the computation of finite-time lyapunov exponents on ... · the ftle to non-euclidean manifolds and...

The computation of finite-time Lyapunov exponents on unstructured meshes and fornon-Euclidean manifoldsFrancois Lekien and Shane D Ross Citation Chaos An Interdisciplinary Journal of Nonlinear Science 20 017505 (2010) doi 10106313278516 View online httpdxdoiorg10106313278516 View Table of Contents httpscitationaiporgcontentaipjournalchaos201ver=pdfcov Published by the AIP Publishing

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Copyright by the American Institute of Physics The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds13Lekien Francois and Ross Shane D Chaos An Interdisciplinary Journal of Nonlinear Science 20 017505 (2010) DOIhttpdxdoiorg1010631327851613

The computation of finite-time Lyapunov exponents on unstructuredmeshes and for non-Euclidean manifolds

Francois Lekien1a and Shane D Ross2b

1Eacutecole Polytechnique Universiteacute Libre de Bruxelles B-1050 Brussels Belgium2Engineering Science and Mechanics Virginia Tech Blacksburg Virginia 24061 USA

Received 4 September 2009 accepted 4 December 2009 published online 8 February 2010

We generalize the concepts of finite-time Lyapunov exponent FTLE and Lagrangian coherentstructures to arbitrary Riemannian manifolds The methods are illustrated for convection cells oncylinders and Moumlbius strips as well as for the splitting of the Antarctic polar vortex in the sphericalstratosphere and a related point vortex model We modify the FTLE computational method andaccommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape aswell as to facilitate dynamic refinement of the FTLE mesh copy 2010 American Institute of Physicsdoi10106313278516

Riemannian manifolds are ubiquitous in science and en-gineering being the more natural mathematical settingfor many dynamical systems For instance transportalong isopycnal surfaces in the ocean and large-scale mix-ing in the atmosphere are processes taking place on acurved manifold not a vector space In this paper wegeneralize the notion of finite-time Lyapunov exponent(FTLE) and Lagrangian coherent structures (LCS) to ar-bitrary Riemannian differentiable manifolds We showthat both notions are independent of the coordinate sys-tem but depend on the chosen metric However we findthat LCS do not depend much on the metric The FTLEmeasures separation and tends to be large and positivealong LCS and very small elsewhere For sufficientlylarge integration times the steep variations of the FTLEfield cannot be modified much by smooth changes in themetric and the LCS remain essentially unchanged Ap-proximating or even ignoring the manifold metric doesnot influence the result for large integration times andtherefore computing the FTLE field on manifolds is ro-bust Aside from these conclusions we present a generalalgorithm for computing the FTLE on manifolds coveredwith meshes of polyhedra The algorithm requires knowl-edge of the mesh nodes the image of the mesh nodesunder the flow as well as information about node neigh-bors (but not the full connectivity) We also used the samealgorithm in Euclidian spaces where the unstructuredmesh permits efficient adaptive refinement for capturingsharp LCS features We illustrate the results and meth-ods on several systems convection cells in a plane on acylinder and on a Moumlbius strip as well as atmospherictransport resulting from the 2002 splitting of the Antarc-tic ozone hole in the spherical stratosphere and a relatedpoint vortex model on the sphere

I INTRODUCTION

Consider an Euclidean domain Rn A typical dy-namical systems is given in the form of a velocity fieldvx t defined on R In this case the trajectories aregiven by the solutions of the ordinary differential equationx=vx t Each trajectory is a function of time but it alsodepends on the initial position x0 and the initial time t0 Howthe trajectory changes as the initial time and the initial posi-tion changes are a central interest in this paper For this rea-son the trajectory which is solution of the initial value prob-lem

x = vxt and xt0 = x0 1

is denoted as xt t0 x0 which emphasizes the explicit de-pendence on the initial condition It is also convenient todefine the flow operator to further highlight the dependenceon the initial position x0 For a given initial time t0 and agiven final time t the flow map is the function

t0t rarr x0 t0

t x0 = xtt0x0 2

Autonomous systems do not depend on time They satisfyt0+T

t+T =t0t for any TR For the associated velocity field

this property translates into vx t=vx For two-dimensional autonomous systems regions of qualitativelydifferent dynamics can be efficiently captured by the stableand unstable invariant manifolds of saddle points in thesystem18 In higher dimensional autonomous systems sepa-ratrices are codimension 1 stable and unstable manifolds ofnormally hyperbolic manifolds1140

In the Poincareacute sections of time-periodic systems lobesbetween the invariant manifolds govern particle transportacross separatrices4847 In this paper we consider the gener-alization of this framework to systems with arbitrary timedependence In this case saddle fixed points are nongenericThe points where the instantaneous velocity vanishes arecalled hyperbolic stagnation points and usually meander intime8 Such points are not trajectories and are not associatedwith hyperbolic invariant manifolds36 The concept of LCS

aElectronic mail lekienulbacbebElectronic mail sdrossvtedu

CHAOS 20 017505 2010

1054-1500201020101750520$3000 copy 2010 American Institute of Physics20 017505-1


12817312576 On Wed 20 Nov 2013 195935

extends the transport framework delineated by invariantmanifolds to time-dependent dynamical systems and to tran-sient processes19205237

How can we define coherent structures without relyingon stagnation points and without using asymptotic notionssuch as convergence hyperbolicity or exponential dichoto-mies Bowman4 provided a robust and general answer Hesuggested to initiate a grid of virtual particles integrate themfor a time on the scale of the observed process and look atthe final separation between initially nearby particles finite-strain maps LCS correspond to curves with locally highrelative dispersion

A remarkable aspect of Bowmanrsquos procedure is its ro-bustness and its ability to uncover structures in the mostcomplex and noisiest data sets It can be improved by replac-ing the linear separation between grid neighbors by a betterstretching estimate Two such modern indicators are thefinite-time Lyapunov exponent FTLE1920 and the finite-sizeLyapunov exponent FSLE2731 The two quantities areclosely related and delineate sharp ridges of high stretchingthat behave almost like material lines5237 In the remainderof this paper we will concentrate on FTLE but most of theresults and methods remain valid for other stretching indica-tors such as FSLE or relative dispersion

For a given time t0 and an initial position x0 the quantityt0

t0+Tx0 is the position after an integration time T To quan-tify particle separation and sensitivity to the initial conditionx0 the FTLE is defined based on the derivative of t0

t0+Tx0with respect to x0 If we make a small perturbation and weconsider the evolution of trajectory starting at the positionx0+u instead of x0 the final position will be

t0

t0+Tx0 + u = t0

t0+Tx0 +t0

t0+T

x0u + Ou2 3

The point x0 is sensitive to the initial condition if t0t0+Tx0

+u separates from t0t0+Tx0 even for small perturbations u

The boundary between regions of qualitatively different dy-namics can therefore be detected as the set of points forwhich t0

t0+T x0u is very large for some direction u Toquantify stretching and sensitivity to initial conditions wedefine the FTLE as

x0t0 =1

Tlnmax

u0

t0

t0+T

x0u

u 4

In the equation above the expression in parentheses givesthe maximum stretching between infinitesimally close trajec-tories The reason for taking the logarithm and dividing bythe magnitude of the integration time T is the parallel that wethen establish with autonomous systems Indeed for a time-independent system we have max

u0t0

t0+T x0u=etu

where is the largest Lyapunov exponent associated with thetrajectory starting at x0

Note that the FTLE x0 t0 is a function of position andtime but it also depends on the chosen integration time TThe latter is not regarded as very large as our aim is toinvestigate transient processes and finite-time mixing2136

Additionally T is used instead of T because computing theFTLE for T0 and T0 produces repulsive LCS and at-tractive LCS respectively and thus this definition facilitatesforward and backward time computations

For a given point x0 the FTLE x0 t0 as definedabove can also be seen as the norm of the differential oft0

t0+T at the point x0 that is

x0t0 =1

Tln Dt0

t0+T 1

Tlnmax

u0

Dt0

t0+Tu

u

5

We will use the expression above to extend the definition ofthe FTLE to non-Euclidean manifolds and to derive an algo-rithm for computing FTLE on an unstructured mesh for Eu-clidean or non-Euclidean manifolds

The LCS are ridges in the FTLE field They are codi-mension 1 manifolds that maximize the FTLE in the trans-verse direction This notion is more formally defined byShadden et al52 and Lekien et al37 The objective is to gen-eralize the concept of stable and unstable invariant manifoldsof hyperbolic invariant objects to time-dependent systemsWhether in the ocean in the atmosphere or any other dy-namical system LCS are barriers to particle transport45

Interlacing of attractive and repulsive LCS gives rise tolobes that govern transport across the regions delineated bythe LCS8 The LCS bound coherent patches of particles thatevolve independently from the surrounding environment andexchange little matter and energy12 Such structures includeoceanic eddies2 separation profiles from a coastline or anairfoil3639 shed vortices over an airfoil6 and wake structuresbehind swimming animals46 Shadden and Taylor53 showedthat LCS indicate stagnation and mixing in blood vesselsTallapragada and Ross57 showed that LCS are also relevantfor nonpassive finite-size particles in fluids where inertialeffects are present

Contrary to classical stable and unstable invariant mani-folds the LCS of time-dependent systems are not perfectbarriers to transport Shadden et al52 and Lekien et al37

showed that the flux through a LCS is usually very small butnot exactly zero As a result one can usually consider LCSas a material barrier with negligible flux but with the possi-bility of occasional bifurcations Invariant manifolds cannotbifurcate and cannot adequately describe transient processesThe weak flux across the LCS of an aperiodic system makesit possible for the LCS to bifurcate and capture finite-timeprocesses

In the ocean bifurcation of the LCS indicates a regimechange between qualitatively different kinds of motion3533

Coulliette et al7 showed that the position of LCS in coastalregions determines whether contaminants will recirculatealong the coast or not Lipinski et al39 showed that LCSbifurcation at the end of an airfoil is responsible for vortexshedding

In an atmospheric context LCS correspond to atmo-spheric transport barriers which separate different regions ofrelatively well-mixed air In this paper we will look intoatmospheric transport and global-scale phenomena such asthe splitting of the Antarctic polar vortex546142 As the atmo-

017505-2 F Lekien and S D Ross Chaos 20 017505 2010


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sphere is global in extent it becomes necessary to considerthe spherical and therefore non-Euclidean nature of theflow

II FINITE-TIME LYAPUNOV EXPONENTSIN EUCLIDEAN SPACES

The computation of the FTLE can be further simplifiedin Euclidean spaces by considering the matrix representationof the differential Dt0

t0+T Let us consider an orthonormalbasis e1 e2 en13 of Rn and the nn matrix M = mij de-fined by mij ei middot Dt0

t0+Te jGiven the column vector U containing the coordinates of

a vector uRn in the orthonormal basis e1 e2 en13 thecolumn vector U=MU gives the coordinates of Dt0

t0+Tuin the same basis As a result we have

x0t0 =1

Tlnmax

u0

Dt0

t0+Tu

u

=1

2Tlnmax

UR0n

UMMU

UU =

1

Tln largest singular value M 6

We first review how the formula above can be used to com-pute FTLE and LCS on a Cartesian mesh We then turn to theimplementation for unstructured lists of triangles in R2 andn-tetrahedra in Rn

A Computation with a Cartesian mesh

Let us consider a Cartesian mesh with grid pointsxijk yijk zijkRn We compute the final positiont0

t0+Txijk yijk zijk= xijk yijk zijk The matrix repre-sentation of the flow differential is given by

M =xi+1jmacrk minus ximinus1jmacrk

2x

xij+1macrk minus xijminus1macrk

2y

macr xijmacrk+1 minus xijmacrkminus1

2z

yi+1jmacrk minus yiminus1jmacrk

2x

yij+1macrk minus yijminus1macrk

2y

macr yijmacrk+1 minus yijmacrkminus1

2z

] ] ]zi+1jmacrk minus ziminus1jmacrk

2x

zij+1macrk minus zijminus1macrk

2y

macr zijmacrk+1 minus zijmacrkminus1

2z

M

+ O 7

where =maxx y z13 is the maximum step sizeOne can then find the largest singular value of the matrix

