16788: 16784: 16775: 16771: 16762: 16758: 16757: 16754: 16742: 16738: 16729: 16725: 16712: 16708: 16699: 16695: 16683: 16679: 16670: 16666: 16665: 16658: 16654: 16642: 16638: 16629: 16625: 16624: 16621: 16609: 16605: 16596: 16591: 16585: 16582: 16581: 16557: 16554: 16553: 16522: 16518: 16517: 16510: 16506: 16505: 16488: 16486: 16483: 16482: 16455: 16452: 16451: 16435: 16432: 16431: 16411: 16408: 16407: 16400: 16369: 16366: 16365: 16349: 16346: 16345: 16321: 16317: 16305: 16301: 16292: 16288: 16279: 16274: 16268: 16265: 16264: 16238: 16234: 16232: 16229: 16228: 16208: 16204: 16195: 16191: 16189: 16186: 16185: 16168: 16164: 16162: 16159: 16158: 16142: 16139: 16138: 16121: 16117: 16115: 16112: 16111: 16095: 16092: 16091: 16051: 16047: 16038: 16034: 16022: 16018: 16009: 16005: 15994: 15991: 15990: 15964: 15960: 15951: 15947: 15938: 15933: 15927: 15924: 15923: 15902: 15898: 15889: 15885: 15883: 15880: 15879: 15852: 15848: 15839: 15838: 15836: 15833: 15832: 15809: 15805: 15796: 15795: 15793: 15790: 15789: 15766: 15762: 15753: 15752: 15750: 15747: 15746: 15709: 15705: 15696: 15692: 15677: 15673: 15664: 15660: 15658: 15644: 15640: 15631: 15627: 15625: 15624: 15614: 15610: 15601: 15597: 15593: 15581: 15577: 15568: 15564: 15549: 15545: 15536: 15532: 15530: 15516: 15512: 15503: 15499: 15497: 15496: 15486: 15482: 15473: 15469: 15465: 15453: 15449: 15440: 15436: 15434: 15431: 15430: 15407: 15403: 15394: 15390: 15377: 15373: 15364: 15360: 15348: 15344: 15335: 15331: 15330: 15320: 15316: 15307: 15303: 15296: 15293: 15281: 15277: 15268: 15264: 15251: 15247: 15238: 15234: 15222: 15218: 15209: 15205: 15204: 15188: 15184: 15175: 15171: 15170: 15167: 15155: 15151: 15142: 15138: 15136: 15133: 15132: 15109: 15105: 15094: 15090: 15089: 15082: 15078: 15077: 15071: 15068: 15067: 15047: 15043: 15034: 15033: 15025: 15021: 15020: 15016: 15014: 15011: 15010: 14984: 14983: 14981: 14978: 14977: 14957: 14953: 14944: 14943: 14934: 14930: 14928: 14925: 14924: 14904: 14900: 14891: 14887: 14882: 14879: 14878: 14875: 14842: 14838: 14829: 14825: 14820: 14817: 14816: 14772: 14769: 14768: 14743: 14740: 14739: 14691: 14690: 14669: 14665: 14646: 14642: 14634: 14630: 14629: 14622: 14618: 14617: 14601: 14597: 14586: 14582: 14581: 14574: 14570: 14569: 14555: 14552: 14549: 14548: 14528: 14524: 14515: 14511: 14506: 14503: 14502: 14455: 14454: 14413: 14409: 14408: 14406: 14383: 14382: 14380: 14377: 14376: 14318: 14315: 14314: 14286: 14282: 14273: 14266: 14262: 14261: 14255: 14252: 14251: 14231: 14224: 14220: 14219: 14213: 14210: 14209: 14192: 14185: 14181: 14180: 14174: 14171: 14170: 14147: 14140: 14136: 14135: 14120: 14109: 14108: 14100: 14096: 14095: 14090: 14072: 14071: 14069: 14066: 14065: 14032: 14028: 14027: 14023: 14014: 14007: 14003: 14002: 13996: 13993: 13992: 13957: 13946: 13945: 13922: 13918: 13905: 13901: 13897: 13893: 13879: 13878: 13876: 13873: 13872: 13839: 13835: 13834: 13830: 13821: 13814: 13810: 13809: 13803: 13800: 13799: 13777: 13770: 13766: 13765: 13755: 13748: 13744: 13743: 13730: 13723: 13719: 13718: 13712: 13709: 13708: 13689: 13682: 13678: 13677: 13671: 13668: 13667: 13641: 13634: 13630: 13629: 13605: 13601: 13588: 13584: 13573: 13569: 13568: 13563: 13545: 13544: 13542: 13539: 13538: 13505: 13501: 13500: 13496: 13487: 13480: 13476: 13475: 13469: 13466: 13465: 13426: 13422: 13409: 13405: 13388: 13387: 13385: 13382: 13381: 13348: 13344: 13343: 13339: 13330: 13323: 13319: 13318: 13312: 13309: 13308: 13283: 13276: 13272: 13271: 13258: 13251: 13247: 13246: 13240: 13237: 13236: 13215: 13208: 13204: 13203: 13197: 13194: 13193: 13171: 13164: 13160: 13159: 13149: 13142: 13138: 13137: 13124: 13117: 13113: 13112: 13102: 13095: 13091: 13090: 13077: 13070: 13066: 13065: 13056: 13053: 13052: 13011: 13007: 13005: 13002: 13001: 12984: 12980: 12978: 12975: 12974: 12954: 12950: 12941: 12937: 12932: 12929: 12928: 12911: 12907: 12905: 12902: 12901: 12882: 12878: 12877: 12870: 12866: 12865: 12856: 12853: 12852: 12833: 12829: 12828: 12821: 12817: 12816: 12810: 12807: 12806: 12802: 12780: 12773: 12769: 12768: 12762: 12759: 12758: 12741: 12737: 12735: 12732: 12731: 12709: 12705: 12703: 12698: 12687: 12683: 12675: 12671: 12670: 12663: 12659: 12658: 12642: 12638: 12637: 12633: 12621: 12611: 12604: 12600: 12599: 12591: 12590: 12588: 12585: 12584: 12563: 12559: 12555: 12547: 12545: 12542: 12541: 12514: 12510: 12508: 12505: 12504: 12490: 12489: 12464: 12460: 12448: 12444: 12440: 12428: 12424: 12416: 12412: 12411: 12404: 12400: 12399: 12383: 12379: 12378: 12374: 12360: 12356: 12348: 12344: 12343: 12336: 12332: 12331: 12315: 12311: 12310: 12306: 12305: 12303: 12296: 12290: 12283: 12279: 12278: 12265: 12264: 12241: 12237: 12225: 12221: 12213: 12209: 12208: 12201: 12197: 12196: 12180: 12176: 12175: 12171: 12169: 12163: 12152: 12147: 12143: 12133: 12126: 12120: 12113: 12109: 12108: 12094: 12092: 12089: 12088: 12067: 12063: 12062: 12055: 12051: 12050: 12044: 12041: 12040: 12013: 12006: 12002: 12001: 11990: 11986: 11974: 11970: 11959: 11952: 11948: 11947: 11936: 11932: 11916: 11915: 11893: 11886: 11882: 11881: 11870: 11866: 11850: 11849: 11842: 11838: 11829: 11822: 11818: 11817: 11811: 11808: 11807: 11789: 11788: 11714: 11710: 11699: 11695: 11687: 11683: 11682: 11675: 11671: 11670: 11654: 11650: 11646: 11644: 11638: 11626: 11622: 11621: 11614: 11606: 11602: 11601: 11594: 11590: 11589: 11576: 11575: 11536: 11532: 11521: 11517: 11509: 11505: 11504: 11497: 11493: 11492: 11476: 11472: 11468: 11466: 11459: 11451: 11447: 11446: 11439: 11435: 11434: 11420: 11418: 11415: 11414: 11393: 11389: 11388: 11381: 11377: 11376: 11370: 11367: 11366: 11347: 11344: 11343: 11326: 11322: 11320: 11317: 11316: 11297: 11293: 11292: 11285: 11281: 11280: 11274: 11271: 11270: 11250: 11246: 11235: 11231: 11230: 11223: 11219: 11218: 11212: 11209: 11208: 11163: 11159: 11158: 11151: 11147: 11146: 11140: 11137: 11136: 11117: 11113: 11112: 11105: 11101: 11100: 11094: 11091: 11090: 11071: 11067: 11066: 11059: 11055: 11054: 11048: 11045: 11044: 11022: 11018: 11017: 11010: 11006: 11005: 10993: 10989: 10988: 10981: 10977: 10976: 10961: 10957: 10956: 10949: 10945: 10944: 10932: 10928: 10927: 10920: 10916: 10915: 10900: 10896: 10895: 10888: 10884: 10883: 10870: 10866: 10858: 10854: 10853: 10846: 10842: 10841: 10820: 10816: 10815: 10808: 10804: 10803: 10797: 10794: 10793: 10781:are called the 10763: 10759: 10758: 10751: 10747: 10746: 10731: 10727: 10726: 10719: 10715: 10714: 10708: 10705: 10704: 10701:rotation tensor 10681: 10677: 10676: 10669: 10665: 10664: 10658: 10655: 10654: 10632: 10628: 10627: 10620: 10616: 10615: 10603: 10599: 10598: 10591: 10587: 10586: 10571: 10567: 10566: 10559: 10555: 10554: 10542: 10538: 10537: 10530: 10526: 10525: 10510: 10506: 10505: 10498: 10494: 10493: 10484: 10481: 10480: 10451: 10447: 10438: 10434: 10429: 10426: 10425: 10418: 10401: 10368: 10364: 10355: 10351: 10349: 10346: 10345: 10325: 10321: 10312: 10308: 10299: 10294: 10288: 10285: 10284: 10281:attracting LCSs 10260: 10256: 10247: 10243: 10241: 10238: 10237: 10217: 10213: 10204: 10200: 10191: 10186: 10180: 10177: 10176: 10152: 10148: 10139: 10135: 10133: 10130: 10129: 10106: 10105: 10100: 10094: 10093: 10085: 10075: 10074: 10050: 10046: 10035: 10031: 10030: 10023: 10019: 10018: 9996: 9992: 9981: 9977: 9976: 9969: 9965: 9964: 9959: 9955: 9945: 9933: 9929: 9918: 9914: 9913: 9906: 9902: 9901: 9895: 9892: 9891: 9870: 9866: 9865: 9858: 9854: 9853: 9847: 9844: 9843: 9790: 9789: 9787: 9784: 9783: 9757: 9756: 9754: 9751: 9750: 9724: 9723: 9721: 9718: 9717: 9687: 9686: 9684: 9681: 9680: 9654: 9653: 9651: 9648: 9647: 9630: 9623: 9619: 9618: 9612: 9609: 9608: 9605: 9584: 9581: 9580: 9564: 9561: 9560: 9534: 9530: 9518: 9514: 9505: 9501: 9492: 9487: 9481: 9478: 9477: 9449: 9445: 9443: 9440: 9439: 9419: 9415: 9406: 9402: 9400: 9397: 9396: 9379: 9375: 9373: 9370: 9369: 9353: 9350: 9349: 9332: 9328: 9326: 9323: 9322: 9306: 9303: 9302: 9277: 9273: 9264: 9260: 9258: 9255: 9254: 9234: 9230: 9221: 9217: 9208: 9203: 9197: 9194: 9193: 9166: 9162: 9153: 9149: 9147: 9144: 9143: 9123: 9119: 9110: 9106: 9104: 9101: 9100: 9091:shrink surfaces 9059: 9055: 9046: 9042: 9030: 9026: 9017: 9013: 9006: 9002: 9000: 8997: 8996: 8970: 8966: 8957: 8953: 8944: 8939: 8933: 8930: 8929: 8908: 8904: 8895: 8891: 8889: 8886: 8885: 8843: 8839: 8830: 8826: 8814: 8810: 8801: 8797: 8790: 8786: 8784: 8781: 8780: 8754: 8750: 8741: 8737: 8728: 8723: 8717: 8714: 8713: 8692: 8688: 8679: 8675: 8673: 8670: 8669: 8642: 8638: 8629: 8628: 8626: 8623: 8622: 8599: 8595: 8586: 8585: 8583: 8580: 8579: 8549: 8546: 8545: 8525: 8521: 8512: 8511: 8509: 8506: 8505: 8482:hyperbolic LCSs 8464: 8460: 8458: 8455: 8454: 8434: 8430: 8421: 8417: 8406: 8402: 8401: 8394: 8390: 8389: 8383: 8380: 8379: 8353: 8352: 8350: 8347: 8346: 8326: 8322: 8313: 8309: 8304: 8301: 8300: 8270: 8269: 8267: 8264: 8263: 8237: 8233: 8224: 8220: 8209: 8205: 8204: 8197: 8193: 8192: 8186: 8183: 8182: 8162: 8158: 8149: 8145: 8140: 8137: 8136: 8110: 8109: 8107: 8104: 8103: 8077: 8073: 8064: 8060: 8049: 8045: 8044: 8037: 8033: 8032: 8026: 8023: 8022: 7991: 7987: 7978: 7974: 7963: 7959: 7958: 7951: 7947: 7946: 7940: 7937: 7936: 7916: 7912: 7903: 7899: 7888: 7884: 7883: 7876: 7872: 7871: 7865: 7862: 7861: 7841: 7837: 7828: 7827: 7825: 7822: 7821: 7801: 7797: 7786: 7782: 7781: 7774: 7770: 7769: 7760: 7757: 7756: 7739: 7735: 7733: 7730: 7729: 7709: 7705: 7694: 7690: 7689: 7682: 7678: 7677: 7668: 7665: 7664: 7644: 7640: 7631: 7630: 7622: 7618: 7617: 7613: 7611: 7608: 7607: 7583: 7579: 7570: 7569: 7561: 7557: 7556: 7552: 7550: 7547: 7546: 7523: 7519: 7510: 7509: 7507: 7504: 7503: 7486: 7482: 7480: 7477: 7476: 7459: 7455: 7453: 7450: 7449: 7429: 7425: 7416: 7412: 7407: 7404: 7403: 7400: 7361: 7358: 7357: 7356:with a varying 7331: 7327: 7312: 7308: 7303: 7300: 7299: 7262: 7258: 7243: 7239: 7234: 7231: 7230: 7205: 7201: 7200: 7187: 7183: 7182: 7168: 7165: 7162: 7161: 7138: 7134: 7132: 7129: 7128: 7102: 7098: 7083: 7079: 7074: 7071: 7070: 7050: 7046: 7037: 7033: 7028: 7025: 7024: 7004: 7001: 7000: 