M that approximates M for a sufficiently small grid spacing Similarly as suggested by Haller20 one can search for the

largest eigenvalue of MM The algorithm above is thereforeidentical to the algorithm proposed as the two-dimensionaldirect Lyapunov exponent DLE of Haller20 the two-dimensional algorithm of Shadden et al52 or then-dimensional algorithm of Lekien et al37

The major advantage of this algorithm is its robustnessRegardless of the grid size an existing LCS passing be-tween two grid points creates a visible shade in the approxi-mated FTLE field As shown in Fig 1 there is more to thisalgorithm than just approximating a derivative with finitedifferences If we were to approximate the derivative of theflow using arbitrary control points we may be missing someLCS passing between grid points Similarly one could takeeach grid point separately and integrate the velocity gradientor get the CauchyndashGreen strain tensor and the FTLE Thiswould also lead to possible LCS obliteration The DLE algo-

rithm estimates total averaged stretching in the mesh Onlygrid points are used in the computation of the derivatives byfinite differences

The disadvantage of the technique is that the FTLE canbe underestimated if the elements are too large One oftenneeds to refine the mesh in order to avoid saturation Theadvantage is that it detects LCS no matter what the distribu-tion of grid points is Even on coarse meshes one is able toidentify a rough estimate of the LCS structure

The singular value decomposition of M always existsand is continuous with respect to any parameter such as thegrid size 10 As a result provided that the largest singularvalue does not vanish its logarithm is also continuous Thisis however the case since the flow is one to one As a resultnone of the column vectors in M are identically zero thiswould violate the uniqueness of solutions This implies thatthe rank of M is at least 1 and its largest singular value isstrictly positive

Note that the FTLE is typically computed at grid resolu-tions much higher than the flow resolution5260 The FTLEindicates long term particle separation and the LCS have a

017505-3 FTLE for non-Euclidean manifolds Chaos 20 017505 2010


12817312576 On Wed 20 Nov 2013 195935

fine foliated structure even for flows defined on a coarsegrid In time-periodic systems the LCS correspond to theinvariant manifold of hyperbolic periodic orbit and they areknown to form tangles and arbitrary small lobes4847 Forarbitrary time dependence the LCS can also form meandersof arbitrary small size provided that the integration time islong enough To avoid saturation the resolution of the meshhas to be adapted to the amplitude of the deformation Con-sequently the mesh resolution depends on the chosen inte-gration time rather than on the resolution or the input veloc-ity One of the contributions of this paper is to generalize theFTLE algorithm to unstructured meshes that can be effi-ciently refined by increasing the resolution only near theLCS

B Computation with an unstructured mesh bdquoin R2hellip

While robust and efficient the computation of FTLE us-ing Cartesian meshes does not translate easily to arbitrarymanifolds For manifolds such as spheres tori and morecomplicated shapes it is usually more convenient to coverthe space with a list of connected polygons Another incen-tive to use unstructured meshes is the need for higher reso-lution To obtain sharper and longer LCS we need tostraddle them with grid points that are as close to each otheras possible With Cartesian meshes it is difficult to refine themesh other than uniformly Using unstructured meshes onecan easily adapt the resolution locally Smaller elements areneeded in regions where FTLE indicate the presence of LCSwhile low FTLE areas can be decimated14

Another reason for computing FTLE on unstructuredmeshes rather than Cartesian meshes is to match the formatof some input velocity fields For instance methods such asnormal mode analysis reconstruct coastal surface currentmaps from radar data and provide reconstructed stream func-tion and velocity potential on meshes of triangles3429 Com-puting FTLE on an unstructured mesh whose boundarymatches that of the input velocity field greatly improves thesmoothness of the FTLE map

We begin by adapting the algorithm described inSec II A for meshes of triangles in R2 We then generalizethis method to meshes of n-tetrahedra covering a subspace ofRn

Consider a domain R2 which is covered by a list ofN non-overlapping triangles For each vertex pi= 1

i 2i

we assume that the image by the flow pi=t0t0+Tpi

= 1i 2

i has been computedPoint pi has an arbitrary 2 number of neighbors from

which we must approximate the matrix of the derivative ofthe flow map Let us consider the N neighbors p j of pi andconstruct the 2N matrices

Y = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 8

and

X = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 9

There does not usually exist a 22 matrix M such thatY =MX but we can compute the least square estimate

M = YXXXminus1 10

which minimizes the L2 error Y minusMX2We can then proceed with the computation of the largest

singular value of M and obtain an estimate of the FTLE atpoint pi of the unstructured mesh There is however a slightmodification that can be made to improve the algorithm Inthe procedure described above each neighbor accounts fortwo equally weighted data in the least square estimate None-theless it is not uncommon for a vertex pi of an unstructuredmesh to have more neighbors in one direction than anotherIn this case the reconstructed matrix M is distorted by datarecorded in directions where the density of grid points ishigher This distortion can be removed by weighting eachequation involved in the least square estimate Let us con-sider the Voronoi cell around vertex pi that is the polygon

pij pi+1 jpiminus1 j

pi jminus1

pi j+1

pprimeij

pprimei+1 j

pprimeiminus1 j

pprimei jminus1

pprimei j+1

FIG 1 Computation of FTLE on a Cartesian mesh Amesh of points is initialized left panel then integratedfor a given integration time T right panel The deriva-tive of the flow at one grid point is evaluated usingfinite differences with other grid points If one were toevaluate the flow derivative using off grid test pointsgray dots a LCS dashed line passing between gridpoints could go unnoticed



12817312576 On Wed 20 Nov 2013 195935

where all the points are closer to vertex pi than any othervertex The area of the Voronoi cell is

Vi =1

8 pj is aneighof pi

Aij 11

where Aij is the area of the two triangles that contain both pi

and p j We can then define

Y =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 12

and

X =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 13

We now have

M = YXXXminus1

M

+ O~

14

and we approximate the FTLE by

pit0 =1

TlnYXXXminus1 15

where the norm of the matrix is defined as its largest singularvalue Note that the factors 8Vi cancel out from the equa-tions hence we can ignore them

C Computation with an unstructured mesh bdquoin Rnhellip

To generalize the computation of FTLE to unstructuredmeshes in Rn we consider a domain Rn that is coveredby a mesh of n-dimensional tetrahedra For each vertexpi= 1

i 2i n

i of the mesh we consider the N neigh-bors p j = 1

j 2j n

j The images of these vertices un-

der the flow map are written as pi=t0t0+Tpi

= 1i 2

i ni and p j=t0

t0+Tp j= 1j 2

j nj

This setting is depicted on Fig 2For each pair of adjacent vertices pi and p j we define

the hypervolume Aij as the sum of the hypervolumes of allthe n-tetrahedra that contain both pi and p j We define thenN matrices

Y =A1i1

1 minus 1i A2i1

2 minus 1i macr ANi1

N minus 1N

A1i21 minus 2

i A2i22 minus 2

i macr ANi2N minus 2

Nmacr ] ]

A1in1 minus n

i A2in2 minus n

i macr ANinN minus n

N

16

and

X =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N17


pit0 =1

TlnYXXXminus1 18

where YXXXminus1 is a nonsingular nn matrix from whichwe can extract the strictly positive maximum singular value

D Example 1 RayleighndashBeacutenard convection cells

To illustrate the use of the algorithm we consider a two-dimensional model of RayleighndashBeacutenard convection cells in-troduced by Solomon and Gollub5655 The stream function isgiven by

xyt = sin13x minus gtsin13y 19

We consider that the lines y=0 and y=1 are material bound-aries and that the stream function delineates an infinite hori-zontal array of convection cells For gt=0 the cells arebounded by vertical segments x=kZ and there is not any

pi

p1

p2

p3

pjpN

pprimei

pprime1

pprime2

pprime3

pprimej

pprimeN

FIG 2 Deformation of an unstructured mesh underthe flow Left panel the node pi has N neighbors p1p2 p j pN Right panel under the action of theflow the node pi moves to the position pi The de-formed edges p1minuspi p2minuspi p jminuspi pN minuspiare used to approximate the deformation tensor



12817312576 On Wed 20 Nov 2013 195935

cell transport When gt is an oscillating function theboundaries become more complex and transport is possiblethrough the lobes of oscillating LCS53336

In this paper we used a quasiperiodic roll motiongt=03 sin413t+01 sin2t The model can be equallyused with a fully aperiodic function gt For instance onecan define gt as a realization of a random process with zeromean and Gaussian covariance in time see Ref 36 orhttpwwwlekiencom~francoissoftwarerndfieldgen

To capture long sharp LCS one typically needs verydense meshes Indeed as illustrated in Fig 3 for a Cartesianmesh the FTLE can be severely underestimated when thegrid size is too small or the integration step T is too largeThe initial element is stretched in a nonlinear way and thefolding of the element can lead to an improper estimate ofthe stretching since only grid points are used to evaluate thedeformation

To solve this problem and obtain sharp LCS it is there-fore necessary to use elements with very small sizes Such ahigh resolution cannot usually be achieved with Cartesianmeshes since one would need to refine the entire mesh Anadvantage of computing FTLE on an unstructured mesh isthe ability to refine the mesh only where needed In this casewe seek high resolution near the FTLE ridges but not every-where Based on an estimate of the FTLE field on a roughmesh one can generate a new mesh of triangles and increasethe resolution only near the LCS

Figure 4 illustrates such a refinement procedure for theRayleighndashBeacutenard convection cells using refinement rulessimilar to that of Garth et al1314 The first row of Fig 4illustrates the preliminary computation The domain is cov-ered with a rough mesh containing about 3000 triangleseach with roughly the same size The upper right panel ofFig 4 shows the resulting FTLE field for the rough mesh Toease the analysis of Fig 4 each triangle is plotted with auniform color corresponding to the maximum FTLE value ofthe triangle nodes

Using the FTLE estimate in the upper right panel ofFig 4 a more appropriate mesh can be designed We identifyall the nodes of the rough mesh for which the FTLE estimateis above 25 All the triangles containing at least one suchnode are labeled for refinement We then require a minimum

triangle diameter of 002 inside the selected zone Note thatthe triangles can still be rather small in regions where theFTLE is below the threshold The actual triangle size variesdepending on the proximity to the refined zone and the prop-erties of the meshing algorithm The resulting mesh has 8000triangles and are shown along with the corresponding FTLEon the second row of Fig 4

Note that the maximum value of the FTLE is higher inthe refined mesh This is a consequence of the saturationphenomena depicted in Fig 3 The stretching induced by aLCS passing between grid points is underevaluated by thefinite mesh size By decreasing the distance between gridpoints near the LCS we refine the estimated FTLE value onthe LCS and hence the maximum FTLE value for the do-main As a result the right panels of Fig 4 have differentlevel sets To compare the FTLE distribution the ranges arehowever kept constant When the maximum level set is in-creased the minimum level set is increased by the sameamount

Based on the new FTLE estimate in the second row ofFig 4 we can continue the refinement process We selected anew FTLE threshold of 29 and set a minimum triangle di-ameter of 001 for triangles above the threshold The result-ing mesh is shown on the third row of Fig 4 and has 20 000triangles The resulting high quality FTLE field should becompared to the relatively low quality result that a 100200Cartesian mesh would give for the same number of ele-ments The last refinement of Fig 4 uses a threshold of 40and a minimum length scale of 0004 The final mesh pro-vides very high FTLE details on a mesh with only 55 000triangles As a comparison Fig 6 of Shadden et al52 showsFTLE on a similar system based on a 160320 Cartesianmesh 51 200 elements The adaptive mesh obviously pro-vides a tighter estimate of the LCS position

An advantage of the refinement method described aboveis that it increases the resolution near the steep FTLE ridgeAs shown by Shadden et al52 the flux through a LCS is assmall as the ridge is steep sharp slope in a transverse direc-tion and the FTLE value is constant along the ridge Thissituation is hardly observable with Cartesian meshes as theresolution of the mesh must be extremely high to avoid spu-rious FTLE oscillation along the ridge With the unstructured

t = t0

t = t1

t = tf(apparent stretching)

t = tf(actual stretching)

pi jpi j

pi j

pi j

pi+1 jpi+1 j

pi+1 j

pi+1 j

pi j+1pi j+1

pi j+1

pi j+1pi+1 j+1pi+1 j+1

pi+1 j+1

pi+1 j+1

pp

p

pFIG 3 Color online Using elements that are too largecan lead to an underestimated FTLE Starting with asquare element at t= t0 the stretching is well capturedby the motion of the four grid points at a later timet= t1 Nevertheless the deformation is nonlinear and theelement can fold at a later time t= tf At this time thealgorithm that uses the position of the four grid pointsto evaluate FTLE will underestimate the actualstretching



12817312576 On Wed 20 Nov 2013 195935

mesh we can concentrate small elements along the ridge andkeep coarser elements elsewhere

Note that the method described above is not designed tocompute FTLE in any subset of an Euclidean space It worksonly for compact submanifolds of Rn with the same dimen-sion n For instance it cannot be used to compute FTLE ona sphere codimension 1 in R3 In Sec III we extend thismethod to arbitrary manifolds and hence make it possible tocompute FTLE on regions such as spheres cylinders andMoumlbius bands

III FINITE-TIME LYAPUNOV EXPONENTSIN RIEMANNIAN MANIFOLDS

A n-dimensional differentiable manifold M is a space inwhich every point has a local neighborhood that looks likeRn More specifically there exist diffeomorphisms

k Uk M rarr Rn 20

such that the finite sequence of open sets Uk covers M Eachcouple Uk k is called a chart and for any point pM we

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

40

35

30

25

20

15

10

05

00

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

45

40

35

30

25

20

15

10

05

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

50

45

40

35

30

25

20

15

10

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

55

50

45

40

35

30

25

20

15

FTLE

FIG 4 Color Computation of FTLE for the aperiodic convection cells of Solomon and Gollub Ref 56 using an unstructured mesh and dynamic meshrefinement The first row is the initial computation rough estimate A mesh with 3000 triangles of equal size upper left panel is used to evaluate the FTLEupper right panel The second row shows the first refinement step The domain is covered by 8000 triangles whose maximum diameter is set to 002 whereverthe FTLE estimate upper right panel is above 25 The third row uses the FTLE estimate from the second row to further refine the mesh The new mesh has20 000 triangles and the maximum triangle diameter is set to 001 for all the points where the previous FTLE estimate was above 29 In the last row a finalmesh of 55 000 triangles is created by setting the maximum diameter to 0004 for all points where the previous FTLE estimate is above 40 The resultingFTLE field shows a crisp LCS with roughly constant FTLE values lower right panel



12817312576 On Wed 20 Nov 2013 195935

can find at least one diffeomorphism from an open neighbor-hood of p to an open set of Rn A set of diffeomorphisms k

whose domains cover M is a called an atlasExample Consider the sphere M= x y zR3 x2

+y2+z2=113 We consider its partition into the followingthree overlapping open sets

U1 = xyz R3x2 + y2 + z2 = 1 and z 1413

U2 = xyz R3x2 + y2 + z2 = 1 and minus 12 z

1213

21

U3 = xyz R3x2 + y2 + z2 = 1 and z minus 1413

and the diffeomorphic maps

1 U1 rarr R2 xyz 1xyz xy

2 U2 rarr R2 xyz 2xyz atan2yxz 22


Each diffeomorphism k maps a subset Uk of the manifold toan open set of R2 and the union of the subsets Uk is thesphere This shows that the unit sphere M is a two-dimensional differentiable manifold

Let us consider a manifold M A dynamical system onM is defined by a flow t0

t MrarrM which for each pointpM returns its image p=t0

t p The differential of t0t

at a point pM is a linear map from TpM the tangentspace at point p to TpM the tangent space at pointp=t0

t pTo define FTLE we need a norm in TpM and in TpM

The computations in the next sections will require also aninner product in the tangent spaces to construct orthonormalbases For this reason we consider that M is a Riemannianmanifold ie it is a differentiable manifold where each tan-gent space TpM is equipped with an inner product and anassociated norm that varies smoothly with p Note that thisis not a limitation as one can always find such a smoothsystem of inner products for a given differentiable manifoldM One way to find such an inner product and make a mani-fold Riemannian is to exploit the Whitney embedding theo-rem Any n-dimensional manifold can be embedded in Rmprovided that mn is large enough6224 Given an embeddingof the manifold in Rm the inner product in the tangent spacescan then be derived from the Euclidean inner product in RmFurthermore we will later show that the choice of the innerproduct does not influence much the position of the resultingLCS Changing the metric results in a different FLTE fieldbut while low FTLE values can change by much the ridgesremain mostly the same

Given a Riemannian manifold M the definition of theFTLE carries over easily We have

pt0 1

TlnDt0

t0+T

=1

Tln max

uTpMu0

Dt0

t0+TuTpM

uTpM 23

where middot TpM and middot TpM are the norms in the tangent

spaces TpM and TpM

A FTLE in manifold coordinates

Note that the definition of the FTLE on a manifold doesnot depend on the atlas or coordinate system used Never-theless when one evaluates the FTLE it is often more con-venient to use local coordinates By the definition of a dif-ferentiable manifold there must be an open set UpMcontaining p on which there exists a diffeomorphism

p Up M rarr Rn 24

Since p is a diffeomorphism it has a C1 inverse

pminus1 Vp Rn rarr M minus1 25

where Vp=pUp is an open set Since p is in the domain ofp it must belong to the image of p

minus1 We denote by= 1

2 n

the unique point of VpRn satisfyingp

minus1=pNote that by definition of the diffeomorphism the de-

rivative Dpminus1 is continuous takes values in TpM and is full

rank The n vectors

e1 Dp

minus110 0 = pminus1

1

=

e2 Dpminus101 0 = p

minus1

2

=

]

en Dpminus100 1 = p

minus1

n

=

26

can therefore be used as a basis of TpM The basis vectorsdo not necessarily have unit norm and are not necessarilyorthogonal Nevertheless M is a Riemannian manifold andthere is an inner product in TpM which gives the ability touse the GramndashSchmidt process and derive an orthonormalbasis e1 e2 en13 from the basis e1 e2 en13

We can proceed similarly for the image p=t0t0+Tp and

TpM the tangent space at point p We denote by

f1 f2 fn13 the orthonormal basis of TpM We then buildthe nn matrix M = mij defined by

mij fi middot Dt0

t0+Te j 27

Since the two bases have been orthonormalized with the ap-propriate inner products the norm of any vector in TpM orin TpM can be computed with its coordinates and we getthe expected FTLE formula



12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1


The only remaining step to compute the FTLE on the Rie-mannian manifold is therefore the evaluation of the matrix Musing the manifold coordinates Indeed in practice one willnot compute the actual flow map t0

t0+T but the ldquocoordinateflow maprdquo defined by

t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29

In the equation above one starts with manifold coordinates and computes the corresponding manifold point p using thechart p

minus1 The flow t0t0+T is applied to p which returns the

image p We then get the coordinates of the image usingthe chart p which might be different from the first chartp Combining all the steps above gives a ldquocoordinate maprdquot0

t0+T that returns the final coordinates as a function of theinitial coordinates The map derivative Dt0

t0+T is the mostreadily available deformation tensor as it only uses coordi-nates How can we derive the matrix M and the FTLE fromDt0

t0+TNote that by definition we have t0

t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30

The GramndashSchmidt process was used to orthonormalize thecolumn vectors of Dp

minus1 and is hence equivalent to the QRdecomposition

Dpminus1 = Qp Rp 31

where Qp is a unitary linear operator that is Qp Rn

rarrTpM such that forall Rn Qp middotQp= middot and Rp isan upper diagonal matrix linear operator from RnrarrRnFrom this point of view the GramndashSchmidt process isequivalent to defining the new basis by

forall j = 1 n e j = Qp1 j 32

where 1 j denotes the vector of Rn that has all componentsequal to zero except the jth that is equal to one

Similarly in TpM the GramndashSchmidt process isequivalent to the QR decomposition

Dpminus1 = Qp Rp 33

where Qp is unitary and Rp is the upper diagonal The or-thonormal basis in TpM is then given by

foralli = 1 n fi = Qp1i 34

Direct computation gives

Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0

t0+Te j = 1i middot RpDt0

t0+TRp

minus11 j 36

As a result the matrix M is given by

M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37

The equation above states that we can compute the FTLE incoordinates From the ldquocoordinate deformation tensorrdquoDt0

t0+T one needs to multiply by Rp on the left and by Rpminus1

on the right to obtain the matrix M whose largest singularvalue gives the FTLE Note that the matrix Rp is obtainedfrom the QR decomposition of the manifold charts Asshown by Dieci and Eirola10 the matrix Rp must then varysmoothly as a function of p

B Orthogonal coordinates

The equation above gives the matrix M from the coordi-nate deformation tensor but it requires the QR decompositionof the derivative of the coordinate map Dminus1 If the manifoldcoordinates are orthogonal the vectors in the basise1 e2 en13 are mutually orthogonal and the matrix Rp isdiagonal We have

M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38

with i= pminus1 i and i= p

minus1 i

C Orthonormal coordinates

If the coordinates are orthonormal we have M =Dt0t0+T

In other words the FTLE can be computed directly as thelargest singular value of the coordinate deformation tensorWe will later use this property to derive an efficient algo-rithm for FTLE computation

D Example 2 Convection cells on a torus

We consider a system similar to the example in Sec II Dbut instead of creating an infinite array of plane convectioncells we wrap four cells around a cylinder More specificallywe consider the cylinder parametrized by



12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39

and the same stream function as in Sec II D that is

12t = sin1 minus gtsin2 +13

2 40

In other words for each point x y z on the cylinder onecan use the diffeomorphism to compute the manifold co-ordinates 1 2 then compute the stream function or thevelocity field using the formula above where the coordinatesx and y of Solomon and Gollub have been replaced by 1 and2

Since we have the parametrization of the manifold wecan determine analytically the local basis For given 1 and2 we have

e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42

The vectors above are orthogonal hence we can computeFTLE using Eq 38 with 1=1= e1= 1

2 and 2=2

= e2= 113

If 1 2 represents the coordinates of a particle thatstarted at time t0 at position 1 2 the deformation inldquomanifold coordinatesrdquo reads

Dt0

t0+T =1

1

1

2

2

1

2

2

43

According to Sec III B the FTLE is given by the largestsingular value of the matrix

M =1

1

2

1

2

2

2

1

2

2

44

To compute the FTLE we can then cover the manifold witha mesh of points each with given 1 and 2 coordinates We

then differentiate numerically the final positions with respectto the initial positions as if the mesh was in R2 The onlydifferences with a computation in Euclidean space are thefactors 13 2 and 2 13 that scale the off-diagonal elements ofM The FTLE and corresponding LCS on the cylinder areshown in Fig 5

E Example 3 Convection cells on a Moumlbius strip

Instead of wrapping an even number of convection cellsaround a cylinder as in the previous example we can arrangean odd number of cells into a continuous velocity field on aMoumlbius strip This construction does not correspond to aknown physical setting but provides a test case to illustrateand test the methods developed above as well as a first in-stance of an FTLE field for a non-orientable manifold Weconsider the Moumlbius strip parametrized by

minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45

and the same streamfunction as in the previous exampleDirect computation shows that the coordinates are also or-thogonal In this case we have 2=2= e2=1 13 The normof e1 depends on the position on the manifold and we have

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47

As a result the deformation can be computed in manifoldcoordinates 1 and 2 and the FTLE is given as the singularvalue of

M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935

Figure 6 shows the corresponding FTLE for positive integra-tion time as well as the attractive and repulsive LCS Notethat the Moumlbius strip is not an orientable surface FTLE andLCS are well defined on any Riemannian differentiablemanifold whether orientable or not

IV FINITE-TIME LYAPUNOV EXPONENTSON DIFFERENTIABLE MANIFOLDS EMBEDDED IN Rm

The materials in Sec III can in principle be used tocompute FTLE in any Riemannian differentiable manifoldbut it requires knowledge of an explicit atlas In practicehowever many manifolds of interest are only described nu-merically The maps in the atlas and hence their derivativesmight not be known analytically For this reason this sectionpresents a more general method for computing FTLE on dif-ferentiable manifolds