6980: 6976: 6959: 6955: 6954: 6941: 6937: 6936: 6922: 6919: 6916: 6915: 6889: 6885: 6870: 6866: 6861: 6858: 6857: 6853: 6818: 6815: 6814: 6797: 6793: 6791: 6788: 6787: 6770: 6766: 6764: 6761: 6760: 6729: 6728: 6715: 6711: 6707: 6703: 6693: 6689: 6680: 6676: 6667: 6663: 6648: 6644: 6632: 6628: 6619: 6615: 6606: 6602: 6587: 6583: 6582: 6580: 6578: 6565: 6561: 6557: 6553: 6543: 6539: 6530: 6526: 6517: 6513: 6498: 6494: 6482: 6478: 6469: 6465: 6456: 6452: 6437: 6433: 6432: 6430: 6428: 6415: 6411: 6407: 6403: 6393: 6389: 6380: 6376: 6367: 6363: 6348: 6344: 6332: 6328: 6319: 6315: 6306: 6302: 6287: 6283: 6282: 6280: 6277: 6276: 6263: 6259: 6255: 6251: 6241: 6237: 6228: 6224: 6215: 6211: 6196: 6192: 6180: 6176: 6167: 6163: 6154: 6150: 6135: 6131: 6130: 6128: 6126: 6113: 6109: 6105: 6101: 6091: 6087: 6078: 6074: 6065: 6061: 6046: 6042: 6030: 6026: 6017: 6013: 6004: 6000: 5985: 5981: 5980: 5978: 5976: 5963: 5959: 5955: 5951: 5941: 5937: 5928: 5924: 5915: 5911: 5896: 5892: 5880: 5876: 5867: 5863: 5854: 5850: 5835: 5831: 5830: 5828: 5825: 5824: 5811: 5807: 5803: 5799: 5789: 5785: 5776: 5772: 5763: 5759: 5744: 5740: 5728: 5724: 5715: 5711: 5702: 5698: 5683: 5679: 5678: 5676: 5674: 5661: 5657: 5653: 5649: 5639: 5635: 5626: 5622: 5613: 5609: 5594: 5590: 5578: 5574: 5565: 5561: 5552: 5548: 5533: 5529: 5528: 5526: 5524: 5511: 5507: 5503: 5499: 5489: 5485: 5476: 5472: 5463: 5459: 5444: 5440: 5428: 5424: 5415: 5411: 5402: 5398: 5383: 5379: 5378: 5376: 5369: 5368: 5356: 5352: 5343: 5336: 5332: 5331: 5322: 5319: 5318: 5298: 5294: 5285: 5281: 5267: 5264: 5263: 5243: 5239: 5230: 5226: 5217: 5213: 5202: 5199: 5198: 5181: 5177: 5175: 5172: 5171: 5151: 5147: 5138: 5134: 5120: 5117: 5116: 5096: 5092: 5083: 5076: 5072: 5071: 5062: 5059: 5058: 5032: 5028: 5027: 5020: 5016: 5015: 5001: 4998: 4995: 4994: 4977: 4973: 4971: 4968: 4967: 4947: 4943: 4932: 4928: 4927: 4920: 4916: 4915: 4901: 4898: 4895: 4894: 4877: 4873: 4871: 4868: 4867: 4828: 4824: 4815: 4811: 4793: 4789: 4780: 4776: 4769: 4764: 4752: 4748: 4737: 4733: 4732: 4725: 4721: 4720: 4706: 4703: 4700: 4699: 4673: 4669: 4660: 4656: 4648: 4636: 4632: 4631: 4624: 4620: 4619: 4614: 4603: 4600: 4599: 4577: 4573: 4564: 4560: 4558: 4546: 4542: 4531: 4527: 4526: 4519: 4515: 4514: 4508: 4505: 4504: 4487: 4483: 4481: 4478: 4477: 4454: 4450: 4449: 4442: 4438: 4437: 4424: 4411: 4407: 4406: 4399: 4395: 4394: 4386: 4382: 4381: 4370: 4366: 4365: 4358: 4354: 4353: 4347: 4344: 4343: 4327: 4326: 4310: 4306: 4291: 4287: 4276: 4272: 4271: 4264: 4260: 4259: 4243: 4239: 4232: 4228: 4226: 4204: 4200: 4193: 4189: 4188: 4175: 4174: 4158: 4154: 4139: 4135: 4124: 4120: 4119: 4112: 4108: 4107: 4088: 4084: 4069: 4065: 4054: 4050: 4049: 4042: 4038: 4037: 4029: 4025: 4023: 4001: 3997: 3990: 3986: 3985: 3972: 3971: 3957: 3953: 3944: 3940: 3931: 3927: 3926: 3922: 3900: 3896: 3889: 3885: 3884: 3870: 3858: 3847: 3838: 3834: 3823: 3819: 3818: 3811: 3807: 3806: 3798: 3796: 3793: 3792: 3773: 3769: 3767: 3764: 3763: 3743: 3739: 3731: 3719: 3715: 3704: 3700: 3699: 3692: 3688: 3687: 3668: 3664: 3655: 3651: 3642: 3637: 3636: 3634: 3631: 3630: 3610: 3606: 3597: 3593: 3579: 3576: 3575: 3558: 3553: 3552: 3550: 3547: 3546: 3526: 3522: 3514: 3511: 3510: 3482: 3479: 3478: 3461: 3457: 3455: 3452: 3451: 3431: 3427: 3419: 3414: 3411: 3410: 3393: 3386: 3382: 3381: 3372: 3369: 3368: 3348: 3344: 3335: 3331: 3317: 3314: 3313: 3288: 3286: 3283: 3282: 3259: 3255: 3246: 3245: 3243: 3240: 3239: 3219: 3215: 3206: 3205: 3203: 3200: 3199: 3179: 3175: 3166: 3165: 3163: 3160: 3159: 3156: 3140:hyperbolic LCSs 3118: 3117: 3108: 3107: 3105: 3102: 3101: 3083: 3081:Hyperbolic LCSs 3062: 3059: 3058: 3042: 3039: 3038: 3022: 3019: 3018: 2951: 2948: 2947: 2898: 2895: 2894: 2878: 2875: 2874: 2858: 2855: 2854: 2804: 2801: 2800: 2774: 2763: 2762: 2753: 2742: 2741: 2728: 2720: 2717: 2716: 2686: 2685: 2677: 2666: 2658: 2640: 2639: 2631: 2617: 2609: 2596: 2595: 2560: 2559: 2557: 2554: 2553: 2520: 2517: 2516: 2499: 2492: 2488: 2487: 2478: 2475: 2474: 2443: 2441: 2438: 2437: 2403: 2400: 2399: 2365: 2362: 2361: 2330: 2327: 2326: 2295: 2292: 2291: 2251: 2250: 2244: 2240: 2192: 2188: 2150: 2149: 2109: 2105: 2067: 2066: 2064: 2061: 2060: 2035: 2031: 1982: 1978: 1968: 1932: 1928: 1879: 1875: 1865: 1842: 1839: 1838: 1807: 1804: 1803: 1772: 1769: 1768: 1749: 1746: 1745: 1720: 1717: 1716: 1694: 1691: 1690: 1665: 1662: 1661: 1644: 1639: 1638: 1630: 1627: 1626: 1574: 1571: 1570: 1549: 1546: 1545: 1523: 1518: 1517: 1475: 1472: 1471: 1467: 1432: 1429: 1428: 1403: 1402: 1400: 1397: 1396: 1386: 1351: 1350: 1348: 1345: 1344: 1318: 1317: 1315: 1312: 1311: 1294: 1293: 1291: 1288: 1287: 1270: 1263: 1259: 1258: 1252: 1249: 1248: 1223: 1222: 1220: 1217: 1216: 1196:. In classical 1177: 1173: 1164: 1163: 1161: 1158: 1157: 1137: 1133: 1124: 1123: 1121: 1118: 1117: 1100: 1099: 1097: 1094: 1093: 1064: 1063: 1061: 1058: 1057: 1030: 1029: 1027: 1024: 1023: 997: 996: 994: 991: 990: 979: 957: 956: 947: 946: 937: 936: 934: 931: 930: 910: 906: 897: 896: 894: 891: 890: 867: 866: 864: 861: 860: 837: 833: 824: 823: 814: 807: 803: 802: 780: 779: 777: 774: 773: 752: 751: 742: 741: 739: 736: 735: 703: 699: 690: 689: 668: 664: 658: 651: 647: 646: 629: 628: 619: 618: 588: 587: 585: 582: 581: 562: 561: 559: 556: 555: 535: 531: 522: 521: 519: 516: 515: 498: 497: 489: 486: 485: 464: 457: 453: 452: 446: 443: 442: 425: 424: 416: 413: 412: 395: 394: 382: 378: 369: 365: 351: 348: 347: 330: 329: 320: 316: 314: 311: 310: 290: 286: 277: 273: 252: 248: 239: 232: 228: 227: 221: 218: 217: 197: 193: 184: 180: 168: 167: 165: 162: 161: 144: 143: 141: 138: 137: 123: 118: 70: 59: 53: 50: 39: 28: 24: 17: 12: 11: 5: 19686: 19676: 19675: 19670: 19665: 19660: 19658:Fluid dynamics 19655: 19641: 19640: 19593: 19532: 19525: 19524: 19523: 19521: 19518: 19515: 19514: 19511:on 2014-10-06. 19492: 19455: 19452:on 2010-06-13. 19436: 19429:Mills, Peter. 19421: 19418:on 2011-05-17. 19402: 19387: 19372: 19369:on 2009-01-07. 19363:"ManGen 1.4.4" 19353: 19312: 19269: 19216: 19164: 19129: 19059: 19001: 18937: 18881: 18821: 18778: 18734: 18715:(7): 574–598. 18696: 18689: 18663: 18628: 18585: 18582:on 2012-07-23. 18576:"LCS Tutorial" 18563: 18516: 18487:Marsden, J. E. 18468: 18453: 18417: 18376: 18338: 18311:(14): 144502. 18295: 18266:(4): 248–277. 18250: 18207: 18172: 18157: 18140:10.1.1.63.4342 18121: 18064: 18029: 17988: 17946: 17905: 17846: 17811: 17776: 17757:(1): 137–162. 17718: 17683: 17632: 17631: 17629: 17626: 17625: 17624: 17619: 17614: 17609: 17604: 17599: 17592: 17589: 17588: 17587: 17580: 17573: 17570: 17563: 17556: 17549: 17533: 17530: 17506: 17502: 17479: 17475: 17454: 17449: 17445: 17441: 17436: 17432: 17428: 17425: 17420: 17416: 17412: 17407: 17403: 17374: 17370: 17362: 17358: 17353: 17333: 17330: 17317: 17312: 17308: 17304: 17297: 17293: 17285: 17281: 17276: 17272: 17269: 17266: 17250: 17247: 17242:Parabolic LCSs 17238: 17237:Parabolic LCSs 17235: 17231:Great Red Spot 17218: 17196: 17192: 17169: 17165: 17144: 17124: 17100: 17096: 17088: 17084: 17079: 17054: 17032: 17027: 17023: 16997: 16993: 16989: 16984: 16980: 16957: 16953: 16949: 16944: 16939: 16935: 16914: 16911: 16908: 16899:Note that for 16886: 16883: 16878: 16874: 16870: 16865: 16861: 16853: 16848: 16844: 16840: 16835: 16831: 16827: 16824: 16819: 16815: 16811: 16806: 16802: 16796: 16791: 16787: 16783: 16778: 16774: 16770: 16765: 16761: 16753: 16750: 16745: 16741: 16737: 16732: 16728: 16720: 16715: 16711: 16707: 16702: 16698: 16694: 16691: 16686: 16682: 16678: 16673: 16669: 16661: 16657: 16653: 16650: 16645: 16641: 16637: 16632: 16628: 16620: 16617: 16612: 16608: 16604: 16599: 16594: 16590: 16567: 16564: 16561: 16541: 16538: 16535: 16532: 16525: 16521: 16513: 16509: 16504: 16500: 16495: 16492: 16465: 16462: 16459: 16439: 16428:Great Red Spot 16415: 16399: 16396: 16373: 16353: 16329: 16324: 16320: 16316: 16313: 16308: 16304: 16300: 16295: 16291: 16287: 16282: 16277: 16273: 16252: 16249: 16246: 16241: 16237: 16216: 16211: 16207: 16203: 16198: 16194: 16171: 16167: 16146: 16124: 16120: 16099: 16085: 16084: 16081:shear surfaces 16068: 16065: 16062: 16059: 16054: 16050: 16046: 16041: 16037: 16033: 16030: 16025: 16021: 16017: 16012: 16008: 16004: 16001: 15998: 15988: 15972: 15967: 15963: 15959: 15954: 15950: 15946: 15941: 15936: 15932: 15921: 15910: 15905: 15901: 15897: 15892: 15888: 15877: 15873: 15872: 15860: 15855: 15851: 15847: 15842: 15829: 15817: 15812: 15808: 15804: 15799: 15786: 15774: 15769: 15765: 15761: 15756: 15743: 15720: 15717: 15712: 15708: 15704: 15699: 15695: 15685: 15680: 15676: 15672: 15667: 15663: 15657: 15652: 15647: 15643: 15639: 15634: 15630: 15622: 15617: 15613: 15609: 15604: 15600: 15592: 15589: 15584: 15580: 15576: 15571: 15567: 15557: 15552: 15548: 15544: 15539: 15535: 15529: 15524: 15519: 15515: 15511: 15506: 15502: 15494: 15489: 15485: 15481: 15476: 15472: 15464: 15461: 15456: 15452: 15448: 15443: 15439: 15418: 15415: 15410: 15406: 15402: 15397: 15393: 15385: 15380: 15376: 15372: 15367: 15363: 15359: 15356: 15351: 15347: 15343: 15338: 15334: 15328: 15323: 15319: 15315: 15310: 15306: 15302: 15299: 15292: 15289: 15284: 15280: 15276: 15271: 15267: 15259: 15254: 15250: 15246: 15241: 15237: 15233: 15230: 15225: 15221: 15217: 15212: 15208: 15202: 15199: 15196: 15191: 15187: 15183: 15178: 15174: 15166: 15163: 15158: 15154: 15150: 15145: 15141: 15117: 15112: 15108: 15104: 15097: 15093: 15085: 15081: 15076: 15055: 15050: 15046: 15042: 15037: 15028: 15024: 15019: 14998: 14995: 14992: 14987: 14965: 14960: 14956: 14952: 14947: 14942: 14937: 14933: 14912: 14907: 14903: 14899: 14894: 14890: 14886: 14874: 14871: 14850: 14845: 14841: 14837: 14832: 14828: 14824: 14791: 14788: 14785: 14782: 14779: 14776: 14756: 14753: 14750: 14747: 14725: 14722: 14719: 14714: 14710: 14707: 14704: 14698: 14695: 14689: 14686: 14683: 14680: 14677: 14672: 14668: 14664: 14661: 14658: 14655: 14652: 14649: 14645: 14637: 14633: 14625: 14621: 14616: 14612: 14609: 14604: 14600: 14596: 14589: 14585: 14577: 14573: 14567: 14564: 14561: 14558: 14536: 14531: 