The rationale for the method is that any differentiablemanifold of dimension n can be embedded in the Euclidian

space Rm provided that m is sufficiently large6224 If webegin with a Riemannian manifold that is a manifold to-gether with an inner product at each point Nash andKuiperrsquos embedding theorem combined with Whitneyrsquos em-bedding theorem guarantees that the manifold can be iso-metrically embedded in Rm4132 The inner product on themanifold then corresponds to the Euclidian inner product inRm In other words it is sufficient to consider only manifoldsembedded in Rm

A Finding the local basis

Consider the node pi of the unstructured mesh As shownin Fig 7 we seek the effect of the flow derivative Dt0

t0+T onvectors in the tangent space Tpi

M but we only have infor-mation about the motion of the mesh node pi and itsneighbors p1 p2 p j pN that are all located on themanifold M

We consider that the manifold M is embedded in Rm

and accordingly we view the node pi and its neighborsp1 p2 p j pN as vectors of Rm The objective is todetermine a basis of Tpi

M seen as embedded in TpiRm

=Rm First we consider the N vectors u j =p j minuspi

j=1 N If the grid size was infinitesimally small the Nvectors of u j would exactly span the n-dimensional manifoldTpi

M In this case we could find an orthonormal basis ofTpi

M using the GramndashSchmidt process Starting with the Nvectors u j Rm we would obtain n nonzero orthonormalvectors ek since Tpi

M has dimension n during the GramndashSchmidt process Nminusn vectors would vanish and be elimi-nated

FIG 5 Color FTLE on a cylinder for a positive integration time upperpanel and a negative integration time middle panel Correspondingattractive and repulsive LCS lower panel enhanced onlineURL httpdxdoiorg101063132785161

FIG 6 Color FTLE upper panel LCS lower panel on a Moumlbius stripenhanced online URL httpdxdoiorg101063132785162



12817312576 On Wed 20 Nov 2013 195935

When the grid size is finite the N vectors u j are notexactly lying in Tpi

M An orthonormal set of vectors thatapproximately spans Tpi

M can however still be obtainedusing a variant of the GramndashSchmidt process

Let us begin by normalizing the N vectors u j and defin-ing u j =u j u j In a regular GramndashSchmidt the first vectorof the orthonormal basis is arbitrary To reconstruct the localbasis from the N vectors u j we need to avoid the ldquodirectionspointing out of Tpi

Mrdquo More specifically for each basis vec-tor candidate u j we consider how well the other vectors alsoalign in this direction The vector ukminus uk middot u ju j representsthe portion of uk that cannot be captured by a projection onu j The quantity

k=1

N

uk minus uk middot u ju j2 49

quantifies how well the set of vectors u1 u2 uN can berepresented by the single vector u j Accordingly we definethe first basis vector as

u1 = arg minuj

k=1

N


To get a second basis vector we proceed by recursion Wehave the direct sum

TpiM = spanu113 Tnminus1 51

where Tnminus1 is a subspace of dimension nminus1 that best fits theremaining Nminus1 vectors u j all but the one that was selectedas u1 As a result we can subtract the projection on u1 fromeach vector u j and consider the new set of vectors

u j2 = 0 if u j minus u j middot u1u1 = 0

u j minus u j middot u1u1

u j minus u j middot u1u1 otherwise 52

The second basis vector is then selected as

u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53

Note that the search above should be restricted to only Nminus1indices k and exclude the index corresponding to the firstbasis vector Nevertheless the first basis vector leads to avanishing u j

2 and will never correspond to the minimum inthe equation above As a result we can keep the N candidatesin the formula above

Applying the method recursively we derive the n stepalgorithm below

Initialization

forall j = 1 N u j1 =

u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM

FIG 7 Approximation of the defor-mation tensor map from Tpi

M toTpi

M using an unstructured mesh ona manifold M embedded in Rm



12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j

i minus u ji middot uiui = 0

u ji minus u j

i middot uiui

u ji minus u j

j middot uiui otherwise

Termination After the n steps the set u1 u1 un13 is anorthonormal basis that spans Tpi

M in the limit of infinitesi-mal mesh elements The residual error

= k=1

N

ukn+12 54

represents how well the n vectors of the orthonormal basiscapture the N input vectors vanishes when the number ofneighbors N is equal to n or when the N vectors u j all lie ina n-dimensional subspace If the residual error is too highthe mesh covering the manifold must be refined

Given the N neighbors p1 p2 pN of a node pi thealgorithm above provides an orthonormal basis of an ap-proximation of Tpi

M According to Sec III C since the ba-sis is orthonormal the FTLE can be computed directly incoordinates We can then process the data gathered on theunstructured mesh using a method similar to that describedin Sec II C In this case the local coordinates are given bythe projection of the vectors p j minuspi on the basis vectors Wewrite

forall j = 1 N forall k = 1 n kj = p j minus pi middot uk 55

and the matrix of initial coordinates is given by

X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56

where Aji is the hypervolume of all the n-tetrahedra thatcontain both pi and p j and Vi is the hypervolume of theVoronoi cell of node pi as in Sec II C

Let us denote the image of p j by p j We can consider theN vectors u j=p jminuspi and apply the variant of the GramndashSchmidt to obtain an orthonormal basis u1 u2 un13 ofTpi

M Defining

forall j = 1 N forall k = 1 n kj = p j minus pi middot uk

57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58

Since the two basis are orthonormal the FTLE at point pi isapproximated by the same formula as in Sec II C that is

pit0 =1

TlnYXXXminus1

=1

Tln largest singular value YXXXminus1

59

B Exploiting the embedding

The algorithm described in Sec IV A evaluates FTLE onany n-dimensional manifold embedded in Rm Reconstruct-ing an orthonormal basis in Tpi

M using the GramndashSchmidtvariant is usually straightforward provided that the mesh el-ements are small enough and well conditioned If the re-sidual error is too large it is always possible to refine themesh covering the manifold

The same remark does not apply to the reconstruction ofthe local basis in Tpi

M Indeed the nodes p j are defined asthe advection of the points p j The triangles can be stronglydeformed by the flow and the vectors u j=p jminuspi might notbe small The LCS are characterized by locally high valuesof FTLE and stretching Consequently the deformation ofthe mesh is high in the very regions where we need thehighest accuracy

It is possible however to skip the reconstruction of thebasis in Tpi

M and replace the nN matrix Y of Sec IV Aby the mN matrix containing the coordinates in Rm inwhich the manifold is embedded More specifically we con-sider the coordinates 1

j 2j m

j of the vectors u j=p jminuspi and we do not project on a basis of Tpi

M Thecorresponding nN matrix is given by

Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60

Since the n-dimensional manifold is embedded in Rm wehave mn and the Whitney embedding theorem guarantees

that we can select m2n6224 The mN matrix Y is there-fore larger than the nN matrix Y Nevertheless the matrix

Y does not require the computation of the local basis at thefinal position pi and is simpler to compute In addition using



12817312576 On Wed 20 Nov 2013 195935

Y leads to a more precise FTLE value as it does not rely onthe reconstruction of the local basis in Tpi

M which can beimprecise due to the stretching of the mesh near the LCS

The computation of the FTLE based on Y is identical to

the computation based on Y The FTLE is given by

pit0 =1


61

where we substituted Y for YTo prove the formula above recall that M is embedded

in Rm As a result the tangent space can be decomposed as

Rm = TpiM Npi

M 62

where NpiM is the normal space Y is the matrix represen-

tation of Dt0t0+T Tpi

MrarrTpiM In the limit of infinitesi-

mally small elements Y is the matrix representation of a mapRmrarrRm that corresponds to the direct sum

Dt0

t0+T Omminusn 63

where Omminusn TpiMrarrNpi

M is the null operator For a meshwhose elements have a finite diameter h the null operatormust be replaced by hW with limhrarr0 hW=Omminusn As a resultthere must be a unitary mm matrix R such that

RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and

M = YXXXminus1 = R YXXXminus1

hWXXXminus1 R 66


RMMR = MM + h2XXminus1XWWXXXminus1

67

hence MM and MM have the same eigenvalues up to aterm that vanishes as the square of the maximum elementdiameter in the mesh ie proportional to the element areaProvided that the mesh is sufficiently fine we do not need toreconstruct the local basis at the point pi We can just list the

coordinates in Rm and use Y instead of Y in the FTLEformula

Note that the same trick does not apply to the initialposition We cannot avoid the reconstruction of the local ba-sis at pi and list the Cartesian coordinates in a mN matrix

X Indeed the product XX would have mminusn eigenvaluesthen vanishes with h and it would be close to singular

It is also worth noting that the product MM we use inthe definition of the FTLE is an approximation of the rightfinite-time CauchyndashGreen deformation tensor Although thisoption has not been exploited yet the FTLE can also bederived using the left CauchyndashGreen deformation tensor

MM52 The same technique described in this section wouldapply to the left CauchyndashGreen deformation tensor Indeedwe have

MM = MM hMXXminus1XW

hWXXXminus1M h2WXXXminus2XW = MM hMP

hPM h2PP 68

with P=WXXXminus1 Compared to MM the matrix MM

has mminusn extra eigenvalues but for sufficiently small h theyare all smaller than the eigenvalues of MM Using theidentity

detA B

C D = detA detD minus CAminus1B 69

we can then show that the matching eigenvalues of MM

and MM differ by a term that vanishes as h2 As a resultwe have also

largest singular value M = largest singular value M

+ Oh2 70

C Example 4 The Antarctic polar vortex splittingevent of 2002

In late September of 2002 over a period of a few daysthe Antarctic ozone hole split in two with one of the daugh-ter fragments subsequently reasserting its position over thepole while the other spread into the midlatitudes The split isclearly evident in the ozone concentrations shown in Fig 8while the splittingreformation sequence is shown in Fig 9A split ozone hole implies a split vortex and thus a suddenstratospheric warming54

These types of warming occur in roughly half of all win-ters in the Arctic23 and are thought to be produced by thedynamical momentum forcing resulting from the breakingand dissipation of planetary-scale Rossby waves in thestratosphere Prior to 2002 however no stratospheric suddenwarming had been observed in the Antarctic where reliablerecords go back half a century The relatively early warmingin September followed by a reforming of the circumpolarvortex was an unprecedented event Because of the impor-tance of the coherence of the stratospheric polar vortex forstratospheric ozone depletion the event made it onto thefront pages of newspapers worldwide and became a focus ofthe atmospheric science community61 Figure 8 shows theAntarctic ozone hole in September 2001 1 year before thesplit September 2002 split ozone hole and September2003 1 year after the split

We view the event as one that caused a clear and iden-tifiable topological bifurcation in the stratospheric flow fieldThus it serves as a prototype atmospheric event on which tostudy the effects of topological bifurcations on mixing andtransport of atmospheric tracers Figure 9 shows the LCSassociated with the splitting event based on the 650 K isen-trope of the National Centers for Environmental PredictionNational Center for Atmospheric Research NCEPNCAR



12817312576 On Wed 20 Nov 2013 195935

reanalysis data28 an analysis and forecast system perform-ing data assimilation using past data from 1948 to thepresent

To compute the FTLE we consider the projection of thepoints of the Southern hemisphere on the tangent plane at theequator In other words for each point p= x y z in theSouthern hemisphere we consider the manifold coordinatesx y and the chart

minus1 D rarr M xy x

y

minus R2 minus x2 minus y2 71

where R is the radius of the Earth and D is a disk of radius RThe above coordinates are not orthogonal and the full equa-tion at the end of Sec III A must be implemented to obtainthe FTLE However instead of computing the 22 matrixrepresentation of Dminus1 we will exploit the embedding Thesphere M is embedded in R3 and the chart derivativeDminus1 TpMrarrTpM can also be seen as a functionDminus1 TpMrarrR3 From this point of view the matrix repre-sentation of the embedded chart is of size 32 and its QRdecomposition is of the form