14527: 14523: 14518: 14514: 14510: 14490: 14484: 14481: 14478: 14475: 14472: 14469: 14464: 14461: 14458: 14452: 14449: 14445: 14442: 14439: 14436: 14433: 14430: 14425: 14422: 14419: 14416: 14412: 14405: 14402: 14399: 14396: 14390: 14387: 14364: 14361: 14358: 14355: 14352: 14349: 14346: 14343: 14340: 14337: 14334: 14331: 14328: 14325: 14322: 14294: 14289: 14285: 14281: 14276: 14269: 14265: 14260: 14234: 14227: 14223: 14218: 14195: 14188: 14184: 14179: 14158: 14155: 14150: 14143: 14139: 14134: 14129: 14126: 14123: 14116: 14113: 14103: 14099: 14093: 14089: 14085: 14079: 14076: 14051: 14048: 14045: 14042: 14035: 14031: 14026: 14022: 14017: 14010: 14006: 14001: 13978: 13975: 13971: 13966: 13963: 13960: 13953: 13950: 13944: 13940: 13936: 13933: 13930: 13925: 13921: 13917: 13914: 13911: 13908: 13904: 13900: 13896: 13892: 13886: 13883: 13858: 13855: 13852: 13849: 13842: 13838: 13833: 13829: 13824: 13817: 13813: 13808: 13785: 13780: 13773: 13769: 13764: 13758: 13751: 13747: 13742: 13738: 13733: 13726: 13722: 13717: 13692: 13685: 13681: 13676: 13652: 13649: 13644: 13637: 13633: 13628: 13623: 13619: 13616: 13613: 13608: 13604: 13600: 13597: 13594: 13591: 13587: 13583: 13576: 13572: 13566: 13562: 13558: 13552: 13549: 13524: 13521: 13518: 13515: 13508: 13504: 13499: 13495: 13490: 13483: 13479: 13474: 13451: 13448: 13444: 13440: 13437: 13434: 13429: 13425: 13421: 13418: 13415: 13412: 13408: 13404: 13401: 13395: 13392: 13367: 13364: 13361: 13358: 13351: 13347: 13342: 13338: 13333: 13326: 13322: 13317: 13286: 13279: 13275: 13270: 13266: 13261: 13254: 13250: 13245: 13218: 13211: 13207: 13202: 13179: 13174: 13167: 13163: 13158: 13152: 13145: 13141: 13136: 13132: 13127: 13120: 13116: 13111: 13105: 13098: 13094: 13089: 13085: 13080: 13073: 13069: 13064: 13060: 13014: 13010: 12987: 12983: 12962: 12957: 12953: 12949: 12944: 12940: 12936: 12914: 12910: 12885: 12881: 12873: 12869: 12864: 12860: 12836: 12832: 12824: 12820: 12815: 12801: 12798: 12783: 12776: 12772: 12767: 12744: 12740: 12712: 12708: 12701: 12696: 12690: 12686: 12678: 12674: 12666: 12662: 12657: 12653: 12650: 12645: 12641: 12636: 12632: 12627: 12624: 12619: 12614: 12607: 12603: 12598: 12594: 12571: 12566: 12562: 12558: 12554: 12550: 12523: 12520: 12517: 12513: 12488: 12485: 12482: 12479: 12475: 12467: 12463: 12457: 12454: 12451: 12447: 12443: 12437: 12431: 12427: 12419: 12415: 12407: 12403: 12398: 12394: 12391: 12386: 12382: 12377: 12373: 12369: 12363: 12359: 12351: 12347: 12339: 12335: 12330: 12326: 12323: 12318: 12314: 12309: 12302: 12299: 12297: 12293: 12286: 12282: 12277: 12273: 12270: 12267: 12266: 12263: 12259: 12255: 12252: 12244: 12240: 12234: 12228: 12224: 12216: 12212: 12204: 12200: 12195: 12191: 12188: 12183: 12179: 12174: 12166: 12161: 12158: 12155: 12151: 12146: 12140: 12137: 12132: 12129: 12127: 12123: 12116: 12112: 12107: 12103: 12100: 12097: 12096: 12070: 12066: 12058: 12054: 12049: 12026: 12022: 12016: 12009: 12005: 12000: 11996: 11993: 11989: 11985: 11980: 11977: 11973: 11968: 11962: 11955: 11951: 11946: 11942: 11939: 11935: 11928: 11925: 11922: 11919: 11914: 11911: 11907: 11902: 11896: 11889: 11885: 11880: 11876: 11873: 11869: 11862: 11859: 11856: 11853: 11848: 11845: 11841: 11837: 11832: 11825: 11821: 11816: 11787: 11784: 11781: 11778: 11775: 11772: 11767: 11764: 11759: 11756: 11753: 11750: 11747: 11744: 11741: 11738: 11735: 11732: 11729: 11725: 11717: 11713: 11707: 11702: 11698: 11690: 11686: 11678: 11674: 11669: 11665: 11662: 11657: 11653: 11649: 11641: 11636: 11632: 11629: 11625: 11620: 11617: 11615: 11609: 11605: 11597: 11593: 11588: 11584: 11581: 11578: 11577: 11574: 11571: 11565: 11562: 11557: 11554: 11551: 11547: 11539: 11535: 11529: 11524: 11520: 11512: 11508: 11500: 11496: 11491: 11487: 11484: 11479: 11475: 11471: 11465: 11462: 11460: 11454: 11450: 11442: 11438: 11433: 11429: 11426: 11423: 11422: 11396: 11392: 11384: 11380: 11375: 11354: 11351: 11329: 11325: 11300: 11296: 11288: 11284: 11279: 11258: 11253: 11249: 11245: 11238: 11234: 11226: 11222: 11217: 11166: 11162: 11154: 11150: 11145: 11120: 11116: 11108: 11104: 11099: 11074: 11070: 11062: 11058: 11053: 11032: 11025: 11021: 11013: 11009: 11004: 10996: 10992: 10984: 10980: 10975: 10971: 10964: 10960: 10952: 10948: 10943: 10935: 10931: 10923: 10919: 10914: 10910: 10903: 10899: 10891: 10887: 10882: 10878: 10873: 10869: 10861: 10857: 10849: 10845: 10840: 10836: 10833: 10830: 10823: 10819: 10811: 10807: 10802: 10766: 10762: 10754: 10750: 10745: 10741: 10734: 10730: 10722: 10718: 10713: 10699:is called the 10684: 10680: 10672: 10668: 10663: 10642: 10635: 10631: 10623: 10619: 10614: 10606: 10602: 10594: 10590: 10585: 10581: 10574: 10570: 10562: 10558: 10553: 10545: 10541: 10533: 10529: 10524: 10520: 10513: 10509: 10501: 10497: 10492: 10488: 10459: 10454: 10450: 10446: 10441: 10437: 10433: 10417: 10414: 10400: 10397: 10376: 10371: 10367: 10363: 10358: 10354: 10333: 10328: 10324: 10320: 10315: 10311: 10307: 10302: 10297: 10293: 10268: 10263: 10259: 10255: 10250: 10246: 10225: 10220: 10216: 10212: 10207: 10203: 10199: 10194: 10189: 10185: 10173:repelling LCSs 10160: 10155: 10151: 10147: 10142: 10138: 10115: 10110: 10104: 10101: 10099: 10096: 10095: 10092: 10089: 10086: 10084: 10081: 10080: 10078: 10073: 10070: 10066: 10062: 10058: 10053: 10049: 10045: 10038: 10034: 10026: 10022: 10017: 10013: 10010: 10007: 10004: 9999: 9995: 9991: 9984: 9980: 9972: 9968: 9963: 9958: 9952: 9949: 9944: 9941: 9936: 9932: 9928: 9921: 9917: 9909: 9905: 9900: 9873: 9869: 9861: 9857: 9852: 9836:null-geodesics 9829:metric tensors 9817:Shearless LCSs 9804: 9801: 9798: 9793: 9771: 9768: 9765: 9760: 9738: 9735: 9732: 9727: 9701: 9698: 9695: 9690: 9668: 9665: 9662: 9657: 9633: 9626: 9622: 9617: 9604: 9601: 9588: 9568: 9542: 9537: 9533: 9529: 9526: 9521: 9517: 9513: 9508: 9504: 9500: 9495: 9490: 9486: 9463: 9460: 9457: 9452: 9448: 9427: 9422: 9418: 9414: 9409: 9405: 9382: 9378: 9357: 9335: 9331: 9310: 9285: 9280: 9276: 9272: 9267: 9263: 9242: 9237: 9233: 9229: 9224: 9220: 9216: 9211: 9206: 9202: 9174: 9169: 9165: 9161: 9156: 9152: 9131: 9126: 9122: 9118: 9113: 9109: 9095: 9094: 9078: 9075: 9071: 9067: 9062: 9058: 9054: 9049: 9045: 9041: 9038: 9033: 9029: 9025: 9020: 9016: 9012: 9009: 9005: 8994: 8978: 8973: 8969: 8965: 8960: 8956: 8952: 8947: 8942: 8938: 8927: 8916: 8911: 8907: 8903: 8898: 8894: 8883: 8879: 8878: 8862: 8859: 8855: 8851: 8846: 8842: 8838: 8833: 8829: 8825: 8822: 8817: 8813: 8809: 8804: 8800: 8796: 8793: 8789: 8778: 8762: 8757: 8753: 8749: 8744: 8740: 8736: 8731: 8726: 8722: 8711: 8700: 8695: 8691: 8687: 8682: 8678: 8667: 8663: 8662: 8650: 8645: 8641: 8637: 8632: 8619: 8607: 8602: 8598: 8594: 8589: 8576: 8565: 8562: 8559: 8556: 8553: 8533: 8528: 8524: 8520: 8515: 8502: 8467: 8463: 8442: 8437: 8433: 8429: 8424: 8420: 8416: 8409: 8405: 8397: 8393: 8388: 8367: 8364: 8361: 8356: 8334: 8329: 8325: 8321: 8316: 8312: 8308: 8284: 8281: 8278: 8273: 8251: 8248: 8245: 8240: 8236: 8232: 8227: 8223: 8219: 8212: 8208: 8200: 8196: 8191: 8170: 8165: 8161: 8157: 8152: 8148: 8144: 8124: 8121: 8118: 8113: 8091: 8088: 8085: 8080: 8076: 8072: 8067: 8063: 8059: 8052: 8048: 8040: 8036: 8031: 8010:(cf. Fig. 7). 7999: 7994: 7990: 7986: 7981: 7977: 7973: 7966: 7962: 7954: 7950: 7945: 7924: 7919: 7915: 7909: 7906: 7902: 7898: 7891: 7887: 7879: 7875: 7870: 7849: 7844: 7840: 7836: 7831: 7809: 7804: 7800: 7796: 7789: 7785: 7777: 7773: 7768: 7764: 7742: 7738: 7717: 7712: 7708: 7704: 7697: 7693: 7685: 7681: 7676: 7672: 7652: 7647: 7643: 7639: 7634: 7625: 7621: 7616: 7591: 7586: 7582: 7578: 7573: 7564: 7560: 7555: 7531: 7526: 7522: 7518: 7513: 7489: 7485: 7462: 7458: 7437: 7432: 7428: 7424: 7419: 7415: 7411: 7399: 7396: 7395: 7394: 7391: 7388: 7381: 7376:are generally 7365: 7345: 7342: 7339: 7334: 7330: 7326: 7323: 7320: 7315: 7311: 7307: 7296: 7276: 7273: 7270: 7265: 7261: 7257: 7254: 7251: 7246: 7242: 7238: 7216: 7213: 7208: 7204: 7198: 7195: 7190: 7186: 7180: 7177: 7174: 7171: 7149: 7146: 7141: 7137: 7116: 7113: 7110: 7105: 7101: 7097: 7094: 7091: 7086: 7082: 7078: 7058: 7053: 7049: 7045: 7040: 7036: 7032: 7008: 6999:under varying 6988: 6983: 6979: 6975: 6970: 6967: 6962: 6958: 6952: 6949: 6944: 6940: 6934: 6931: 6928: 6925: 6903: 6900: 6897: 6892: 6888: 6884: 6881: 6878: 6873: 6869: 6865: 6852: 6849: 6828: 6825: 6822: 6800: 6796: 6773: 6769: 6738: 6733: 6724: 6718: 6714: 6710: 6706: 6701: 6696: 6692: 6688: 6683: 6679: 6675: 6670: 6666: 6662: 6659: 6656: 6651: 6647: 6643: 6640: 6635: 6631: 6627: 6622: 6618: 6614: 6609: 6605: 6601: 6598: 6595: 6590: 6586: 6579: 6574: 6568: 6564: 6560: 6556: 6551: 6546: 6542: 6538: 6533: 6529: 6525: 6520: 6516: 6512: 6509: 6506: 6501: 6497: 6493: 6490: 6485: 6481: 6477: 6472: 6468: 6464: 6459: 6455: 6451: 6448: 6445: 6440: 6436: 6429: 6424: 6418: 6414: 6410: 6406: 6401: 6396: 6392: 6388: 6383: 6379: 6375: 6370: 6366: 6362: 6359: 6356: 6351: 6347: 6343: 6340: 6335: 6331: 6327: 6322: 6318: 6314: 6309: 6305: 6301: 6298: 6295: 6290: 6286: 6279: 6278: 6272: 6266: 6262: 6258: 6254: 6249: 6244: 6240: 6236: 6231: 6227: 6223: 6218: 6214: 6210: 6207: 6204: 6199: 6195: 6191: 6188: 6183: 6179: 6175: 6170: 6166: 6162: 6157: 6153: 6149: 6146: 6143: 6138: 6134: 6127: 6122: 6116: 6112: 6108: 6104: 6099: 6094: 6090: 6086: 6081: 6077: 6073: 6068: 6064: 6060: 6057: 6054: 6049: 6045: 6041: 6038: 6033: 6029: 6025: 6020: 6016: 6012: 6007: 6003: 5999: 5996: 5993: 5988: 5984: 5977: 5972: 5966: 5962: 5958: 5954: 5949: 5944: 5940: 5936: 5931: 5927: 5923: 5918: 5914: 5910: 5907: 5904: 5899: 5895: 5891: 5888: 5883: 5879: 5875: 5870: 5866: 5862: 5857: 5853: 5849: 5846: 5843: 5838: 5834: 5827: 5826: 5820: 5814: 5810: 5806: 5802: 5797: 5792: 5788: 5784: 5779: 5775: 5771: 5766: 5762: 5758: 5755: 5752: 5747: 5743: 5739: 5736: 5731: 5727: 5723: 5718: 5714: 5710: 5705: 5701: 5697: 5694: 5691: 5686: 5682: 5675: 5670: 5664: 5660: 5656: 5652: 5647: 5642: 5638: 5634: 5629: 5625: 5621: 5616: 5612: 5608: 5605: 5602: 5597: 5593: 5589: 5586: 5581: 5577: 5573: 5568: 5564: 5560: 5555: 5551: 5547: 5544: 5541: 5536: 5532: 