Dminus1 =1 0

0 1

xR2 minus x2 minus y2

yR2 minus x2 minus y2

= QR 72

where Q is a matrix of size 32 satisfying QQ= I and R isan upper triangular matrix of size 22 Note that the ldquore-duced QR decompositionrdquo above is inspired by the GramndashSchmidt process and does not match the usual QR decompo-sition where Q would be of size 33 and R would be ofsize 32 The desired matrices above can however easilybe derived either from the GramndashSchmidt process the modi-fied GramndashSchmidt process or from the standard full QRfactorization16

The upper triangular matrix R can be computed analyti-cally and depends on the coordinates x and y Its inverse canalso be computed analytically and according to Sec III Athe FTLE is given by the largest singular value of the matrix

RxyMRminus1x0y0 73

where x0 y0 are the initial coordinates x y are the finalcoordinates and M is the matrix containing the finite differ-ences of manifold coordinates In other words we keep trackof the manifold coordinates x y and we compute finite dif-ferences as if the space was a plane The only modificationdue to the manifold chart is the multiplication by Rx y onthe left and by Rminus1x0 y0 on the right prior to computingthe singular value

Figure 9 reveals that the computed LCS match the ma-terial border of the ozone hole It has been recognized thattransport in the stratosphere is dominated by advection fromlarge-scale structures and that on a time scale of days toweeks the transport is quasi-horizontal along isentropicsurfaces22 ie on two-dimensional spherical layers like theone shown in Fig 9

The bifurcating LCS structure is noteworthy in severalways The ridges show details of the lobe dynamics associ-ated with the structures something that would be more dif-ficult to obtain from an analysis of potential vorticity fieldsbecause vortex boundaries defined for example as potentialvorticity contours or regions of enhanced potential vorticitygradients are less accurate Although qualitatively the pictureof the vortex breakup and of the large-scale filamentation inthe LCS figure is similar to what can be seen in the potentialvorticity field the LCS are more suitable for calculations oftransport eg by lobe dynamics for which accurate bound-aries and small-scale details of the time-dependent vortexboundaries are needed As showed by Green et al17 anddrsquoOvidio et al9 defining LCS based on Lagrangian criteriainstead of potential vorticity or other Eulerian quantities be-comes essential for fluids with strong time dependence or inturbulent regions

Figure 10 shows how a new LCS appears in the center ofthe polar vortex prior to the splitting and separates particlesbased on which daughter vortex they will end up in

We finish this study by discussing the effect of the ma-trix R in the computation of the FTLE For this exampleignoring the matrices Rx y and Rx0 y0 does not makeany visible difference in the computed LCS The value of the

FIG 8 Color The splitting of the Antarctic ozone hole September 2002 as evident in total column ozone concentrations September 2001 and 2003 areshown for comparison from Ref 61 An example of the computed LCS around the time of the splitting and reformation event is shown in Fig 9 enhancedonline URL httpdxdoiorg101063132785163



12817312576 On Wed 20 Nov 2013 195935

FTLE does change but the variations are small for pointswhere the FTLE is high the resulting ridges are almost iden-tical The differences get much higher when the chart deriva-tive has higher variations for example considering the flownear surface topography or along subsurfaces eg invariantmanifolds of high curvature However as our calculationswere limited to one chart specifically the southern hemi-

sphere below the 10S latitude we did not see large variationsbut note that the FTLE in the chosen chart is clearly incor-rect as we approach the equator

This effect can be understood by analyzing the structureof the matrices R and M Near a LCS the finite-time coor-dinate deformation tensor M has an exponentially growingeigenvalue and an exponentially decaying eigenvalue Theseparation between the eigenvalues grows exponentially intime and there are several orders of magnitude between thetwo when we integrate particles for 8 days In contrast thematrices R are bounded regardless of the integration timeConsequently a large eigenvalue in M induces a large eigen-value in Rx yMRx0 y0 The actual value of the FTLEchanges if we ignore R but provided that the integrationtime is sufficiently large the ridges will remain the same Animportant corollary is that the LCS computation is robust toerror in manifold modelization

D Example 5 A multivortex flow on a sphere

In Sec IV C we studied the 2002 splitting of the Ant-arctic polar vortex using an explicit atlas of the sphericalEarth An alternate method for computing FTLE in this sys-tem is to cover the Earth with an unstructured mesh that weconsider embedded in R3 see eg left panel of Fig 11 Aswe follow the mesh nodes in their motion the algorithm inSec IV B provides an estimate of the FTLE for each of thenodes In this setting the finite differences are taken withrespect to the edges of the two-dimensional meshes but thedifferentiated functions are unprojected coordinates in R3 Inother words the matrix elements of X 2N are differencesbetween manifold coordinates projection on the recon-structed local basis but the matrix elements of Y 3N aredifferences in Cartesian x y and z coordinates

By and large this technique is more convenient and ef-ficient It does not require knowledge of analytical equationsof the charts nor the analytical computation of the tangentbundle Furthermore it is simpler to implement and opti-mize When applied to the NCEPNCAR reanalysis data inSec IV C we obtain the same FTLE field in less computa-tion time

To illustrate the method on a different but related sys-tem we consider chaotic advection in a multivortex flow ona sphere This system is a semi-analytical model that offers auseful paradigm for understanding how complicated spatialstructures can arise and evolve such as the polar vortex splitAs a first attempt to model this event we consider the mo-tion of a passive tracer due to velocity field produced by fourself-advecting point vortices all constrained to the surface ofthe sphere for additional details see eg Ref 42 We con-sider this the simplest model for capturing some of the dy-namical features of the event

Additional important geophysical effects such asrotation2643 or vertical density stratification further compli-cate these dynamical processes but are not considered here

The dynamics of point vortices moving on the surface ofa sphere is not as well understood as the corresponding pla-nar problem3 However the model is very relevant both in

FIG 9 Color The left column shows the superposition of the attractiveand repulsive LCS on the 650 K isentrope on the days surrounding theAntarctic polar vortex splitting event of September 2002 based on NCEPNCAR reanalysis data The attracting repelling curves analogous to un-stable stable manifolds are shown in blue red Before and after thesplitting event in late September we see an isolated blob of air bounded byLCS curves slowly rotating over Antarctica In the days leading up to thesplitting LCS curves form inside the vortex The vortex pinches off sendingthe northwestern part of the ozone hole off into the midlatitudes while thesouthwestern portion goes back to its regular position over AntarcticaNote the formation of lobe at the edges where chaotic stirring occurs acrossthe LCS The right column shows the corresponding daily ozoneconcentration based on NASA TOMS satellite data enhanced onlineURL httpdxdoiorg101063132785164



12817312576 On Wed 20 Nov 2013 195935

atmospheric and oceanographic settings when one considerslarge-scale phenomena where the spherical geometry ofEarthrsquos surface becomes important The full spherical geom-etry as opposed to tangent plane approximations is particu-larly important when considering global streamline patternsgenerated by a given vorticity distribution since the Poincareacuteindex theorem provides an important constraint on allowablepatterns30 These patterns in turn provide the dynamicaltemplates by which one can begin to understand the chaoticadvection of particles in a vortex-dominated flow a topicclosely related to the dynamics of the point vortices

Our interest is the simple case of four identical vorticesinitially on a constant latitudinal ring of colatitude 0 withrespect to the pole When the vortices are evenly spacedalong the ring the configuration is known to be a relativeequilibrium configuration38

We perturb the configuration symmetrically We definethe perturbation parameter which takes the configurationfrom its initial square shape =0 to a rectangle 0ultimately to the singular limit of a two-vortex configuration=1 The vortex motion consists of two frequencies whichin general are incommensurate

FIG 10 Color Prior to the splitting event repulsive LCS appear in the core of the polar vortex indicating the onset of a bifurcation and a separation linebetween particles that will end up in different vortices To verify the dynamics we initiated two parcels of particles on 20 September one on each side of thenascent LCS The simulation shows that the green parcel northwest of the LCS will remain in the core vortex while the purple parcel southeast of the LCSis dragged into the secondary vortex and disintegrate at higher latitude enhanced online URL httpdxdoiorg101063132785165

FIG 11 Color Left Unstructured mesh on the Earth Center FTLE for T=10Trel forward time Right FTLE for T=minus10Trel backward time



12817312576 On Wed 20 Nov 2013 195935

For a given 0 the parameter naturally divides thephase space into two regimes in which one would expect theadvected particle motion to share certain dynamical charac-teristics

For an unperturbed ring of N vortices on the sphere for4N6 there is a colatitudinal region of instability whichopens up in the equatorial zone whereas for N7 the ring isunstable at all latitudes38 In particular for N=4 the ring isstable if 0cosminus11 3 ie 0547deg

The dynamics of the four identical vortices can be de-composed into a superposition of relative vortex motionalong with a global rotation of the whole system around thecenter of vorticity The relative motion is periodic with pe-riod Trel and frequency rel=213 Trel The global rotation canbe characterized by measuring the angular displacement 0

about the center of vorticity between two configurationsseparated by time Trel The frequency of the global rotationcan be defined as glob=0 Trel The pair of frequenciesrel glob depend on the parameters 0 and are gener-ally incommensurate implying that the overall vortex mo-tion is quasiperiodic

Without loss of generality we align the center of vortic-ity with the south polar axis The global rotation of the vor-tices is then around the z-axis with frequency glob In therotating frame the particle motion corresponds to a periodi-cally forced Hamiltonian dynamical system that depends ingeneral on the pair of parameters 0 42

For illustration we consider a ring of vortices initially atlatitude 0=30deg and with =054 which roughly approxi-mates the extent of the Antarctic polar vortex In Fig 11 we

show the FTLE field for the resulting particle velocity fieldand the superimposed attraction and repulsion LCS inFig 12 One notices the prominent rotating saddle point atthe pole with undulating stable and unstable manifoldswhich is also apparent in the real data Fig 9

Advection in a multivortex flow on a sphere provides animportant link between simple dynamical systems modelsand much more complicated models of particle advection inglobal geophysical flows such as the polar vortex2354 Tak-ing the point of view of building dynamically consistentsimple models we can add additional vortices of variousstrengths along with realistic rotation models all of whichavoids the trouble of interpreting results belonging only to akinematic model22 Techniques developed and tested in thesesimpler models for quantifyingmdashand perhaps eveninfluencingmdashtransport can be useful for more realistic mod-els of global-scale geophysical phenomena such as pollutiondispersion in the atmosphere and ocean352 polar vortexbreak-up6123 and large-scale transport of aerobiota includ-ing airborne plant pathogens and other invasive species12551

V CONCLUSIONS

This paper introduces numerical techniques for resolvingand representing evolving and likely convolutednminus1-dimensional LCS in nonautonomous vector fields onn-dimensional Riemannian manifolds where the vector fieldis known either analytically or from measured data

We have shown that LCS exist for systems on any Rie-mannian differentiable manifold whether orientable or notWhile our first algorithm for computing FTLE requiresknowledge of charts and chart derivatives and is more theo-retical in nature we have developed a general algorithm forcomputing FTLE on Riemannian manifolds covered bymeshes of polyhedra Only the initial positions the final po-sitions and a link list of node neighbors are required to usethe algorithm

In addition to providing FTLE fields on Riemannianmanifolds we also showed that the unstructured algorithmcan be efficiently used in Euclidean spaces where it gives theopportunity to refine the mesh near the LCS while not in-creasing the number of elements in other regions Adaptivemeshing is an essential step in achieving FTLE computationin high-dimensional spaces at a sufficiently high resolutionHigh resolution FTLE fields are important for accurate auto-mated ridge extraction algorithms

In this paper we have applied the developed methods tocompute FTLE in two-dimensional Riemannian manifoldsHowever the methods also apply to the computation ofFTLE for three-dimensional flows In this case the LCS aretwo-dimensional surfaces More generally computation ofFTLE for a dynamical system on a n-dimensional manifoldleads to the identification of LCS of dimension nminus137 Alibrary encapsulating all the methods described in this paperis available at httpwwwlekiencom~francoissoftwarelibunsdle The library can compute FTLE on unstructuredmeshes in spaces of any dimension using an embedding ornot

FIG 12 Color Superposition of the attractive and repulsive LCS for aperturbed four vortex ring near the South Pole As in the LCS shown inFig 9 from atmospheric reanalysis data the model shows a hyperbolicregion and the formation of filamentary structures that extendover large portions of the sphere enhanced onlineURL httpdxdoiorg101063132785166