5525: 5520: 5514: 5510: 5506: 5502: 5497: 5492: 5488: 5484: 5479: 5475: 5471: 5466: 5462: 5458: 5455: 5452: 5447: 5443: 5439: 5436: 5431: 5427: 5423: 5418: 5414: 5410: 5405: 5401: 5397: 5394: 5391: 5386: 5382: 5375: 5374: 5372: 5367: 5364: 5359: 5355: 5351: 5346: 5339: 5335: 5330: 5326: 5306: 5301: 5297: 5293: 5288: 5284: 5280: 5277: 5274: 5271: 5251: 5246: 5242: 5238: 5233: 5229: 5225: 5220: 5216: 5212: 5209: 5206: 5184: 5180: 5159: 5154: 5150: 5146: 5141: 5137: 5133: 5130: 5127: 5124: 5104: 5099: 5095: 5091: 5086: 5079: 5075: 5070: 5066: 5035: 5031: 5023: 5019: 5013: 5010: 5007: 5004: 4980: 4976: 4955: 4950: 4946: 4942: 4935: 4931: 4923: 4919: 4913: 4910: 4907: 4904: 4880: 4876: 4839: 4836: 4831: 4827: 4823: 4818: 4814: 4810: 4807: 4801: 4796: 4792: 4788: 4783: 4779: 4775: 4772: 4768: 4763: 4760: 4755: 4751: 4747: 4740: 4736: 4728: 4724: 4718: 4715: 4712: 4709: 4681: 4676: 4672: 4668: 4663: 4659: 4655: 4651: 4647: 4639: 4635: 4627: 4623: 4618: 4613: 4610: 4607: 4585: 4580: 4576: 4572: 4567: 4563: 4557: 4554: 4549: 4545: 4541: 4534: 4530: 4522: 4518: 4513: 4490: 4486: 4457: 4453: 4445: 4441: 4436: 4432: 4427: 4422: 4414: 4410: 4402: 4398: 4393: 4389: 4385: 4380: 4373: 4369: 4361: 4357: 4352: 4322: 4318: 4313: 4309: 4305: 4302: 4299: 4294: 4290: 4286: 4279: 4275: 4267: 4263: 4258: 4254: 4251: 4246: 4242: 4238: 4235: 4231: 4223: 4220: 4216: 4212: 4207: 4203: 4199: 4196: 4192: 4187: 4183: 4180: 4178: 4176: 4170: 4166: 4161: 4157: 4153: 4150: 4147: 4142: 4138: 4134: 4127: 4123: 4115: 4111: 4106: 4102: 4099: 4096: 4091: 4087: 4083: 4080: 4077: 4072: 4068: 4064: 4057: 4053: 4045: 4041: 4036: 4032: 4028: 4020: 4017: 4013: 4009: 4004: 4000: 3996: 3993: 3989: 3984: 3980: 3977: 3975: 3973: 3969: 3965: 3960: 3956: 3952: 3947: 3943: 3939: 3934: 3930: 3925: 3919: 3916: 3912: 3908: 3903: 3899: 3895: 3892: 3888: 3883: 3877: 3874: 3867: 3864: 3861: 3857: 3853: 3850: 3848: 3846: 3841: 3837: 3833: 3826: 3822: 3814: 3810: 3805: 3801: 3800: 3776: 3772: 3751: 3746: 3742: 3738: 3734: 3730: 3727: 3722: 3718: 3714: 3707: 3703: 3695: 3691: 3686: 3682: 3679: 3676: 3671: 3667: 3663: 3658: 3654: 3650: 3645: 3640: 3618: 3613: 3609: 3605: 3600: 3596: 3592: 3589: 3586: 3583: 3561: 3556: 3534: 3529: 3525: 3521: 3518: 3498: 3495: 3492: 3489: 3486: 3464: 3460: 3439: 3434: 3430: 3426: 3422: 3418: 3396: 3389: 3385: 3380: 3376: 3356: 3351: 3347: 3343: 3338: 3334: 3330: 3327: 3324: 3321: 3301: 3298: 3295: 3291: 3267: 3262: 3258: 3254: 3249: 3227: 3222: 3218: 3214: 3209: 3187: 3182: 3178: 3174: 3169: 3155: 3152: 3121: 3116: 3111: 3095:attracting LCS 3082: 3079: 3066: 3046: 3026: 3006: 3003: 3000: 2997: 2994: 2991: 2988: 2985: 2982: 2979: 2976: 2973: 2970: 2967: 2964: 2961: 2958: 2955: 2935: 2932: 2929: 2926: 2923: 2920: 2917: 2914: 2911: 2908: 2905: 2902: 2882: 2862: 2816: 2812: 2808: 2788: 2785: 2782: 2777: 2772: 2769: 2766: 2761: 2756: 2751: 2748: 2745: 2740: 2735: 2732: 2727: 2724: 2698: 2695: 2690: 2683: 2680: 2676: 2673: 2670: 2667: 2664: 2661: 2657: 2654: 2651: 2648: 2645: 2642: 2641: 2637: 2634: 2630: 2627: 2624: 2621: 2618: 2615: 2612: 2608: 2605: 2602: 2601: 2599: 2594: 2591: 2588: 2585: 2582: 2579: 2576: 2573: 2567: 2564: 2539: 2536: 2533: 2530: 2527: 2524: 2502: 2495: 2491: 2486: 2482: 2462: 2459: 2456: 2453: 2450: 2446: 2425: 2422: 2419: 2416: 2413: 2410: 2407: 2384: 2381: 2378: 2375: 2372: 2369: 2349: 2346: 2343: 2340: 2337: 2334: 2314: 2311: 2308: 2305: 2302: 2299: 2273: 2270: 2267: 2264: 2258: 2255: 2247: 2243: 2239: 2236: 2233: 2230: 2227: 2224: 2221: 2218: 2215: 2212: 2209: 2206: 2203: 2200: 2195: 2191: 2187: 2184: 2181: 2178: 2175: 2172: 2169: 2166: 2163: 2157: 2154: 2147: 2144: 2141: 2138: 2135: 2132: 2129: 2126: 2123: 2120: 2117: 2112: 2108: 2104: 2101: 2098: 2095: 2092: 2089: 2086: 2083: 2080: 2074: 2071: 2048: 2044: 2038: 2034: 2030: 2027: 2024: 2021: 2018: 2015: 2012: 2009: 2006: 2003: 2000: 1997: 1994: 1991: 1988: 1985: 1981: 1975: 1972: 1967: 1964: 1961: 1958: 1955: 1952: 1949: 1945: 1941: 1935: 1931: 1927: 1924: 1921: 1918: 1915: 1912: 1909: 1906: 1903: 1900: 1897: 1894: 1891: 1888: 1885: 1882: 1878: 1872: 1869: 1864: 1861: 1858: 1855: 1852: 1849: 1846: 1826: 1823: 1820: 1817: 1814: 1811: 1791: 1788: 1785: 1782: 1779: 1776: 1753: 1733: 1730: 1727: 1724: 1704: 1701: 1698: 1678: 1675: 1672: 1669: 1647: 1642: 1637: 1634: 1614: 1611: 1608: 1605: 1602: 1599: 1596: 1593: 1590: 1587: 1584: 1581: 1578: 1553: 1531: 1526: 1521: 1516: 1513: 1510: 1507: 1503: 1500: 1497: 1494: 1491: 1488: 1485: 1482: 1479: 1466: 1463: 1436: 1406: 1385: 1382: 1354: 1332: 1329: 1326: 1321: 1297: 1273: 1266: 1262: 1257: 1226: 1185: 1180: 1176: 1172: 1167: 1145: 1140: 1136: 1132: 1127: 1103: 1078: 1075: 1072: 1067: 1033: 1011: 1008: 1005: 1000: 978: 975: 960: 955: 950: 945: 940: 918: 913: 909: 905: 900: 870: 848: 845: 840: 836: 832: 827: 822: 817: 810: 806: 801: 797: 794: 791: 788: 783: 770:fluid dynamics 755: 750: 745: 714: 711: 706: 702: 698: 693: 688: 685: 682: 679: 674: 671: 667: 661: 654: 650: 645: 641: 638: 632: 627: 622: 617: 614: 611: 608: 605: 602: 599: 596: 591: 565: 543: 538: 534: 530: 525: 501: 496: 493: 482:diffeomorphism 467: 460: 456: 451: 428: 423: 420: 398: 393: 390: 385: 381: 377: 372: 368: 364: 361: 358: 355: 333: 328: 323: 319: 298: 293: 289: 285: 280: 276: 272: 269: 266: 263: 260: 255: 251: 247: 242: 235: 231: 226: 205: 200: 196: 192: 187: 183: 179: 176: 171: 147: 122: 119: 117: 114: 72: 71: 31: 29: 22: 15: 9: 6: 4: 3: 2: 19685: 19674: 19671: 19669: 19666: 19664: 19661: 19659: 19656: 19654: 19651: 19650: 19648: 19637: 19633: 19629: 19625: 19620: 19615: 19611: 19607: 19603: 19599: 19594: 19590: 19586: 19582: 19578: 19574: 19570: 19566: 19562: 19558: 19553: 19552: 19547: 19546: 19545: 19539: 19535: 19507: 19503: 19499: 19495: 19489: 19485: 19481: 19477: 19473: 19466: 19459: 19451: 19447: 19440: 19432: 19425: 19417: 19413: 19406: 19398: 19391: 19383: 19376: 19368: 19364: 19357: 19348: 19343: 19339: 19335: 19331: 19327: 19323: 19316: 19308: 19304: 19300: 19296: 19292: 19288: 19285:(1): 017514. 19284: 19280: 19273: 19265: 19261: 19257: 19253: 19249: 19245: 19240: 19235: 19231: 19227: 19220: 19212: 19208: 19204: 19200: 19195: 19190: 19186: 19182: 19175: 19173: 19171: 19169: 19160: 19156: 19152: 19148: 19144: 19140: 19133: 19125: 19121: 19117: 19113: 19109: 19105: 19100: 19095: 19091: 19087: 19080: 19078: 19076: 19074: 19072: 19070: 19068: 19066: 19064: 19055: 19051: 19047: 19043: 19039: 19035: 19030: 19025: 19021: 19017: 19010: 19008: 19006: 18997: 18993: 18989: 18985: 18981: 18977: 18972: 18967: 18963: 18959: 18952: 18950: 18948: 18946: 18944: 18942: 18933: 18929: 18925: 18921: 18917: 18913: 18908: 18903: 18899: 18892: 18890: 18888: 18886: 18877: 18873: 18869: 18865: 18861: 18857: 18852: 18847: 18843: 18836: 18834: 18832: 18830: 18828: 18826: 18817: 18813: 18809: 18805: 18801: 18797: 18794:(1): 013128. 18793: 18789: 18782: 18773: 18768: 18764: 18760: 18756: 18752: 18748: 18741: 18739: 18730: 18726: 18722: 18718: 18714: 18710: 18703: 18701: 18692: 18686: 18682: 18678: 18674: 18667: 18659: 18655: 18651: 18647: 18643: 18639: 18632: 18624: 18620: 18616: 18612: 18608: 18604: 18601:(1): 017504. 18600: 18596: 18589: 18581: 18577: 18570: 18568: 18559: 18555: 18551: 18547: 18544:(6): 065404. 18543: 18539: 18532: 18525: 18523: 18521: 18512: 18508: 18504: 18500: 18496: 18492: 18488: 18481: 18479: 18477: 18475: 18473: 18464: 18460: 18456: 18450: 18446: 18442: 18437: 18432: 18428: 18421: 18412: 18407: 18403: 18399: 18395: 18391: 18387: 18380: 18372: 18368: 18364: 18360: 18356: 18352: 18345: 18343: 18334: 18330: 18326: 18322: 18318: 18314: 18310: 18306: 18299: 18291: 18287: 18282: 18277: 18273: 18269: 18265: 18261: 18254: 18246: 18242: 18238: 18234: 18230: 18226: 18222: 18218: 18211: 18203: 18199: 18195: 18191: 18187: 18183: 18176: 18168: 18164: 18160: 18154: 18150: 18146: 18141: 18136: 18133:. p. 1. 18132: 18125: 18117: 18113: 18108: 18103: 18099: 18095: 18091: 18087: 18084:(3): 033122. 18083: 18079: 18075: 18068: 18060: 18056: 18052: 18048: 18044: 18040: 18033: 18024: 18019: 18015: 18011: 18007: 18003: 17999: 17992: 17983: 17978: 17974: 17970: 17966: 17962: 17958: 17950: 17941: 17936: 17932: 17928: 17924: 17920: 17916: 17909: 17901: 17897: 17892: 17887: 17882: 17877: 17873: 17869: 17865: 17861: 17857: 17850: 17842: 17838: 17834: 17830: 17826: 17822: 17815: 17807: 17803: 17799: 17795: 17791: 17787: 17780: 17772: 17768: 17764: 17760: 17756: 17752: 17745: 17743: 17741: 17739: 17737: 17735: 17733: 17731: 17729: 17727: 17725: 17723: 17714: 17710: 17706: 17702: 17698: 17694: 17693:Physics Today 17687: 17679: 17675: 17671: 17667: 17663: 17659: 17652: 17650: 17648: 17646: 17644: 17642: 17640: 17638: 17633: 17623: 17620: 17618: 17615: 17613: 17610: 17608: 17605: 17603: 17600: 17598: 17595: 17594: 17585: 17581: 17578: 17574: 17571: 17568: 17564: 17561: 17557: 17554: 17550: 17547: 17543: 17542: 17541: 17540:calculation: 17539: 17524: 17520: 17504: 17500: 17477: 17473: 17447: 17443: 17434: 17430: 17426: 17418: 17414: 17405: 17401: 17390: 17372: 17368: 17360: 17356: 17351: 17343: 17342:metric tensor 17340: 17329: 17310: 17306: 17295: 17291: 17283: 17279: 17274: 17270: 17267: 17264: 17256: 17246: 17243: 17234: 17232: 17216: 17194: 17190: 17167: 17163: 17142: 17122: 17098: 17094: 17086: 17082: 17077: 17068: 17052: 17030: 17025: 17021: 17011: 16995: 16991: 16982: 16978: 16955: 16951: 16942: 16937: 16933: 16912: 16909: 16906: 16897: 16884: 16876: 16872: 16863: 16859: 16846: 16842: 16833: 16829: 16825: 16817: 16813: 16804: 16800: 16789: 16785: 16776: 16772: 16768: 16763: 16759: 16751: 16743: 16739: 16730: 16726: 16713: 16709: 16700: 16696: 16692: 16684: 16680: 16671: 16667: 16659: 16655: 16651: 16643: 16639: 16630: 16626: 16618: 16610: 16606: 16597: 16592: 16588: 16579: 16565: 16562: 16559: 16536: 16533: 16530: 16523: 16519: 16511: 16507: 16502: 16493: 16490: 16463: 16460: 16457: 16437: 16429: 16413: 16404: 16390: 16386: 16343: 16327: 16318: 16314: 16306: 16302: 16293: 16289: 16285: 16275: 16271: 16247: 16239: 16235: 16209: 16205: 