12817312576 On Wed 20 Nov 2013 195935

We also note that the present paper focuses on comput-ing the FTLE but the definitions and methods extend easilyto FSLE27319 In the finite time approach we evaluate thestretching of the elements for a prescribed finite horizon intime In the finite size scheme one marks the time it takes toreach a prescribed stretching ratio The techniques developedin this paper can be applied equally to FTLE or FSLE

Further developments will include application to higherdimensional manifolds relevant in mechanics such as prod-ucts of the Lie groups SO3 SE3 and their tangentbundles With the appropriate visualization tools computa-tions in these phase spaces can aid the search for dynamicalstructure in mechanical and biomechanical data from experi-ments or computational models such as separatrices betweenstable and unstable motions441549585950 Additionally itmay be of interest to detect LCS structure on high-dimensional invariant manifolds of possibly high curvaturewhich are commonly found in phase space such as energymanifolds center manifolds and stable and unstable mani-folds

1Aylor D E ldquoSpread of plant disease on a continental scale Role of aerialdispersal of pathogensrdquo Ecology 84 1989ndash1997 2003

2Beron-Vera F J Olascoaga M J and Goni G J ldquoOceanic mesoscaleeddies as revealed by Lagrangian coherent structuresrdquo Geophys ResLett 35 L12603 doi1010292008GL033957 2008

3Boatto S and Pierrehumbert R ldquoDynamics of a passive tracer in thevelocity field of four identical point vorticesrdquo J Fluid Mech 394 137ndash174 1999

4Bowman K P ldquoManifold geometry and mixing in observed atmosphericflowsrdquo preprint 1999

5Camassa R and Wiggins S ldquoChaotic advection in a RayleighndashBeacutenardflowrdquo Phys Rev A 43 774ndash797 1991

6Cardwell B M and Mohseni K ldquoVortex shedding over a two-dimensional airfoil Where the particles come fromrdquo AIAA J 46 545ndash547 2008

7Coulliette C Lekien F Paduan J D Haller G and Marsden J EldquoOptimal pollution mitigation in Monterey Bay based on coastal radar dataand nonlinear dynamicsrdquo Environ Sci Technol 41 6562ndash6572 2007

8Coulliette C and Wiggins S ldquoIntergyre transport in a wind-drivenquasigeostrophic double gyre An application of lobe dynamicsrdquo Nonlin-ear Processes Geophys 8 69ndash94 2001

9drsquoOvidio F Isern-Fontanet J Loacutepez C Hernaacutendez-Garciacutea E andGarciacutea-Ladona E ldquoComparison between Eulerian diagnostics and finite-size Lyapunov exponents computed from altimetry in the Algerian basinrdquoDeep-Sea Res Part I 56 15ndash31 2009

10Dieci L and Eirola T ldquoOn smooth decompositions of matricesrdquo SIAMJ Matrix Anal Appl 20 800ndash819 1999

11Fenichel N ldquoPersistence and smoothness of invariant manifolds forflowsrdquo Indiana Univ Math J 21 193 1971

12Froyland G and Padberg K ldquoAlmost-invariant sets and invariantmanifoldsmdashConnecting probabilistic and geometric descriptions of coher-ent structures in flowsrdquo Physica D 238 1507ndash1523 2009

13Garth C Gerhardt F Tricoche X and Hagen H ldquoEfficient computa-tion and visualization of coherent structures in fluid flow applicationsrdquoIEEE Trans Vis Comput Graph 13 1464ndash1471 2007

14Garth C Wiebel A Tricoche X Joy K and Scheuermann G ldquoLa-grangian visualization of flow-embedded surface structuresrdquo ComputGraph Forum 27 1007ndash1014 2008

15Gawlik E S Marsden J E Du Toit P C and Campagnola S ldquoLa-grangian coherent structures in the planar elliptic restricted three-bodyproblemrdquo Celest Mech Dyn Astron 103 227ndash249 2009

16Golub G H and Van Loan C F Matrix Computations 3rd ed JohnsHopkins University Press Baltimore MD 1996

17Green M A Rowley C W and Haller G ldquoDetection of Lagrangiancoherent structures in three-dimensional turbulencerdquo J Fluid Mech 572111ndash120 2007

18Guckenheimer J and Holmes P J Nonlinear Oscillations Dynamical

Systems and Bifurcations of Vector Fields Springer-Verlag New York1983

19Haller G ldquoFinding finite-time invariant manifolds in two-dimensionalvelocity fieldsrdquo Chaos 10 99ndash108 2000

20Haller G ldquoDistinguished material surfaces and coherent structures in 3Dfluid flowsrdquo Physica D 149 248ndash277 2001

21Haller G and Poje A C ldquoFinite-time transport in aperiodic flowsrdquoPhysica D 119 352ndash380 1998

22Haynes P ldquoTransport stirring and mixing in the atmosphererdquo in Pro-ceedings of the NATO Advanced Study Institute on Mixing Chaos andTurbulence Cargese Corse France 7ndash20 July 1996 edited by Chateacute Hand Villermaux E Kluwer Dordrecht 1999 pp 229ndash272

23Haynes P ldquoStratospheric dynamicsrdquo Annu Rev Fluid Mech 37 263ndash293 2005

24Hirsch M W Differential Topology Springer New York 1976 p 3325Isard S A Gage S H Comtois P and Russo J H ldquoPrinciples of the

atmospheric pathway for invasive species applied to soybean rustrdquo Bio-Science 55 851ndash861 2005

26Jamaloodeen M I and Newton P K ldquoThe N-vortex problem on a rotat-ing sphere II Heterogeneous Platonic solid equilibriardquo Proc R SocLondon Ser A 462 3277ndash3299 2006

27Joseph B and Legras B ldquoRelation between kinematic boundaries stir-ring and barriers for the Antarctic polar vortexrdquo J Atmos Sci 59 1198ndash1212 2002

28Kalnay E Kanamitsu M Kistler R Collins W Deaven D GandinL Iredell M Saha S White G Woollen J Zhu Y Chelliah MEbisuzaki W Higgins W Janowiak J Mo K C Ropelewski CWang J Leetmaa A Reynolds R Jenne R and Joseph D ldquoTheNCEPNCAR 40-year reanalysis projectrdquo Bull Am Meteorol Soc 77437ndash471 1996

29Kaplan D M and Lekien F ldquoSpatial interpolation and filtering of sur-face current data based on open-boundary modal analysisrdquo J GeophysRes Oceans 112 C12007 doi1010292006JC003984 2007

30Kidambi R and Newton P K ldquoStreamline topologies for integrablevortex motion on a sphererdquo Physica D 140 95ndash125 2000

31Koh T and Legras B ldquoHyperbolic lines and the stratospheric polar vor-texrdquo Chaos 12 382ndash394 2002

32Kuiper N H ldquoOn C1-isometric embeddingsrdquo Proc K Ned Akad WetSer A Math Sci 58 545ndash556 1955

33Lekien F and Coulliette C ldquoChaotic stirring in quasi-turbulent flowsrdquoPhilos Trans R Soc London Ser A 365 3061ndash3084 2007

34Lekien F Coulliette C Bank R and Marsden J E ldquoOpen-boundarymodal analysis Interpolation extrapolation and filteringrdquo J GeophysRes Oceans 109 C12004 doi1010292004JC002323 2004

35Lekien F Coulliette C Mariano A J Ryan E H Shay L K HallerG and Marsden J E ldquoPollution release tied to invariant manifolds Acase study for the coast of Floridardquo Physica D 210 1ndash20 2005 seewwwlekiencom~francoispapersrsmas

36Lekien F and Haller G ldquoUnsteady flow separation on slip boundariesrdquoPhys Fluids 20 097101 2008

37Lekien F Shadden S C and Marsden J E ldquoLagrangian coherentstructures in n-dimensional systemsrdquo J Math Phys 48 065404 2007see wwwlekiencom~francoispaperslcs3d

38Lim C C Montaldi J and Roberts M ldquoRelative equilibria of pointvortices on the sphererdquo Physica D 148 97ndash135 2001

39Lipinski D Cardwell B and Mohseni K ldquoA Lagrangian analysis of atwo-dimensional airfoil with vortex sheddingrdquo J Phys A Math Theor41 344011 2008

40Mantildeeacute R ldquoPersistent manifolds are normally hyperbolicrdquo Trans AmMath Soc 246 261ndash283 1978

41Nash J ldquoC1-isometric imbeddingsrdquo Ann Math 60 383ndash396 195442Newton P K and Ross S D ldquoChaotic advection in the restricted four-

vortex problem on a sphererdquo Physica D 223 36ndash53 200643Newton P K and Shokraneh H ldquoThe N-vortex problem on a rotating

sphere I Multi-frequency configurationsrdquo Proc R Soc London Ser A462 149ndash169 2006

44Norris J A Marsh A P Granata K P and Ross S D ldquoRevisiting thestability of 2D passive biped walking Local behaviorrdquo Physica D 2373038ndash3045 2008

45Olascoaga M J Rypina I I Brown M G Beron-Vera F J KocakH Brand L E Halliwell G R and Shay L K ldquoPersistent transportbarrier on the West Florida Shelfrdquo Geophys Res Lett 33 L22603doi1010292006GL027800 2006

46Peng J F Dabiri J O Madden P G and Lauder G V ldquoNon-invasive



12817312576 On Wed 20 Nov 2013 195935

measurement of instantaneous forces during aquatic locomotion A casestudy of the bluegill sunfish pectoral finrdquo J Exp Biol 210 685ndash6982007

47Rom-Kedar V and Wiggins S ldquoTransport in two-dimensional mapsConcepts examples and a comparison of the theory of Rom-Kedar andWiggins with the Markov model of MacKay Meiss Ott and PercivalrdquoPhysica D 51 248ndash266 1991

48Rom-Kedar V and Wiggins S ldquoTransport in 2-dimensional mapsrdquoArch Ration Mech Anal 109 239ndash298 1990

49Ross S D ldquoThe interplanetary transport networkrdquo Am Sci 94 230ndash237 2006

50Ross S D Tanaka M L and Senatore C ldquoDetecting dynamicalboundaries from kinematic data in biomechanicsrdquo Chaos 20 0175072010

51Schmale D G Shields E J and Bergstrom G C ldquoNight-time sporedeposition of the Fusarium head blight pathogen Gibberella zeaerdquo Can JPlant Pathol 28 100ndash108 2006

52Shadden S C Lekien F and Marsden J E ldquoDefinition and propertiesof Lagrangian coherent structures from finite-time Lyapunov exponents intwo-dimensional aperiodic flowsrdquo Physica D 212 271ndash304 2005

53Shadden S C and Taylor C A ldquoCharacterization of coherent structuresin the cardiovascular systemrdquo Ann Biomed Eng 36 1152ndash1162 2008

54Simmons A Hortal M Kelly G McNally A Untch A and UppalaS ldquoECMWF analyses and forecasts of stratospheric winter polar vortex

breakup September 2002 in the Southern Hemisphere and related eventsrdquoJ Atmos Sci 62 668ndash689 2005

55Solomon T H and Gollub J P ldquoChaotic particle-transport in time-dependent RayleighndashBeacutenard convectionrdquo Phys Rev A 38 6280ndash62861988

56Solomon T H and Gollub J P ldquoPassive transport in steady RayleighndashBeacutenard convectionrdquo Phys Fluids 31 1372ndash1379 1988

57Tallapragada P and Ross S D ldquoParticle segregation by Stokes numberfor small neutrally buoyant spheres in a fluidrdquo Phys Rev E 78 0363082008

58Tanaka M L and Ross S D ldquoSeparatrices and basins of stability fromtime series data An application to biodynamicsrdquo Nonlinear Dyn 581ndash21 2009

59Tanaka M L Nussbaum M A and Ross S D ldquoEvaluation of thethreshold of stability for the human spinerdquo J Biomech 42 1017ndash10222009

60Tew Kai E Rossi V Sudre J Weimerskirch H Lopez C Hernandez-Garcia E Marsac F and Garccedilon V ldquoTop marine predators track La-grangian coherent structuresrdquo Proc Natl Acad Sci USA 106 8245ndash8250 2009