16196: 16192: 16165: 16118: 16082: 16066: 16063: 16052: 16048: 16039: 16035: 16031: 16023: 16019: 16010: 16006: 16002: 15989: 15986: 15965: 15961: 15952: 15948: 15944: 15934: 15930: 15922: 15903: 15899: 15890: 15886: 15878: 15874: 15853: 15849: 15810: 15806: 15767: 15763: 15740: 15734: 15731: 15718: 15710: 15706: 15697: 15693: 15678: 15674: 15665: 15661: 15655: 15645: 15641: 15632: 15628: 15615: 15611: 15602: 15598: 15590: 15582: 15578: 15569: 15565: 15550: 15546: 15537: 15533: 15527: 15517: 15513: 15504: 15500: 15487: 15483: 15474: 15470: 15462: 15454: 15450: 15441: 15437: 15416: 15408: 15404: 15395: 15391: 15378: 15374: 15365: 15361: 15357: 15349: 15345: 15336: 15332: 15321: 15317: 15308: 15304: 15300: 15297: 15290: 15282: 15278: 15269: 15265: 15252: 15248: 15239: 15235: 15231: 15223: 15219: 15210: 15206: 15200: 15197: 15189: 15185: 15176: 15172: 15164: 15156: 15152: 15143: 15139: 15129: 15110: 15106: 15095: 15091: 15083: 15079: 15074: 15048: 15044: 15026: 15022: 15017: 14993: 14958: 14954: 14940: 14935: 14931: 14905: 14901: 14897: 14892: 14888: 14866: 14862: 14843: 14839: 14835: 14830: 14826: 14807: 14803: 14786: 14783: 14780: 14774: 14751: 14745: 14736: 14723: 14720: 14717: 14712: 14705: 14693: 14687: 14681: 14678: 14670: 14666: 14662: 14659: 14653: 14647: 14643: 14635: 14631: 14623: 14619: 14614: 14610: 14602: 14598: 14587: 14583: 14575: 14571: 14529: 14525: 14521: 14516: 14512: 14488: 14476: 14470: 14450: 14447: 14440: 14437: 14434: 14428: 14420: 14414: 14410: 14403: 14397: 14385: 14359: 14356: 14353: 14347: 14344: 14338: 14332: 14329: 14326: 14320: 14312: 14308: 14287: 14283: 14274: 14267: 14263: 14258: 14250: 14232: 14225: 14221: 14193: 14186: 14182: 14156: 14153: 14148: 14141: 14137: 14127: 14124: 14121: 14111: 14101: 14097: 14091: 14083: 14077: 14074: 14063: 14046: 14040: 14033: 14029: 14024: 14020: 14015: 14008: 14004: 13989: 13976: 13973: 13969: 13964: 13961: 13958: 13948: 13942: 13938: 13934: 13931: 13923: 13919: 13915: 13912: 13906: 13902: 13898: 13894: 13890: 13884: 13881: 13870: 13853: 13847: 13840: 13836: 13827: 13822: 13815: 13811: 13796: 13783: 13778: 13771: 13767: 13756: 13749: 13745: 13736: 13731: 13724: 13720: 13715: 13706: 13690: 13683: 13679: 13674: 13664: 13650: 13647: 13642: 13635: 13631: 13626: 13621: 13617: 13614: 13606: 13602: 13598: 13595: 13589: 13585: 13581: 13574: 13570: 13564: 13560: 13556: 13550: 13547: 13536: 13519: 13513: 13506: 13502: 13493: 13488: 13481: 13477: 13472: 13462: 13449: 13446: 13442: 13438: 13435: 13427: 13423: 13419: 13416: 13410: 13406: 13402: 13399: 13393: 13390: 13379: 13362: 13356: 13349: 13345: 13336: 13331: 13324: 13320: 13315: 13306: 13302: 13284: 13277: 13273: 13268: 13264: 13259: 13252: 13248: 13243: 13234: 13216: 13209: 13205: 13200: 13190: 13177: 13172: 13165: 13161: 13156: 13150: 13143: 13139: 13134: 13130: 13125: 13118: 13114: 13109: 13103: 13096: 13092: 13087: 13083: 13078: 13071: 13067: 13062: 13050: 13041: 13032: 13028: 13012: 13008: 12985: 12981: 12955: 12951: 12947: 12942: 12938: 12912: 12908: 12883: 12879: 12871: 12867: 12862: 12834: 12830: 12822: 12818: 12813: 12797: 12781: 12774: 12770: 12765: 12742: 12738: 12728: 12710: 12706: 12699: 12694: 12688: 12684: 12676: 12672: 12664: 12660: 12655: 12648: 12643: 12639: 12634: 12630: 12625: 12622: 12617: 12612: 12605: 12601: 12596: 12592: 12569: 12564: 12560: 12556: 12552: 12539: 12521: 12518: 12515: 12511: 12486: 12483: 12480: 12477: 12473: 12465: 12461: 12455: 12452: 12449: 12445: 12441: 12435: 12429: 12425: 12417: 12413: 12405: 12401: 12396: 12389: 12384: 12380: 12375: 12371: 12367: 12361: 12357: 12349: 12345: 12337: 12333: 12328: 12321: 12316: 12312: 12307: 12300: 12298: 12291: 12284: 12280: 12275: 12271: 12268: 12261: 12257: 12253: 12250: 12242: 12238: 12232: 12226: 12222: 12214: 12210: 12202: 12198: 12193: 12186: 12181: 12177: 12172: 12164: 12159: 12156: 12153: 12149: 12144: 12138: 12135: 12130: 12128: 12121: 12114: 12110: 12105: 12101: 12098: 12086: 12068: 12064: 12056: 12052: 12047: 12037: 12024: 12020: 12014: 12007: 12003: 11998: 11994: 11991: 11987: 11983: 11978: 11975: 11971: 11966: 11960: 11953: 11949: 11944: 11940: 11937: 11933: 11912: 11909: 11905: 11900: 11894: 11887: 11883: 11878: 11874: 11871: 11867: 11846: 11843: 11839: 11835: 11830: 11823: 11819: 11814: 11805: 11802: 11785: 11779: 11776: 11773: 11765: 11762: 11754: 11751: 11748: 11742: 11736: 11733: 11730: 11723: 11715: 11711: 11700: 11696: 11688: 11684: 11676: 11672: 11667: 11660: 11655: 11651: 11639: 11634: 11630: 11627: 11623: 11618: 11616: 11607: 11603: 11595: 11591: 11586: 11582: 11579: 11572: 11569: 11563: 11560: 11555: 11552: 11549: 11545: 11537: 11533: 11522: 11518: 11510: 11506: 11498: 11494: 11489: 11482: 11477: 11473: 11463: 11461: 11452: 11448: 11440: 11436: 11431: 11427: 11424: 11412: 11394: 11390: 11382: 11378: 11373: 11352: 11349: 11327: 11323: 11298: 11294: 11286: 11282: 11277: 11251: 11247: 11236: 11232: 11224: 11220: 11215: 11207: 11199: 11194: 11186: 11182: 11164: 11160: 11152: 11148: 11143: 11118: 11114: 11106: 11102: 11097: 11072: 11068: 11060: 11056: 11051: 11030: 11023: 11019: 11011: 11007: 11002: 10994: 10990: 10982: 10978: 10973: 10969: 10962: 10958: 10950: 10946: 10941: 10933: 10929: 10921: 10917: 10912: 10908: 10901: 10897: 10889: 10885: 10880: 10871: 10859: 10855: 10847: 10843: 10838: 10828: 10821: 10817: 10809: 10805: 10800: 10790: 10788: 10784: 10764: 10760: 10752: 10748: 10743: 10739: 10732: 10728: 10720: 10716: 10711: 10702: 10682: 10678: 10670: 10666: 10661: 10640: 10633: 10629: 10621: 10617: 10612: 10604: 10600: 10592: 10588: 10583: 10579: 10572: 10568: 10560: 10556: 10551: 10543: 10539: 10531: 10527: 10522: 10518: 10511: 10507: 10499: 10495: 10490: 10478: 10476: 10471: 10452: 10448: 10444: 10439: 10435: 10423: 10413: 10411: 10407: 10399:Elliptic LCSs 10396: 10394: 10389: 10369: 10365: 10356: 10352: 10326: 10322: 10313: 10309: 10305: 10295: 10291: 10282: 10261: 10257: 10248: 10244: 10218: 10214: 10205: 10201: 10197: 10187: 10183: 10174: 10153: 10149: 10140: 10136: 10126: 10113: 10108: 10102: 10097: 10090: 10087: 10082: 10076: 10071: 10064: 10060: 10051: 10047: 10036: 10032: 10024: 10020: 10015: 10008: 9997: 9993: 9982: 9978: 9970: 9966: 9961: 9956: 9950: 9947: 9942: 9934: 9930: 9919: 9915: 9907: 9903: 9898: 9889: 9871: 9867: 9859: 9855: 9850: 9841: 9837: 9832: 9830: 9826: 9825:Elliptic LCSs 9822: 9818: 9799: 9766: 9733: 9715: 9696: 9663: 9631: 9624: 9620: 9615: 9600: 9558: 9553: 9540: 9531: 9527: 9519: 9515: 9506: 9502: 9498: 9488: 9484: 9475: 9458: 9450: 9446: 9420: 9416: 9407: 9403: 9376: 9329: 9278: 9274: 9265: 9261: 9235: 9231: 9222: 9218: 9214: 9204: 9200: 9190: 9186: 9167: 9163: 9154: 9150: 9124: 9120: 9111: 9107: 9092: 9076: 9073: 9069: 9060: 9056: 9047: 9043: 9039: 9031: 9027: 9018: 9014: 9010: 9003: 8995: 8992: 8971: 8967: 8958: 8954: 8950: 8940: 8936: 8928: 8909: 8905: 8896: 8892: 8884: 8880: 8876: 8860: 8857: 8853: 8844: 8840: 8831: 8827: 8823: 8815: 8811: 8802: 8798: 8794: 8787: 8779: 8776: 8775:stretch lines 8755: 8751: 8742: 8738: 8734: 8724: 8720: 8712: 8693: 8689: 8680: 8676: 8668: 8664: 8643: 8639: 8600: 8596: 8563: 8560: 8557: 8554: 8551: 8526: 8522: 8499: 8493: 8491: 8485: 8483: 8465: 8461: 8435: 8431: 8427: 8422: 8418: 8407: 8403: 8395: 8391: 8386: 8362: 8327: 8323: 8319: 8314: 8310: 8298: 8279: 8262:signals that 8249: 8246: 8238: 8234: 8230: 8225: 8221: 8210: 8206: 8198: 8194: 8189: 8181:. Similarly, 8163: 8159: 8155: 8150: 8146: 8119: 8089: 8086: 8078: 8074: 8070: 8065: 8061: 8050: 8046: 8038: 8034: 8029: 8015: 8011: 7992: 7988: 7984: 7979: 7975: 7964: 7960: 7952: 7948: 7943: 7917: 7913: 7907: 7904: 7900: 7889: 7885: 7877: 7873: 7868: 7842: 7838: 7802: 7798: 7787: 7783: 7775: 7771: 7766: 7755:normal under 7740: 7736: 7710: 7706: 7695: 7691: 7683: 7679: 7674: 7645: 7641: 7623: 7619: 7614: 7605: 7604:tangent space 7584: 7580: 7562: 7558: 7553: 7545: 7544:tangent space 7524: 7520: 7487: 7483: 7460: 7456: 7430: 7426: 7422: 7417: 7413: 7392: 7389: 7386: 7382: 7379: 7363: 7340: 7337: 7332: 7328: 7324: 7321: 7318: 7313: 7309: 7297: 7294: 7293: 7292: 7289: 7271: 7268: 7263: 7259: 7255: 7252: 7249: 7244: 7240: 7214: 7211: 7206: 7202: 7196: 7193: 7188: 7184: 7147: 7144: 7139: 7135: 7111: 7108: 7103: 7099: 7095: 7092: 7089: 7084: 7080: 7051: 7047: 7043: 7038: 7034: 7022: 7006: 6981: 6977: 6968: 6965: 6960: 6956: 6950: 6947: 6942: 6938: 6898: 6895: 6890: 6886: 6882: 6879: 6876: 6871: 6867: 6844: 6840: 6826: 6823: 6820: 6798: 6794: 6771: 6767: 6753: 6749: 6736: 6731: 6722: 6716: 6712: 6708: 6704: 6694: 6690: 6686: 6681: 6677: 6673: 6668: 6664: 6660: 6657: 6649: 6645: 6641: 6633: 6629: 6625: 6620: 6616: 6612: 6607: 6603: 6599: 6596: 6588: 6584: 6572: 6566: 6562: 6558: 6554: 6544: 6540: 6536: 6531: 6527: 6523: 6518: 6514: 6510: 6507: 6499: 6495: 6491: 6483: 6479: 6475: 6470: 6466: 6462: 6457: 6453: 6449: 6446: 6438: 6434: 6422: 6416: 6412: 6408: 6404: 6394: 6390: 6386: 6381: 6377: 6373: 6368: 6364: 6360: 6357: 6349: 6345: 6341: 6333: 6329: 6325: 6320: 6316: 6312: 6307: 6303: 6299: 6296: 6288: 6284: 6270: 6264: 6260: 6256: 6252: 6242: 6238: 6234: 6229: 6225: 6221: 6216: 6212: 6208: 6205: 6197: 6193: 6189: 6181: 6177: 6173: 6168: 6164: 6160: 6155: 6151: 6147: 6144: 6136: 6132: 6120: 6114: 6110: 6106: 6102: 6092: 6088: 6084: 6079: 6075: 6071: 6066: 6062: 6058: 6055: 6047: 6043: 6039: 6031: 6027: 6023: 6018: 6014: 6010: 6005: 6001: 5997: 5994: 5986: 5982: 5970: 5964: 5960: 5956: 5952: 5942: 5938: 5934: 5929: 5925: 5921: 5916: 5912: 5908: 5905: 5897: 5893: 5889: 5881: 5877: 5873: 5868: 5864: 5860: 5855: 5851: 5847: 5844: 5836: 5832: 5818: 5812: 5808: 5804: 5800: 5790: 5786: 5782: 5777: 5773: 5769: 5764: 5760: 5756: 5753: 5745: 5741: 5737: 5729: 5725: 5721: 5716: 5712: 5708: 5703: 5699: 5695: 5692: 5684: 5680: 5668: 5662: 5658: 5654: 5650: 5640: 5636: 5632: 5627: 5623: 5619: 5614: 5610: 5606: 5603: 5595: 5591: 5587: 5579: 5575: 5571: 5566: 5562: 5558: 5553: 5549: 5545: 5542: 5534: 5530: 5518: 5512: 5508: 5504: 5500: 5490: 5486: 5482: 5477: 5473: 5469: 5464: 5460: 5456: 5453: 5445: 5441: 5437: 5429: 5425: 5421: 5416: 5412: 5408: 5403: 5399: 5395: 5392: 5384: 5380: 5370: 5365: 5357: 5353: 5344: 5337: 5333: 5328: 5299: 5295: 5291: 5286: 5282: 5278: 5275: 5269: 5244: 5240: 5236: 5231: 5227: 5223: 5218: 5214: 5207: 5204: 5182: 5178: 5152: 5148: 5144: 5139: 5135: 5131: 5128: 5122: 5097: 5093: 5084: 5077: 5073: 5068: 5056: 5051: 5033: 5029: 5021: 5017: 4978: 4974: 4948: 4944: 4933: 4929: 4921: 4917: 4878: 4874: 4865: 4864: 4854: 4850: 4837: 4829: 4825: 4816: 4812: 4808: 4805: 4794: 4790: 4786: 4781: 4777: 4770: 4766: 4761: 4753: 4749: 4738: 4734: 4726: 4722: 4697: 4695: 4674: 4670: 4666: 4661: 4657: 4649: 4637: 4633: 4625: 4621: 4616: 4611: 4608: 4578: 4574: 4565: 4561: 4555: 4547: 4543: 4532: 4528: 4520: 4516: 4511: 4488: 4484: 4475: 4455: 4451: 4443: 4439: 4434: 4425: 4420: 4412: 4408: 4400: 4396: 4391: 4383: 4378: 4371: 4367: 4359: 4355: 4350: 4340: 4320: 4311: 4307: 4300: 4292: 4288: 4277: 4273: 4265: 4261: 4256: 4252: 4244: 4240: 4233: 4229: 4221: 4218: 4214: 4205: 4201: 4194: 4190: 4181: 4179: 4168: 4159: 4155: 4148: 4140: 4136: 4125: 4121: 4113: 4109: 4104: 4097: 4089: 4085: 4078: 4070: 4066: 4055: 4051: 4043: 4039: 4034: 4026: 4018: 4015: 4011: 4002: 3998: 3991: 3987: 3978: 3976: 3967: 3958: 3954: 3950: 3945: 3941: 3932: 3928: 3923: 3917: 3914: 3910: 3901: 3897: 3890: 3886: 3875: 3872: 3865: 3859: 3851: 3849: 3839: 3835: 3824: 3820: 3812: 3808: 3803: 3790: 3774: 3770: 3744: 3740: 3732: 3728: 3720: 3716: 3705: 3701: 3693: 3689: 3684: 3677: 3669: 3665: 3661: 3656: 3652: 3643: 3638: 3611: 3607: 3603: 3598: 3594: 3590: 3587: 3581: 3559: 3527: 3523: 3516: 3496: 3493: 3490: 3487: 3484: 3462: 3458: 3432: 3428: 3420: 3416: 3394: 3387: 3383: 3378: 3349: 3345: 3341: 3336: 3332: 3328: 3325: 3319: 3296: 3289: 3279: 3260: 3256: 3220: 3216: 3180: 3176: 3151: 3149: 3145: 3141: 3137: 3136:repelling LCS 3114: 3100: 3096: 3087: 3078: 3064: 3044: 3024: 3001: 2995: 2992: 2986: 2980: 2974: 2968: 2965: 2959: 2953: 2930: 2924: 2921: 2918: 2912: 2906: 2903: 2900: 2880: 2860: 2852: 2847: 2843: 2833: 2829: 2810: 2786: 2783: 2775: 2767: 2759: 2754: 2746: 2733: 2730: 2725: 2722: 2714: 2709: 2696: 2693: 2688: 2681: 2678: 2674: 2671: 2668: 2662: 2659: 2655: 2652: 2649: 2646: 2643: 2635: 2632: 2628: 2625: 2622: 2619: 2613: 2610: 2606: 2603: 2597: 2592: 2586: 2583: 2580: 2574: 2571: 2565: 2562: 2551: 2534: 2531: 2528: 2522: 2500: 2493: 2489: 2484: 2457: 2454: 2451: 2444: 2420: 2417: 2414: 2408: 2396: 2379: 2376: 2373: 2367: 2344: 2341: 2338: 2332: 2309: 2306: 2303: 2297: 2289: 2284: 2271: 2265: 2256: 2253: 2245: 2237: 2231: 2228: 2222: 2216: 2210: 2207: 2204: 2198: 2193: 2185: 2179: 2176: 2170: 2167: 2164: 2152: 2145: 2139: 2133: 2127: 2124: 2121: 2115: 2110: 2102: 2096: 2093: 2087: 2084: 2081: 2069: 2046: 2042: 2036: 2025: 2022: 2019: 2013: 2004: 1998: 1995: 1992: 1986: 1979: 1973: 1970: 1965: 1959: 1956: 1953: 1947: 1943: 1939: 1933: 1922: 1919: 1916: 1910: 1901: 1895: 1892: 1889: 1883: 1876: 1870: 1867: 1862: 1856: 1853: 1850: 1844: 1821: 1818: 1815: 1809: 1786: 1783: 1780: 1774: 1767: 1751: 1728: 1722: 1702: 1699: 1696: 1673: 1667: 1645: 1635: 1632: 1612: 1606: 1600: 1597: 1594: 1588: 1582: 1579: 1576: 1568: 1551: 1544:the open set 1542: 1529: 1524: 1514: 1511: 1508: 1505: 1501: 1495: 1492: 1489: 1483: 1480: 1477: 1457: 1453: 1451: 1434: 1426: 1422: 1395: 1391: 1381: 1378: 1377:more strongly 1374: 1368: 1327: 1271: 1264: 1260: 1255: 1246: 1242: 1214: 1209: 1207: 1203: 1199: 1178: 1174: 1138: 1134: 1091: 1073: 1056: 1052: 1047: 1006: 983: 974: 953: 943: 911: 907: 888: 887: 838: 834: 815: 808: 804: 799: 795: 789: 771: 748: 734: 730: 725: 704: 700: 686: 680: 672: 669: 659: 652: 648: 643: 636: 625: 615: 609: 606: 603: 594: 579: 536: 532: 494: 491: 483: 465: 458: 454: 449: 421: 418: 411:for any time 391: 383: 379: 375: 370: 366: 362: 359: 353: 326: 321: 317: 291: 287: 283: 278: 274: 270: 267: 261: 253: 249: 245: 240: 233: 229: 224: 198: 194: 190: 185: 181: 174: 136: 127: 113: 110: 106: 102: 99: 95: 91: 87: 78: 68: 65: 57: 47: 43: 38: 36: 30: 21: 20: 19668:Chaos theory 19601: 19597: 19567:(1): 39–58. 19564: 19560: 19542: 19541: 19540:profile for 19537: 19506:the original 19471: 19458: 19450:the original 19439: 19424: 19416:the original 19405: 19390: 19375: 19367:the original 19356: 19329: 19325: 19315: 19282: 19278: 19272: 19229: 19225: 19219: 19187:(1): 69–89. 19184: 19180: 19145:(20): 1680. 19142: 19138: 19132: 19089: 19085: 19019: 19015: 18961: 18957: 18897: 18841: 18791: 18787: 18781: 18754: 18750: 18712: 18708: 18672: 18666: 18644:(18): 1475. 18641: 18637: 18631: 18598: 18594: 18588: 18580:the original 18541: 18537: 18494: 18490: 18426: 18420: 18393: 18389: 18379: 18354: 18350: 18308: 18304: 18298: 18263: 18259: 18253: 18220: 18216: 18210: 18185: 18181: 18175: 18130: 18124: 18081: 18077: 18067: 18042: 18038: 18032: 18005: 18001: 17991: 17967:(23): 6171. 17964: 17960: 17949: 17922: 17918: 17908: 17863: 17859: 17849: 17824: 17820: 17814: 17789: 17785: 17779: 17754: 17750: 17696: 17692: 17686: 17664:(3–4): 352. 17661: 17657: 17602:Chaos theory 17535: 17527:Hadjighasem. 17391: 17335: 17252: 17241: 17240: 17233:of Jupiter. 17066: 17012: 16898: 16580: 16479: 16341: 16088: 16080: 15984: 15732: 15130: 14876: 14813: 14737: 14310: 14306: 14248: 14064: 13990: 13871: 13797: 13707: 13665: 13537: 13463: 13380: 13191: 13051: 13047: 12803: 12729: 12087: 12038: 11806: 11803: 11413: 11205: 11203: 10791: 10479: 10472: 10422:elliptic LCS 10421: 10419: 10402: 10392: 10390: 10280: 10172: 10127: 9890: 9833: 9824: 9820: 9816: 9713: 9606: 9556: 9554: 9476: 9299: 9098: 9090: 8991:shrink lines 8990: 8874: 8774: 8486: 8481: 8296: 8020: 7401: 7384: 7377: 7290: 7020: 6854: 6758: 5052: 4861: 4859: 4698: 4472:denotes the 4341: 3791: 3280: 3157: 3139: 3135: 3094: 3092: 2873:-frame to a 2850: 2839: 2799:holds, with 2710: 2552: 2397: 2285: 1543: 1468: 1387: 1376: 1372: 1369: 1240: 1210: 1089: 1051:inequalities 1048: 988: 884: 726: 580: 132: 111: 107: 103: 94:trajectories 89: 85: 84: 60: 51: 32: 19604:: 111–120. 19412:"cuda_ftle" 19181:SIAM Review 19092:: 136–173. 18107:10919/24411 18045:: 230–239. 17925:(17): n/a. 17584:source code 17577:source code 17567:source code 17565:cuda_ftle ( 17560:source code 17553:source code 17546:source code 15985:shear lines 9888:defined as 8666:Attracting 3509:, and with 3099:phase space 1837:defined as 1766:strain rate 1394:phase space 1373:exceptional 1241:exceptional 733:phase space 135:phase space 19663:Turbulence 19647:Categories 19332:(2): 753. 19099:1506.04061 19029:1510.05367 18971:1503.05970 18757:(4): 439. 18008:(6): n/a. 17628:References 17597:Turbulence 17339:Lorentzian 9821:Strainless 8882:Repelling 1450:autonomous 1425:autonomous 1421:autonomous 1388:Classical 1245:continuity 1206:attractors 1090:attracting 578:, the set 19614:CiteSeerX 19502:1612-3786 19264:119570289 19239:1308.2352 19194:1408.5594 19022:: 70–93. 18907:1308.6136 18851:1306.6497 18431:CiteSeerX 18276:CiteSeerX 18135:CiteSeerX 17792:: 47–60. 17699:(2): 41. 17431:λ 17402:λ 17217:λ 17143:λ 17123:λ 17053:λ 17031:± 17026:λ 17022:η 16983:± 16979:η 16943:± 16938:λ 16934:η 16907:λ 16860:ξ 16830:λ 16826:− 16801:λ 16773:λ 16769:− 16760:λ 16752:± 16727:ξ 16697:λ 16693:− 16668:λ 16656:λ 16652:− 16627:λ 16598:± 16593:λ 16589:η 16560:λ 16534:λ 16531:− 16458:λ 16438:λ 16414:λ 16372:Π 16352:Π 16323:Π 16315:× 16294:± 16281:′ 16197:± 16170:Π 16145:Π 16123:Π 16098:Π 16061:⟩ 16040:± 16011:± 16003:× 16000:∇ 15997:⟨ 15953:± 15949:η 15940:′ 15891:± 15876:Elliptic 15694:ξ 15662:λ 15629:λ 15599:λ 15591:± 15566:ξ 15534:λ 15501:λ 15471:λ 15442:± 15392:ξ 15362:λ 15358:− 15333:λ 15305:λ 15301:− 15291:± 15266:ξ 15236:λ 15232:− 15207:λ 15198:− 15173:λ 15144:± 15140:η 15075:σ 14941:∈ 14697:¯ 14694:ω 14688:− 14648:ω 14615:∫ 14429:ω 14411:∫ 14389:¯ 14386:ω 14345:× 14342:∇ 14321:ω 14259:ψ 14217:Φ 14178:Φ 14154:β 14133:Φ 14115:¯ 14088:Φ 14078:˙ 14075:β 14041:β 14030:β 14000:Θ 13974:α 13952:¯ 13943:− 13885:˙ 13882:α 13848:α 13837:α 13832:∂ 13807:Φ 13763:Θ 13741:Φ 13551:˙ 13498:∇ 13394:˙ 13341:∇ 13059:∇ 12859:∇ 12766:θ 12730:The time 12707:λ 12685:ξ 12652:∇ 12640:ξ 12512:ϵ 12481:≠ 12446:ϵ 12426:ξ 12393:∇ 12381:ξ 12372:− 12358:ξ 12325:∇ 12313:ξ 12276:θ 12272:⁡ 12251:− 12239:λ 12223:ξ 12190:∇ 12178:ξ 12150:∑ 12106:θ 12102:⁡ 11999:θ 11995:⁡ 11984:⁡ 11976:− 11945:θ 11941:⁡ 11910:π 11879:θ 11875:⁡ 11847:− 11815:θ 11712:λ 11706:⟩ 11697:ξ 11664:∇ 11652:ξ 11648:⟨ 11628:− 11587:θ 11583:⁡ 11534:λ 11528:⟩ 11519:ξ 11486:∇ 11474:ξ 11470:⟨ 11432:θ 11428:⁡ 11353:π 11216:θ 10877:∇ 10835:∇ 10487:∇ 10353:λ 10310:ξ 10301:′ 10245:λ 10202:ξ 10193:′ 10137:ξ 10088:− 10069:Ω 10012:Ω 10009:− 10006:Ω 9800:ϵ 9767:ϵ 9734:ϵ 9697:ϵ 9664:ϵ 9587:Π 9567:Π 9536:Π 9528:× 9503:ξ 9494:′ 9404:ξ 9381:Π 9356:Π 9334:Π 9309:Π 9262:λ 9219:ξ 9210:′ 9151:λ 9108:λ 9044:ξ 9015:ξ 9011:× 9008:∇ 8955:ξ 8946:′ 8893:ξ 8828:ξ 8799:ξ 8795:× 8792:∇ 8739:ξ 8730:′ 8677:ξ 8387:ρ 8190:ρ 8030:ρ 7944:σ 7869:ρ 7763:∇ 7671:∇ 7385:incorrect 6824:× 6768:δ 6713:δ 6691:δ 6687:− 6642:− 6630:δ 6563:δ 6541:δ 6537:− 6492:− 6480:δ 6413:δ 6391:δ 6387:− 6342:− 6330:δ 6261:δ 6239:δ 6235:− 6190:− 6178:δ 6111:δ 6089:δ 6085:− 6040:− 6028:δ 5961:δ 5939:δ 5935:− 5890:− 5878:δ 5809:δ 5787:δ 5783:− 5738:− 5726:δ 5659:δ 5637:δ 5633:− 5588:− 5576:δ 5509:δ 5487:δ 5483:− 5438:− 5426:δ 5366:≈ 5325:∇ 5065:∇ 4813:λ 4809:⁡ 4787:− 4667:− 4617:δ 4612:⁡ 4562:λ 4512:δ 4431:∇ 4388:∇ 4301:ξ 4234:ξ 4195:ξ 4149:ξ 4101:∇ 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8788:⟨ 8661:for n=3 8618:for n=2 8578:ODE for 4503:is just 4321:⟩ 4230:⟨ 4169:⟩ 4027:⟨ 1200:theory, 19636:1074531 19606:Bibcode 19569:Bibcode 19534:Scholia 19431:"CTRAJ" 19334:Bibcode 19287:Bibcode 19244:Bibcode 19147:Bibcode 19104:Bibcode 19034:Bibcode 18976:Bibcode 18912:Bibcode 18856:Bibcode 18796:Bibcode 18759:Bibcode 18717:Bibcode 18646:Bibcode 18603:Bibcode 18546:Bibcode 18499:Bibcode 18463:7716670 18398:Bibcode 18359:Bibcode 18313:Bibcode 18268:Bibcode 18225:Bibcode 18190:Bibcode 18167:8190391 18086:Bibcode 18047:Bibcode 18010:Bibcode 17969:Bibcode 17927:Bibcode 17891:3323984 17868:Bibcode 17829:Bibcode 17794:Bibcode 17759:Bibcode 17701:Bibcode 17666:Bibcode 13231:is the 12900:in the 12536:is the 9842:tensor 4696:(FTLE) 3477:, with 19634: 19616: 19587: 19536:has a 19500: 19490: 19305: 19262: 19232:: R4. 