61The Antarctic Vortex Splitting Event edited by Shepherd T and PlumbA special issue of J Atmos Sci 62 3 2005

62Whitney H ldquoThe self-intersections of a smooth n-manifold in 2n-spacerdquoAnn Math 45 220ndash293 1944



12817312576 On Wed 20 Nov 2013 195935

The computation of finite-time Lyapunov exponents on unstructuredmeshes and for non-Euclidean manifolds

Francois Lekien1a and Shane D Ross2b

1Eacutecole Polytechnique Universiteacute Libre de Bruxelles B-1050 Brussels Belgium2Engineering Science and Mechanics Virginia Tech Blacksburg Virginia 24061 USA

Received 4 September 2009 accepted 4 December 2009 published online 8 February 2010

We generalize the concepts of finite-time Lyapunov exponent FTLE and Lagrangian coherentstructures to arbitrary Riemannian manifolds The methods are illustrated for convection cells oncylinders and Moumlbius strips as well as for the splitting of the Antarctic polar vortex in the sphericalstratosphere and a related point vortex model We modify the FTLE computational method andaccommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape aswell as to facilitate dynamic refinement of the FTLE mesh copy 2010 American Institute of Physicsdoi10106313278516

Riemannian manifolds are ubiquitous in science and en-gineering being the more natural mathematical settingfor many dynamical systems For instance transportalong isopycnal surfaces in the ocean and large-scale mix-ing in the atmosphere are processes taking place on acurved manifold not a vector space In this paper wegeneralize the notion of finite-time Lyapunov exponent(FTLE) and Lagrangian coherent structures (LCS) to ar-bitrary Riemannian differentiable manifolds We showthat both notions are independent of the coordinate sys-tem but depend on the chosen metric However we findthat LCS do not depend much on the metric The FTLEmeasures separation and tends to be large and positivealong LCS and very small elsewhere For sufficientlylarge integration times the steep variations of the FTLEfield cannot be modified much by smooth changes in themetric and the LCS remain essentially unchanged Ap-proximating or even ignoring the manifold metric doesnot influence the result for large integration times andtherefore computing the FTLE field on manifolds is ro-bust Aside from these conclusions we present a generalalgorithm for computing the FTLE on manifolds coveredwith meshes of polyhedra The algorithm requires knowl-edge of the mesh nodes the image of the mesh nodesunder the flow as well as information about node neigh-bors (but not the full connectivity) We also used the samealgorithm in Euclidian spaces where the unstructuredmesh permits efficient adaptive refinement for capturingsharp LCS features We illustrate the results and meth-ods on several systems convection cells in a plane on acylinder and on a Moumlbius strip as well as atmospherictransport resulting from the 2002 splitting of the Antarc-tic ozone hole in the spherical stratosphere and a relatedpoint vortex model on the sphere

I INTRODUCTION

Consider an Euclidean domain Rn A typical dy-namical systems is given in the form of a velocity fieldvx t defined on R In this case the trajectories aregiven by the solutions of the ordinary differential equationx=vx t Each trajectory is a function of time but it alsodepends on the initial position x0 and the initial time t0 Howthe trajectory changes as the initial time and the initial posi-tion changes are a central interest in this paper For this rea-son the trajectory which is solution of the initial value prob-lem

x = vxt and xt0 = x0 1

is denoted as xt t0 x0 which emphasizes the explicit de-pendence on the initial condition It is also convenient todefine the flow operator to further highlight the dependenceon the initial position x0 For a given initial time t0 and agiven final time t the flow map is the function

t0t rarr x0 t0

t x0 = xtt0x0 2

Autonomous systems do not depend on time They satisfyt0+T

t+T =t0t for any TR For the associated velocity field

this property translates into vx t=vx For two-dimensional autonomous systems regions of qualitativelydifferent dynamics can be efficiently captured by the stableand unstable invariant manifolds of saddle points in thesystem18 In higher dimensional autonomous systems sepa-ratrices are codimension 1 stable and unstable manifolds ofnormally hyperbolic manifolds1140

In the Poincareacute sections of time-periodic systems lobesbetween the invariant manifolds govern particle transportacross separatrices4847 In this paper we consider the gener-alization of this framework to systems with arbitrary timedependence In this case saddle fixed points are nongenericThe points where the instantaneous velocity vanishes arecalled hyperbolic stagnation points and usually meander intime8 Such points are not trajectories and are not associatedwith hyperbolic invariant manifolds36 The concept of LCS

aElectronic mail lekienulbacbebElectronic mail sdrossvtedu

CHAOS 20 017505 2010

1054-1500201020101750520$3000 copy 2010 American Institute of Physics20 017505-1


12817312576 On Wed 20 Nov 2013 195935







t0

t0+Tx0 + u = t0

t0+Tx0 +t0

t0+T

x0u + Ou2 3





x0t0 =1

Tlnmax

u0

t0

t0+T

x0u

u 4


u0t0

t0+T x0u=etu






x0t0 =1

Tln Dt0

t0+T 1

Tlnmax

u0

Dt0

t0+Tu

u

5











12817312576 On Wed 20 Nov 2013 195935








x0t0 =1

Tlnmax

u0

Dt0

t0+Tu

u

=1

2Tlnmax

UR0n

UMMU

UU =

1







2x


2y


2z


2x


2y


2z


2x


2y


2z

M

+ O 7











12817312576 On Wed 20 Nov 2013 195935







i 2i


= 1i 2



Y = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 8

and

X = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 9


M = YXXXminus1 10




pi jminus1

pi j+1

pprimeij

pprimei+1 j

pprimeiminus1 j

pprimei jminus1

pprimei j+1




12817312576 On Wed 20 Nov 2013 195935


Vi =1

8 pj is aneighof pi

Aij 11



Y =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 12

and

X =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 13

We now have

M = YXXXminus1

M

+ O~

14


pit0 =1

TlnYXXXminus1 15




i 2i n


j 2j n



= 1i 2

i ni and p j=t0

t0+Tp j= 1j 2

j nj



Y =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N

16

and

X =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N17


pit0 =1

TlnYXXXminus1 18






pi

p1

p2

p3

pjpN

pprimei

pprime1

pprime2

pprime3

pprimej

pprimeN




12817312576 On Wed 20 Nov 2013 195935











t = t0

t = t1



pi jpi j

pi j

pi j

pi+1 jpi+1 j

pi+1 j

pi+1 j

pi j+1pi j+1

pi j+1


pi+1 j+1

pi+1 j+1

pp

p




12817312576 On Wed 20 Nov 2013 195935





k Uk M rarr Rn 20


0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

40

35

30

25

20

15

10

05

00

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

45

40

35

30

25

20

15

10

05

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

50

45

40

35

30

25

20

15

10

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

55

50

45

40

35

30

25

20

15

FTLE




12817312576 On Wed 20 Nov 2013 195935




U1 = xyz R3x2 + y2 + z2 = 1 and z 1413


1213

21














pt0 1

TlnDt0

t0+T

=1

Tln max

uTpMu0

Dt0

t0+TuTpM

uTpM 23


spaces TpM and TpM



p Up M rarr Rn 24





2 n




rank The n vectors

e1 Dp


1

=

e2 Dpminus101 0 = p

minus1

2

=

]

en Dpminus100 1 = p

minus1

n

=

26





mij fi middot Dt0

t0+Te j 27




12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1




t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29







t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30



Dpminus1 = Qp Rp 31






Dpminus1 = Qp Rp 33




Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0


t0+TRp

minus11 j 36


M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37






M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38


minus1 i








12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935







t0

t0+Tx0 + u = t0

t0+Tx0 +t0

t0+T

x0u + Ou2 3





x0t0 =1

Tlnmax

u0

t0

t0+T

x0u

u 4


u0t0

t0+T x0u=etu






x0t0 =1

Tln Dt0

t0+T 1

Tlnmax

u0

Dt0

t0+Tu

u

5











12817312576 On Wed 20 Nov 2013 195935








x0t0 =1

Tlnmax

u0

Dt0

t0+Tu

u

=1

2Tlnmax

UR0n

UMMU

UU =

1







2x


2y


2z


2x


2y


2z


2x


2y


2z

M

+ O 7











12817312576 On Wed 20 Nov 2013 195935







i 2i


= 1i 2



Y = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 8

and

X = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 9


M = YXXXminus1 10




pi jminus1

pi j+1

pprimeij

pprimei+1 j

pprimeiminus1 j

pprimei jminus1

pprimei j+1




12817312576 On Wed 20 Nov 2013 195935


Vi =1

8 pj is aneighof pi

Aij 11



Y =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 12

and

X =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 13

We now have

M = YXXXminus1

M

+ O~

14


pit0 =1

TlnYXXXminus1 15




i 2i n


j 2j n



= 1i 2

i ni and p j=t0

t0+Tp j= 1j 2

j nj



Y =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N

16

and

X =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N17


pit0 =1

TlnYXXXminus1 18






pi

p1

p2

p3

pjpN

pprimei

pprime1

pprime2

pprime3

pprimej

pprimeN




12817312576 On Wed 20 Nov 2013 195935











t = t0

t = t1



pi jpi j

pi j

pi j

pi+1 jpi+1 j

pi+1 j

pi+1 j

pi j+1pi j+1

pi j+1


pi+1 j+1

pi+1 j+1

pp

p




12817312576 On Wed 20 Nov 2013 195935





k Uk M rarr Rn 20


0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

40

35

30

25

20

15

10

05

00

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

45

40

35

30

25

20

15

10

05

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

50

45

40

35

30

25

20

15

10

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

55

50

45

40

35

30

25

20

15

FTLE




12817312576 On Wed 20 Nov 2013 195935




U1 = xyz R3x2 + y2 + z2 = 1 and z 1413


1213

21














pt0 1

TlnDt0

t0+T

=1

Tln max

uTpMu0

Dt0

t0+TuTpM

uTpM 23


spaces TpM and TpM



p Up M rarr Rn 24





2 n




rank The n vectors

e1 Dp


1

=

e2 Dpminus101 0 = p

minus1

2

=

]

en Dpminus100 1 = p

minus1

n

=

26





mij fi middot Dt0

t0+Te j 27




12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1




t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29







t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30



Dpminus1 = Qp Rp 31






Dpminus1 = Qp Rp 33




Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0


t0+TRp

minus11 j 36


M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37






M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38


minus1 i








12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935








x0t0 =1

Tlnmax

u0

Dt0

t0+Tu

u

=1

2Tlnmax

UR0n

UMMU

UU =

1







2x


2y


2z


2x


2y


2z


2x


2y


2z

M

+ O 7











12817312576 On Wed 20 Nov 2013 195935







i 2i


= 1i 2



Y = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 8

and

X = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 9


M = YXXXminus1 10




pi jminus1

pi j+1

pprimeij

pprimei+1 j

pprimeiminus1 j

pprimei jminus1

pprimei j+1




12817312576 On Wed 20 Nov 2013 195935


Vi =1

8 pj is aneighof pi

Aij 11



Y =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 12

and

X =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 13

We now have

M = YXXXminus1

M

+ O~

14


pit0 =1

TlnYXXXminus1 15




i 2i n


j 2j n



= 1i 2

i ni and p j=t0

t0+Tp j= 1j 2

j nj



Y =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N

16

and

X =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N17


pit0 =1

TlnYXXXminus1 18






pi

p1

p2

p3

pjpN

pprimei

pprime1

pprime2

pprime3

pprimej

pprimeN




12817312576 On Wed 20 Nov 2013 195935











t = t0

t = t1



pi jpi j

pi j

pi j

pi+1 jpi+1 j

pi+1 j

pi+1 j

pi j+1pi j+1

pi j+1


pi+1 j+1

pi+1 j+1

pp

p




12817312576 On Wed 20 Nov 2013 195935





k Uk M rarr Rn 20


0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

40

35

30

25

20

15

10

05

00

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

45

40

35

30

25

20

15

10

05

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

50

45

40

35

30

25

20

15

10

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

55

50

45

40

35

30

25

20

15

FTLE




12817312576 On Wed 20 Nov 2013 195935




U1 = xyz R3x2 + y2 + z2 = 1 and z 1413


1213

21














pt0 1

TlnDt0

t0+T

=1

Tln max

uTpMu0

Dt0

t0+TuTpM

uTpM 23


spaces TpM and TpM



p Up M rarr Rn 24





2 n




rank The n vectors

e1 Dp


1

=

e2 Dpminus101 0 = p

minus1

2

=

]