19209: 19122: 19052: 18994: 18930: 18874: 18814: 18687: 18621: 18461: 18451: 18433: 18331: 18278: 18243: 18165: 18155: 18137: 18114: 17898: 17888: 12503:where 7475:, let 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Let 2840:Since 1625:where 1419:of an 727:is an 19632:S2CID 19585:S2CID 19538:topic 19509:(PDF) 19468:(PDF) 19260:S2CID 19234:arXiv 19207:S2CID 19189:arXiv 19120:S2CID 19094:arXiv 19050:S2CID 19024:arXiv 18992:S2CID 18966:arXiv 18928:S2CID 18902:arXiv 18872:S2CID 18846:arXiv 18534:(PDF) 18459:S2CID 18241:S2CID 18163:S2CID 17572:CTRAJ 14738:with 9838:of a 1286:over 883:as a 480:is a 133:On a 96:in a 19498:ISSN 19488:ISBN 19303:PMID 18812:PMID 18685:ISBN 18619:PMID 18449:ISBN 18329:PMID 18153:ISBN 18112:PMID 17896:PMID 17492:and 17182:and 16563:> 15742:LCS 14311:LAVD 13464:and 13303:and 13000:and 10785:and 8544:for 8501:LCS 8247:< 8087:> 3488:< 2849:for 2784:< 2290:for 90:LCSs 19624:doi 19602:572 19577:doi 19480:doi 19342:doi 19295:doi 19252:doi 19230:731 19199:doi 19155:doi 19143:241 19112:doi 19090:795 19042:doi 18984:doi 18962:315 18920:doi 18864:doi 18804:doi 18767:doi 18755:241 18725:doi 18713:240 18677:doi 18654:doi 18642:241 18611:doi 18554:doi 18507:doi 18495:212 18441:doi 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19191:: 19161:. 19157:: 19149:: 19126:. 19114:: 19106:: 19096:: 19056:. 19044:: 19036:: 19026:: 18998:. 18986:: 18978:: 18968:: 18934:. 18922:: 18914:: 18904:: 18878:. 18866:: 18858:: 18848:: 18818:. 18806:: 18798:: 18775:. 18769:: 18761:: 18731:. 18727:: 18719:: 18693:. 18679:: 18660:. 18656:: 18648:: 18625:. 18613:: 18605:: 18560:. 18556:: 18548:: 18513:. 18509:: 18501:: 18465:. 18443:: 18414:. 18408:: 18400:: 18373:. 18369:: 18361:: 18335:. 18323:: 18315:: 18292:. 18288:: 18270:: 18247:. 18235:: 18227:: 18204:. 18200:: 18192:: 18169:. 18147:: 18118:. 18104:: 18096:: 18088:: 18061:. 18057:: 18049:: 18026:. 18020:: 18012:: 17985:. 17979:: 17971:: 17943:. 17937:: 17929:: 17902:. 17878:: 17870:: 17843:. 17839:: 17831:: 17808:. 17804:: 17796:: 17773:. 17769:: 17761:: 17715:. 17711:: 17703:: 17680:. 17676:: 17668:: 17586:) 17579:) 17569:) 17562:) 17555:) 17548:) 17505:1 17501:t 17478:0 17474:t 17453:) 17448:0 17444:x 17440:( 17435:2 17427:= 17424:) 17419:0 17415:x 17411:( 17406:1 17373:1 17369:t 17361:0 17357:t 17352:D 17316:) 17311:0 17307:x 17303:( 17296:1 17292:t 17284:0 17280:t 17275:E 17271:L 17268:T 17265:F 17195:1 17191:t 17168:0 17164:t 17099:1 17095:t 17087:0 17083:t 17078:F 16996:0 16992:x 16988:( 16956:0 16952:x 16948:( 16913:1 16910:= 16885:, 16882:) 16877:0 16873:x 16869:( 16864:2 16852:) 16847:0 16843:x 16839:( 16834:1 16823:) 16818:0 16814:x 16810:( 16805:2 16795:) 16790:0 16786:x 16782:( 16777:1 16764:2 16749:) 16744:0 16740:x 16736:( 16731:1 16719:) 16714:0 16710:x 16706:( 16701:1 16690:) 16685:0 16681:x 16677:( 16672:2 16660:2 16649:) 16644:0 16640:x 16636:( 16631:2 16616:) 16611:0 16607:x 16603:( 16566:0 16540:) 16537:I 16524:1 16520:t 16512:0 16508:t 16503:C 16499:( 16494:2 16491:1 16464:1 16461:= 16328:, 16319:n 16312:) 16307:0 16303:x 16299:( 16290:n 16286:= 16276:0 16272:x 16251:) 16248:s 16245:( 16240:0 16236:x 16215:) 16210:0 16206:x 16202:( 16193:n 16166:n 16119:n 16079:( 16067:0 16064:= 16058:) 16053:0 16049:x 16045:( 16036:n 16032:, 16029:) 16024:0 16020:x 16016:( 16007:n 15987:) 15983:( 15971:) 15966:0 15962:x 15958:( 15945:= 15935:0 15931:x 15909:) 15904:0 15900:x 15896:( 15887:n 15859:) 15854:0 15850:t 15846:( 15841:M 15816:) 15811:0 15807:t 15803:( 15798:M 15773:) 15768:0 15764:t 15760:( 15755:M 15719:, 15716:) 15711:0 15707:x 15703:( 15698:3 15684:) 15679:0 15675:x 15671:( 15666:3 15656:+ 15651:) 15646:0 15642:x 15638:( 15633:1 15621:) 15616:0 15612:x 15608:( 15603:3 15588:) 15583:0 15579:x 15575:( 15570:1 15556:) 15551:0 15547:x 15543:( 15538:3 15528:+ 15523:) 15518:0 15514:x 15510:( 15505:1 15493:) 15488:0 15484:x 15480:( 15475:1 15463:= 15460:) 15455:0 15451:x 15447:( 15438:n 15417:, 15414:) 15409:0 15405:x 15401:( 15396:2 15384:) 15379:0 15375:x 15371:( 15366:1 15355:) 15350:0 15346:x 15342:( 15337:2 15327:) 15322:0 15318:x 15314:( 15309:1 15298:1 15288:) 15283:0 15279:x 15275:( 15270:1 15258:) 15253:0 15249:x 15245:( 15240:1 15229:) 15224:0 15220:x 15216:( 15211:2 15201:1 15195:) 15190:0 15186:x 15182:( 15177:2 15162:) 15157:0 15153:x 15149:( 15116:) 15111:0 15107:x 15103:( 15096:1 15092:t 15084:0 15080:t 15054:) 15049:0 15045:t 15041:( 15036:M 15027:0 15023:x 15018:T 14997:) 14994:t 14991:( 14986:M 14964:) 14959:0 14955:t 14951:( 14946:M 14936:0 14932:x 14911:] 14906:1 14902:t 14898:, 14893:0 14889:t 14885:[ 14849:] 14844:1 14840:t 14836:, 14831:0 14827:t 14823:[ 14790:) 14787:t 14784:, 14781:x 14778:( 14775:v 14755:) 14752:t 14749:( 14746:U 14724:, 14721:s 14718:d 14713:| 14709:) 14706:s 14703:( 14685:) 14682:s 14679:, 14676:) 14671:0 14667:x 14663:; 14660:s 14657:( 14654:x 14651:( 14644:| 14636:1 14632:t 14624:0 14620:t 14608:) 14603:0 14599:x 14595:( 14588:1 14584:t 14576:0 14572:t 14566:D 14563:V 14560:A 14557:L 14535:] 14530:1 14526:t 14522:, 14517:0 14513:t 14509:[ 14489:, 14483:) 14480:) 14477:t 14474:( 14471:U 14468:( 14463:l 14460:o 14457:v 14451:V 14448:d 14444:) 14441:t 14438:, 14435:x 14432:( 14424:) 14421:t 14418:( 14415:U 14404:= 14401:) 14398:t 14395:( 14363:) 14360:t 14357:, 14354:x 14351:( 14348:v 14339:= 14336:) 14333:t 14330:, 14327:x 14324:( 14309:( 14293:) 14288:0 14284:x 14280:( 14275:t 14268:0 14264:t 14233:t 14226:0 14222:t 14194:t 14187:0 14183:t 14157:. 14149:t 14142:0 14138:t 14128:) 14125:t 14122:( 14112:W 14102:0 14098:t 14092:t 14084:= 14050:) 14047:t 14044:( 14034:0 14025:D 14021:= 14016:t 14009:0 14005:t 13977:. 13970:] 13965:) 13962:t 13959:( 13949:W 13939:) 13935:t 13932:, 13929:) 13924:0 13920:x 13916:; 13913:t 13910:( 13907:x 13903:( 13899:W 13895:[ 13891:= 13857:) 13854:t 13851:( 13841:0 13828:= 13823:t 13816:0 13812:t 13784:. 13779:t 13772:0 13768:t 13757:t 13750:0 13746:t 13737:= 13732:t 13725:0 13721:t 13716:O 13691:t 13684:0 13680:t 13675:O 13651:. 13648:b 13643:t 13636:0 13632:t 13627:O 13622:) 13618:t 13615:, 13612:) 13607:0 13603:x 13599:; 13596:t 13593:( 13590:x 13586:( 13582:S 13575:0 13571:t 13565:t 13561:O 13557:= 13548:b 13523:) 13520:t 13517:( 13514:b 13507:0 13503:b 13494:= 13489:t 13482:0 13478:t 13473:M 13450:, 13447:a 13443:) 13439:t 13436:, 13433:) 13428:0 13424:x 13420:; 13417:t 13414:( 13411:x 13407:( 13403:W 13400:= 13391:a 13366:) 13363:t 13360:( 13357:a 13350:0 13346:a 13337:= 13332:t 13325:0 13321:t 13316:O 13285:t 13278:0 13274:t 13269:N 13265:, 13260:t 13253:0 13249:t 13244:M 13217:t 13210:0 13206:t 13201:O 13178:, 13173:t 13166:0 13162:t 13157:O 13151:t 13144:0 13140:t 13135:N 13131:= 13126:t 13119:0 13115:t 13110:M 13104:t 13097:0 13093:t 13088:O 13084:= 13079:t 13072:0 13068:t 13063:F 13013:1 13009:t 12986:0 12982:t 12961:] 12956:1 12952:t 12948:, 12943:0 12939:t 12935:[ 12913:2 12909:L 12884:1 12880:t 12872:0 12868:t 12863:F 12835:1 12831:t 12823:0 12819:t 12814:R 12782:t 12775:0 12771:t 12743:0 12739:t 12711:k 12700:/ 12689:k 12677:1 12673:t 12665:0 12661:t 12656:F 12649:, 12644:j 12631:= 12626:k 12623:j 12618:] 12613:t 12606:0 12602:t 12597:K 12593:[ 12570:} 12565:k 12561:e 12557:{ 12553:= 12549:e 12522:k 12519:j 12516:i 12487:, 12484:j 12478:i 12474:, 12466:k 12462:e 12456:k 12453:j 12450:i 12442:2 12430:i 12418:1 12414:t 12406:0 12402:t 12397:F 12390:, 12385:j 12362:j 12350:1 12346:t 12338:0 12334:t 12329:F 12322:, 12317:i 12301:= 12292:t 12285:0 12281:t 12262:, 12258:) 12254:1 12243:i 12227:i 12215:1 12211:t 12203:0 12199:t 12194:F 12187:, 12182:i 12165:3 12160:1 12157:= 12154:i 12145:( 12139:2 12136:1 12131:= 12122:t 12115:0 12111:t 12069:1 12065:t 12057:0 12053:t 12048:C 12025:. 12021:) 12015:t 12008:0 12004:t 11988:( 11979:1 11967:) 11961:t 11954:0 11950:t 11934:( 11927:n 11924:g 11921:i 11918:s 11913:+ 11906:] 11901:) 11895:t 11888:0 11884:t 11868:( 11861:n 11858:g 11855:i 11852:s 11844:1 11840:[ 11836:= 11831:t 11824:0 11820:t 11786:, 11783:) 11780:1 11777:, 11774:2 11771:( 11766:r 11763:o 11758:) 11755:2 11752:, 11749:1 11746:( 11743:= 11740:) 11737:j 11734:, 11731:i 11728:( 11724:, 11716:j 11701:j 11689:1 11685:t 11677:0 11673:t 11668:F 11661:, 11656:i 11640:j 11635:) 11631:1 11624:( 11619:= 11608:1 11604:t 11596:0 11592:t 11573:, 11570:2 11564:r 11561:o 11556:1 11553:= 11550:i 11546:, 11538:i 11523:i 11511:1 11507:t 11499:0 11495:t 11490:F 11483:, 11478:i 11464:= 11453:1 11449:t 11441:0 11437:t 11395:1 11391:t 11383:0 11379:t 11374:C 11350:2 11328:0 11324:x 11299:1 11295:t 11287:0 11283:t 11278:R 11257:) 11252:0 11248:x 11244:( 11237:1 11233:t 11225:0 11221:t 11165:1 11161:t 11153:0 11149:t 11144:R 11119:1 11115:t 11107:0 11103:t 11098:R 11073:1 11069:t 11061:0 11057:t 11052:C 11031:, 11024:1 11020:t 11012:0 11008:t 11003:V 10995:1 10991:t 10983:0 10979:t 10974:V 10970:= 10963:1 10959:t 10951:0 10947:t 10942:U 10934:1 10930:t 10922:0 10918:t 10913:U 10909:= 10902:1 10898:t 10890:0 10886:t 10881:F 10872:T 10868:] 10860:1 10856:t 10848:0 10844:t 10839:F 10832:[ 10829:= 10822:1 10818:t 10810:0 10806:t 10801:C 10765:1 10761:t 10753:0 10749:t 10744:V 10740:, 10733:1 10729:t 10721:0 10717:t 10712:U 10683:1 10679:t 10671:0 10667:t 10662:R 10641:, 10634:1 10630:t 10622:0 10618:t 10613:R 10605:1 10601:t 10593:0 10589:t 10584:V 10580:= 10573:1 10569:t 10561:0 10557:t 10552:U 10544:1 10540:t 10532:0 10528:t 10523:R 10519:= 10512:1 10508:t 10500:0 10496:t 10491:F 10458:] 10453:1 10449:t 10445:, 10440:0 10436:t 10432:[ 10375:) 10370:0 10366:x 10362:( 10357:1 10332:) 10327:0 10323:x 10319:( 10314:2 10306:= 10296:0 10292:x 10267:) 10262:0 10258:x 10254:( 10249:2 10224:) 10219:0 10215:x 10211:( 10206:1 10198:= 10188:0 10184:x 10159:) 10154:0 10150:x 10146:( 10141:i 10114:. 10109:) 10103:0 10098:1 10091:1 10083:0 10077:( 10072:= 10065:, 10061:] 10057:) 10052:0 10048:x 10044:( 10037:1 10033:t 10025:0 10021:t 10016:C 10003:) 9998:0 9994:x 9990:( 9983:1 9979:t 9971:0 9967:t 9962:C 9957:[ 9951:2 9948:1 9943:= 9940:) 9935:0 9931:x 9927:( 9920:1 9916:t 9908:0 9904:t 9899:D 9872:1 9868:t 9860:0 9856:t 9851:D 9803:) 9797:( 9792:O 9770:) 9764:( 9759:O 9737:) 9731:( 9726:O 9700:) 9694:( 9689:O 9667:) 9661:( 9656:O 9632:t 9625:0 9621:t 9616:F 9541:, 9532:n 9525:) 9520:0 9516:x 9512:( 9507:3 9499:= 9489:0 9485:x 9462:) 9459:s 9456:( 9451:0 9447:x 9426:) 9421:0 9417:x 9413:( 9408:3 9377:n 9330:n 9284:) 9279:0 9275:x 9271:( 9266:2 9241:) 9236:0 9232:x 9228:( 9223:1 9215:= 9205:0 9201:x 9173:) 9168:0 9164:x 9160:( 9155:1 9130:) 9125:0 9121:x 9117:( 9112:2 9089:( 9077:0 9074:= 9066:) 9061:0 9057:x 9053:( 9048:3 9040:, 9037:) 9032:0 9028:x 9024:( 9019:3 8989:( 8977:) 8972:0 8968:x 8964:( 8959:1 8951:= 8941:0 8937:x 8915:) 8910:0 8906:x 8902:( 8897:n 8873:( 8861:0 8858:= 8850:) 8845:0 8841:x 8837:( 8832:1 8824:, 8821:) 8816:0 8812:x 8808:( 8803:1 8777:) 8773:( 8761:) 8756:0 8752:x 8748:( 8743:2 8735:= 8725:0 8721:x 8699:) 8694:0 8690:x 8686:( 8681:1 8649:) 8644:0 8640:t 8636:( 8631:M 8606:) 8601:0 8597:t 8593:( 8588:M 8564:3 8561:, 8558:2 8555:= 8552:n 8532:) 8527:0 8523:t 8519:( 8514:M 8466:0 8462:n 8441:) 8436:0 8432:n 8428:, 8423:0 8419:x 8415:( 8408:1 8404:t 8396:0 8392:t 8366:) 8363:t 8360:( 8355:M 8333:] 8328:1 8324:t 8320:, 8315:0 8311:t 8307:[ 8283:) 8280:t 8277:( 8272:M 8250:1 8244:) 8239:0 8235:n 8231:, 8226:0 8222:x 8218:( 8211:1 8207:t 8199:0 8195:t 8169:] 8164:1 8160:t 8156:, 8151:0 8147:t 8143:[ 8123:) 8120:t 8117:( 8112:M 8090:1 8084:) 8079:0 8075:n 8071:, 8066:0 8062:x 8058:( 8051:1 8047:t 8039:0 8035:t 7998:) 7993:0 7989:n 7985:, 7980:0 7976:x 7972:( 7965:1 7961:t 7953:0 7949:t 7923:) 7918:0 7914:n 7908:, 7905:0 7901:x 7897:( 7890:1 7886:t 7878:0 7874:t 7848:) 7843:1 7839:t 7835:( 7830:M 7808:) 7803:0 7799:x 7795:( 7788:1 7784:t 7776:0 7772:t 7767:F 7741:0 7737:n 7716:) 7711:0 7707:x 7703:( 7696:1 7692:t 7684:0 7680:t 7675:F 7651:) 7646:1 7642:t 7638:( 7633:M 7624:1 7620:x 7615:T 7590:) 7585:0 7581:t 7577:( 7572:M 7563:0 7559:x 7554:T 7530:) 7525:0 7521:t 7517:( 7512:M 7488:0 7484:n 7461:0 7457:x 7436:] 7431:1 7427:t 7423:, 7418:0 7414:t 7410:[ 7364:T 7344:] 7341:T 7338:+ 7333:1 7329:t 7325:, 7322:T 7319:+ 7314:0 7310:t 7306:[ 7275:] 7272:T 7269:+ 7264:1 7260:t 7256:, 7253:T 7250:+ 7245:0 7241:t 7237:[ 7215:T 7212:+ 7207:1 7203:t 7197:T 7194:+ 7189:0 7185:t 7179:E 7176:L 7173:T 7170:F 7148:T 7145:+ 7140:0 7136:t 7115:] 7112:T 7109:+ 7104:1 7100:t 7096:, 7093:T 7090:+ 7085:0 7081:t 7077:[ 7057:] 7052:1 7048:t 7044:, 7039:0 7035:t 7031:[ 7007:T 6987:) 6982:0 6978:x 6974:( 6969:T 6966:+ 6961:1 6957:t 6951:T 6948:+ 6943:0 6939:t 6933:E 6930:L 6927:T 6924:F 6902:] 6899:T 6896:+ 6891:1 6887:t 6883:, 6880:T 6877:+ 6872:0 6868:t 6864:[ 6827:2 6821:2 6799:i 6795:x 6772:i 6737:, 6732:) 6723:| 6717:3 6709:2 6705:| 6700:) 6695:3 6682:0 6678:x 6674:, 6669:0 6665:t 6661:; 6658:t 6655:( 6650:3 6646:x 6639:) 6634:3 6626:+ 6621:0 6617:x 6613:, 6608:0 6604:t 6600:; 6597:t 6594:( 6589:3 6585:x 6573:| 6567:2 6559:2 6555:| 6550:) 6545:2 6532:0 6528:x 6524:, 6519:0 6515:t 6511:; 6508:t 6505:( 6500:3 6496:x 6489:) 6484:2 6476:+ 6471:0 6467:x 6463:, 6458:0 6454:t 6450:; 6447:t 6444:( 6439:3 6435:x 6423:| 6417:1 6409:2 6405:| 6400:) 6395:1 6382:0 6378:x 6374:, 6369:0 6365:t 6361:; 6358:t 6355:( 6350:3 6346:x 6339:) 6334:1 6326:+ 6321:0 6317:x 6313:, 6308:0 6304:t 6300:; 6297:t 6294:( 6289:3 6285:x 6271:| 6265:3 6257:2 6253:| 6248:) 6243:3 6230:0 6226:x 6222:, 6217:0 6213:t 6209:; 6206:t 6203:( 6198:2 6194:x 6187:) 6182:3 6174:+ 6169:0 6165:x 6161:, 6156:0 6152:t 6148:; 6145:t 6142:( 6137:2 6133:x 6121:| 6115:2 6107:2 6103:| 6098:) 6093:2 6080:0 6076:x 6072:, 6067:0 6063:t 6059:; 6056:t 6053:( 6048:2 6044:x 6037:) 6032:2 6024:+ 6019:0 6015:x 6011:, 6006:0 6002:t 5998:; 5995:t 5992:( 5987:2 5983:x 5971:| 5965:1 5957:2 5953:| 5948:) 5943:1 5930:0 5926:x 5922:, 5917:0 5913:t 5909:; 5906:t 5903:( 5898:2 5894:x 5887:) 5882:1 5874:+ 5869:0 5865:x 5861:, 5856:0 5852:t 5848:; 5845:t 5842:( 5837:2 5833:x 5819:| 5813:3 5805:2 5801:| 5796:) 5791:3 5778:0 5774:x 5770:, 5765:0 5761:t 5757:; 5754:t 5751:( 5746:1 5742:x 5735:) 5730:3 5722:+ 5717:0 5713:x 5709:, 5704:0 5700:t 5696:; 5693:t 5690:( 5685:1 5681:x 5669:| 5663:2 5655:2 5651:| 5646:) 5641:2 5628:0 5624:x 5620:, 5615:0 5611:t 5607:; 5604:t 5601:( 5596:1 5592:x 5585:) 5580:2 5572:+ 5567:0 5563:x 5559:, 5554:0 5550:t 5546:; 5543:t 5540:( 5535:1 5531:x 5519:| 5513:1 5505:2 5501:| 5496:) 5491:1 5478:0 5474:x 5470:, 5465:0 5461:t 5457:; 5454:t 5451:( 5446:1 5442:x 5435:) 5430:1 5422:+ 5417:0 5413:x 5409:, 5404:0 5400:t 5396:; 5393:t 5390:( 5385:1 5381:x 5371:( 5363:) 5358:0 5354:x 5350:( 5345:t 5338:0 5334:t 5329:F 5305:) 5300:0 5296:x 5292:, 5287:0 5283:t 5279:; 5276:t 5273:( 5270:x 5250:) 5245:3 5241:x 5237:, 5232:2 5228:x 5224:, 5219:1 5215:x 5211:( 5208:= 5205:x 5183:0 5179:x 5158:) 5153:0 5149:x 5145:, 5140:0 5136:t 5132:; 5129:t 5126:( 5123:x 5103:) 5098:0 5094:x 5090:( 5085:t 5078:0 5074:t 5069:F 5034:0 5030:t 5022:1 5018:t 5012:E 5009:L 5006:T 5003:F 4979:1 4975:t 4954:) 4949:0 4945:x 4941:( 4934:1 4930:t 4922:0 4918:t 4912:E 4909:L 4906:T 4903:F 4879:0 4875:t 4838:. 4835:) 4830:0 4826:x 4822:( 4817:n 4800:) 4795:0 4791:t 4782:1 4778:t 4774:( 4771:2 4767:1 4762:= 4759:) 4754:0 4750:x 4746:( 4739:1 4735:t 4727:0 4723:t 4717:E 4714:L 4711:T 4708:F 4680:) 4675:0 4671:t 4662:1 4658:t 4654:( 4650:/ 4646:) 4638:1 4634:t 4626:0 4622:t 4606:( 4584:) 4579:0 4575:x 4571:( 4566:n 4556:= 4553:) 4548:0 4544:x 4540:( 4533:1 4529:t 4521:0 4517:t 4489:0 4485:x 4456:1 4452:t 4444:0 4440:t 4435:F 4426:T 4421:] 4413:1 4409:t 4401:0 4397:t 4392:F 4384:[ 4379:= 4372:1 4368:t 4360:0 4356:t 4351:C 4317:) 4312:0 4308:t 4304:( 4298:) 4293:0 4289:x 4285:( 4278:1 4274:t 4266:0 4262:t 4257:C 4253:, 4250:) 4245:0 4241:t 4237:( 4222:1 4219:= 4215:| 4211:) 4206:0 4202:t 4198:( 4191:| 4182:= 4165:) 4160:0 4156:t 4152:( 4146:) 4141:0 4137:x 4133:( 4126:1 4122:t 4114:0 4110:t 4105:F 4098:, 4095:) 4090:0 4086:t 4082:( 4076:) 4071:0 4067:x 4063:( 4056:1 4052:t 4044:0 4040:t 4035:F 4019:1 4016:= 4012:| 4008:) 4003:0 3999:t 3995:( 3988:| 3979:= 3968:| 3964:) 3959:0 3955:x 3951:; 3946:1 3942:t 3938:( 3924:| 3918:1 3915:= 3911:| 3907:) 3902:0 3898:t 3894:( 3887:| 3873:1 3866:0 3852:= 3845:) 3840:0 3836:x 3832:( 3825:1 3821:t 3813:0 3809:t 3775:0 3771:x 3750:) 3745:0 3741:t 3737:( 3726:) 3721:0 3717:x 3713:( 3706:1 3702:t 3694:0 3690:t 3685:F 3678:= 3675:) 3670:0 3666:x 3662:; 3657:1 3653:t 3649:( 3617:) 3612:0 3608:x 3604:, 3599:0 3595:t 3591:, 3588:t 3585:( 3582:x 3560:n 3555:R 3533:) 3528:0 3524:t 3520:( 3497:1 3485:0 3463:0 3459:x 3438:) 3433:0 3429:t 3425:( 3395:t 3388:0 3384:t 3379:F 3355:) 3350:0 3346:x 3342:, 3337:0 3333:t 3329:, 3326:t 3323:( 3320:x 3300:) 3297:t 3294:( 3266:) 3261:0 3257:t 3253:( 3248:M 3226:) 3221:0 3217:t 3213:( 3208:M 3186:) 3181:0 3177:t 3173:( 3168:M 3120:I 3110:P 3065:y 3045:y 3025:x 3005:) 3002:t 2999:( 2996:b 2993:+ 2990:) 2987:t 2984:( 2981:y 2978:) 2975:t 2972:( 2969:Q 2966:= 2963:) 2960:t 2957:( 2954:x 2934:) 2931:t 2928:( 2925:b 2922:+ 2919:y 2916:) 2913:t 2910:( 2907:Q 2904:= 2901:x 2881:y 2861:x 2815:| 2807:| 2787:0 2781:) 2776:2 2771:| 2768:W 2765:| 2755:2 2750:| 2747:S 2744:| 2739:( 2734:2 2731:1 2726:= 2723:q 2697:, 2694:x 2689:) 2682:t 2679:4 2663:t 2660:4 2650:+ 2647:2 2636:t 2633:4 2623:+ 2620:2 2614:t 2611:4 2598:( 2593:= 2590:) 2587:t 2584:, 2581:x 2578:( 2575:v 2572:= 2563:x 2538:) 2535:t 2532:, 2529:x 2526:( 2523:v 2501:t 2494:0 2490:t 2485:F 2461:) 2458:t 2455:, 2452:y 2449:( 2445:W 2424:) 2421:t 2418:, 2415:x 2412:( 2409:v 2383:) 2380:t 2377:, 2374:x 2371:( 2368:W 2348:) 2345:t 2342:, 2339:x 2336:( 2333:S 2313:) 2310:t 2307:, 2304:x 2301:( 2298:S 2272:. 2269:) 2266:t 2263:( 2254:Q 2246:T 2242:) 2238:t 2235:( 2232:Q 2226:) 2223:t 2220:( 2217:Q 2214:) 2211:t 2208:, 2205:x 2202:( 2199:S 2194:T 2190:) 2186:t 2183:( 2180:Q 2177:= 2174:) 2171:t 2168:, 2165:y 2162:( 2153:W 2146:, 2143:) 2140:t 2137:( 2134:Q 2131:) 2128:t 2125:, 2122:x 2119:( 2116:S 2111:T 2107:) 2103:t 2100:( 2097:Q 2094:= 2091:) 2088:t 2085:, 2082:y 2079:( 2070:S 2047:, 2043:) 2037:T 2033:) 2029:) 2026:t 2023:, 2020:x 2017:( 2014:v 2008:( 2002:) 1999:t 1996:, 1993:x 1990:( 1987:v 1980:( 1974:2 1971:1 1966:= 1963:) 1960:t 1957:, 1954:x 1951:( 1948:W 1944:, 1940:) 1934:T 1930:) 1926:) 1923:t 1920:, 1917:x 1914:( 1911:v 1905:( 1902:+ 1899:) 1896:t 1893:, 1890:x 1887:( 1884:v 1877:( 1871:2 1868:1 1863:= 1860:) 1857:t 1854:, 1851:x 1848:( 1845:S 1825:) 1822:t 1819:, 1816:x 1813:( 1810:W 1790:) 1787:t 1784:, 1781:x 1778:( 1775:S 1752:3 1732:) 1729:t 1726:( 1723:b 1703:3 1697:3 1677:) 1674:t 1671:( 1668:Q 1646:3 1641:R 1633:y 1613:, 1610:) 1607:t 1604:( 1601:b 1598:+ 1595:y 1592:) 1589:t 1586:( 1583:Q 1580:= 1577:x 1552:U 1530:, 1525:3 1520:R 1512:U 1506:x 1502:, 1499:) 1496:t 1493:, 1490:x 1487:( 1484:v 1481:= 1478:v 1435:I 1405:P 1353:I 1331:) 1328:t 1325:( 1320:M 1296:I 1272:t 1265:0 1261:t 1256:F 1225:I 1184:) 1179:1 1175:t 1171:( 1166:M 1144:) 1139:0 1135:t 1131:( 1126:M 1102:I 1077:) 1074:t 1071:( 1066:M 1032:I 1010:) 1007:t 1004:( 999:M 959:I 949:P 939:M 917:) 912:0 908:t 904:( 899:M 869:M 847:) 844:) 839:0 835:t 831:( 826:M 821:( 816:t 809:0 805:t 800:F 796:= 793:) 790:t 787:( 782:M 754:I 744:P 713:} 710:) 705:0 701:t 697:( 692:M 684:) 681:x 678:( 673:1 666:] 660:t 653:0 649:t 644:F 640:[ 631:I 621:P 613:) 610:t 607:, 604:x 601:( 598:{ 595:= 590:M 564:P 542:) 537:0 533:t 529:( 524:M 500:I 492:t 466:t 459:0 455:t 450:F 427:I 419:t 397:P 389:) 384:0 380:x 376:, 371:0 367:t 363:, 360:t 357:( 354:x 332:P 322:0 318:x 297:) 292:0 288:x 284:, 279:0 275:t 271:, 268:t 265:( 262:x 254:0 250:x 241:t 234:0 230:t 225:F 204:] 199:1 195:t 191:, 186:0 182:t 178:[ 175:= 170:I 146:P 88:( 67:) 61:( 56:) 52:( 48:.

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Index