en Dpminus100 1 = p

minus1

n

=

26





mij fi middot Dt0

t0+Te j 27




12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1




t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29







t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30



Dpminus1 = Qp Rp 31






Dpminus1 = Qp Rp 33




Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0


t0+TRp

minus11 j 36


M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37






M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38


minus1 i








12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935







i 2i


= 1i 2



Y = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 8

and

X = 11 minus 1

i 11 minus 1

imacr 1

N minus 1N

21 minus 2

i 21 minus 2

imacr 2

N minus 2N 9


M = YXXXminus1 10




pi jminus1

pi j+1

pprimeij

pprimei+1 j

pprimeiminus1 j

pprimei jminus1

pprimei j+1




12817312576 On Wed 20 Nov 2013 195935


Vi =1

8 pj is aneighof pi

Aij 11



Y =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 12

and

X =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 13

We now have

M = YXXXminus1

M

+ O~

14


pit0 =1

TlnYXXXminus1 15




i 2i n


j 2j n



= 1i 2

i ni and p j=t0

t0+Tp j= 1j 2

j nj



Y =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N

16

and

X =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N17


pit0 =1

TlnYXXXminus1 18






pi

p1

p2

p3

pjpN

pprimei

pprime1

pprime2

pprime3

pprimej

pprimeN




12817312576 On Wed 20 Nov 2013 195935











t = t0

t = t1



pi jpi j

pi j

pi j

pi+1 jpi+1 j

pi+1 j

pi+1 j

pi j+1pi j+1

pi j+1


pi+1 j+1

pi+1 j+1

pp

p




12817312576 On Wed 20 Nov 2013 195935





k Uk M rarr Rn 20


0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

40

35

30

25

20

15

10

05

00

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

45

40

35

30

25

20

15

10

05

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

50

45

40

35

30

25

20

15

10

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

55

50

45

40

35

30

25

20

15

FTLE




12817312576 On Wed 20 Nov 2013 195935




U1 = xyz R3x2 + y2 + z2 = 1 and z 1413


1213

21














pt0 1

TlnDt0

t0+T

=1

Tln max

uTpMu0

Dt0

t0+TuTpM

uTpM 23


spaces TpM and TpM



p Up M rarr Rn 24





2 n




rank The n vectors

e1 Dp


1

=

e2 Dpminus101 0 = p

minus1

2

=

]

en Dpminus100 1 = p

minus1

n

=

26





mij fi middot Dt0

t0+Te j 27




12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1




t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29







t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30



Dpminus1 = Qp Rp 31






Dpminus1 = Qp Rp 33




Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0


t0+TRp

minus11 j 36


M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37






M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38


minus1 i








12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935


Vi =1

8 pj is aneighof pi

Aij 11



Y =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 12

and

X =A1i

8Vi1

1 minus 1i macr

ANi

8Vi1

N minus 1N

A1i

8Vi2

1 minus 2i macr

ANi

8Vi2

N minus 2N 13

We now have

M = YXXXminus1

M

+ O~

14


pit0 =1

TlnYXXXminus1 15




i 2i n


j 2j n



= 1i 2

i ni and p j=t0

t0+Tp j= 1j 2

j nj



Y =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N

16

and

X =A1i1

1 minus 1i A2i1


N minus 1N

A1i21 minus 2

i A2i22 minus 2


Nmacr ] ]

A1in1 minus n

i A2in2 minus n


N17


pit0 =1

TlnYXXXminus1 18






pi

p1

p2

p3

pjpN

pprimei

pprime1

pprime2

pprime3

pprimej

pprimeN




12817312576 On Wed 20 Nov 2013 195935











t = t0

t = t1



pi jpi j

pi j

pi j

pi+1 jpi+1 j

pi+1 j

pi+1 j

pi j+1pi j+1

pi j+1


pi+1 j+1

pi+1 j+1

pp

p




12817312576 On Wed 20 Nov 2013 195935





k Uk M rarr Rn 20


0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

40

35

30

25

20

15

10

05

00

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

45

40

35

30

25

20

15

10

05

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

50

45

40

35

30

25

20

15

10

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

55

50

45

40

35

30

25

20

15

FTLE




12817312576 On Wed 20 Nov 2013 195935




U1 = xyz R3x2 + y2 + z2 = 1 and z 1413


1213

21














pt0 1

TlnDt0

t0+T

=1

Tln max

uTpMu0

Dt0

t0+TuTpM

uTpM 23


spaces TpM and TpM



p Up M rarr Rn 24





2 n




rank The n vectors

e1 Dp


1

=

e2 Dpminus101 0 = p

minus1

2

=

]

en Dpminus100 1 = p

minus1

n

=

26





mij fi middot Dt0

t0+Te j 27




12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1




t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29







t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30



Dpminus1 = Qp Rp 31






Dpminus1 = Qp Rp 33




Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0


t0+TRp

minus11 j 36


M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37






M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38


minus1 i








12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935











t = t0

t = t1



pi jpi j

pi j

pi j

pi+1 jpi+1 j

pi+1 j

pi+1 j

pi j+1pi j+1

pi j+1


pi+1 j+1

pi+1 j+1

pp

p




12817312576 On Wed 20 Nov 2013 195935





k Uk M rarr Rn 20


0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

40

35

30

25

20

15

10

05

00

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

45

40

35

30

25

20

15

10

05

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

50

45

40

35

30

25

20

15

10

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

55

50

45

40

35

30

25

20

15

FTLE




12817312576 On Wed 20 Nov 2013 195935




U1 = xyz R3x2 + y2 + z2 = 1 and z 1413


1213

21














pt0 1

TlnDt0

t0+T

=1

Tln max

uTpMu0

Dt0

t0+TuTpM

uTpM 23


spaces TpM and TpM



p Up M rarr Rn 24





2 n




rank The n vectors

e1 Dp


1

=

e2 Dpminus101 0 = p

minus1

2

=

]

en Dpminus100 1 = p

minus1

n

=

26





mij fi middot Dt0

t0+Te j 27




12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1




t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29







t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30



Dpminus1 = Qp Rp 31






Dpminus1 = Qp Rp 33




Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0


t0+TRp

minus11 j 36


M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37






M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38


minus1 i








12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935





k Uk M rarr Rn 20


0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

40

35

30

25

20

15

10

05

00

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

45

40

35

30

25

20

15

10

05

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

50

45

40

35

30

25

20

15

10

FTLE

0 05 1 15 20

02

04

06

08

1

0 05 1 15 20

02

04

06

08

1

55

50

45

40

35

30

25

20

15

FTLE




12817312576 On Wed 20 Nov 2013 195935




U1 = xyz R3x2 + y2 + z2 = 1 and z 1413


1213

21














pt0 1

TlnDt0

t0+T

=1

Tln max

uTpMu0

Dt0

t0+TuTpM

uTpM 23


spaces TpM and TpM



p Up M rarr Rn 24





2 n




rank The n vectors

e1 Dp


1

=

e2 Dpminus101 0 = p

minus1

2

=

]

en Dpminus100 1 = p

minus1

n

=

26





mij fi middot Dt0

t0+Te j 27




12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1




t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29







t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30



Dpminus1 = Qp Rp 31






Dpminus1 = Qp Rp 33




Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0


t0+TRp

minus11 j 36


M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37






M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38


minus1 i








12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935




U1 = xyz R3x2 + y2 + z2 = 1 and z 1413


1213

21














pt0 1

TlnDt0

t0+T

=1

Tln max

uTpMu0

Dt0

t0+TuTpM

uTpM 23


spaces TpM and TpM



p Up M rarr Rn 24





2 n




rank The n vectors

e1 Dp


1

=

e2 Dpminus101 0 = p

minus1

2

=

]

en Dpminus100 1 = p

minus1

n

=

26





mij fi middot Dt0

t0+Te j 27




12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1




t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29







t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30



Dpminus1 = Qp Rp 31






Dpminus1 = Qp Rp 33




Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0


t0+TRp

minus11 j 36


M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37






M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38


minus1 i








12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935

x0t0 =1

Tlnmax

UR0n

MUU

=1

2Tlnmax

UR0n

UMMU

UU =

1




t0

t0+T Vp rarr Vp pt0

t0+Tpminus1 29







t0+T=pminus1

t0t0+T p

hence

Dt0

t0+T = Dpminus1

Dt0

t0+T

Dpp 30



Dpminus1 = Qp Rp 31






Dpminus1 = Qp Rp 33




Dt0

t0+Te j = Dpminus1

Dt0

t0+TRp

minus11 j 35

and

mij = fi middot Dt0


t0+TRp

minus11 j 36


M = RpDt0

t0+TRpminus1 = Rp

1

1

1

2macr

2

n

2

1

2

2macr

2

n

] ] ]

n

1

n

2

macr

n

n

Rpminus1

37






M =1

1

11

minus1 11

22

minus1macr 1

2

nn

minus1

22

11

minus1 22

22

minus1macr 2

2

nn

minus1

] ] ]

n

n

11

minus1 n

n

22

minus1macr n

n

nn

minus1 38


minus1 i








12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935

minus1 0413 minus13

213

2 rarr R3

12 cos1

2sin

1

22

13 39



2 40



e1 = Dminus110 =1

2minus sin

1

2cos

1

20 41

and

e2 = Dminus101 = 001

13 42


2 and 2=2

= e2= 113


Dt0

t0+T =1

1

1

2

2

1

2

2

43


M =1

1

2

1

2

2

2

1

2

2

44





minus1 0313 minus13

213

2 rarr R3

12 1 +2

13cos

1

3cos

21

3

1 +2

13cos

1

3sin

21

3

2

13sin

1

3

45


1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

46

and

1 = e112 =2

2

9132 +4

91 +

2

13cos

1

32

47


M =2

2

9132 +4

91 +

2

13cos

1

32

22

9132 +4

91 +

2

13cos

1

32

1

1 2

2

9132 +4

91 +

2

13cos

1

32 1

213

1

13

1

22

9132 +4

91 +

2

13cos

1

32

2

12

2

48



12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935






















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935






k=1

N



u1 = arg minuj

k=1

N









u2 = arg minuj

2

k=1

N

uk2 minus uk

2 middot u j2u j

22 53




Initialization


u j

u j

Loop for i=1 n

ui = arg minuj

i

k=1

N

uki minus uk

i middot u jiu j

i2

forall j = 1 macr N

pi

p1 p2pj

v1

v2

vj

pprimei

pprime1

pprime2

pprimej

vprime1

vprime2

vprimej

M

TpiM

TpprimeiM


M toTpi




12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935

u ji+1 = 0 if u j


u ji minus u j

i middot uiui

u ji minus u j




= k=1

N

ukn+12 54






X =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 56



M Defining


57


Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vin

1 A2i

8Vin

2macr

ANi

8Vin

N 58


pit0 =1

TlnYXXXminus1

=1


59








j 2j m



Y =A1i

8Vi1

1 A2i

8Vi1

2macr

ANi

8Vi1

N

A1i

8Vi2

1 A2i

8Vi2

2macr

ANi

8Vi2

N

] ] ]

A1i

8Vim

1 A2i

8Vim

2macr

ANi

8Vim

N 60






12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935





pit0 =1


61



Rm = TpiM Npi

M 62





Dt0

t0+T Omminusn 63



RYR = Y

hW 64

Let us define

M = YXXXminus1 65

and


hWXXXminus1 R 66



67









hPM h2PP 68



detA B





+ Oh2 70







12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935




y



Dminus1 =1 0

0 1



= QR 72



RxyMRminus1x0y0 73









12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935













12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935








12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935









V CONCLUSIONS








12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935




















































12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935





















12817312576 On Wed 20 Nov 2013 195935

the computation of finite-time lyapunov exponents on ... · the ftle to non-euclidean manifolds and...

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