A Variational Theory of Hyperbolic Lagrangian Coherent Structures George Haller ∗ Department of Mechanical Engineering, Mcgill University, 817 Sherbrooke Ave

Physica D 240 (2011) 574–598

Contents lists available at ScienceDirect

Physica D

journal homepage: www.elsevier.com/locate/physd

A variational theory of hyperbolic Lagrangian Coherent Structures George Haller ∗ Department of Mechanical Engineering, McGill University, 817 Sherbrooke Ave. West, Montreal, Quebec H3A 2K6, Canada Department of Mathematics and Statistics, McGill University, 817 Sherbrooke Ave. West, Montreal, Quebec H3A 2K6, Canada article info a b s t r a c t

Article history: We develop a mathematical theory that clarifies the relationship between observable Lagrangian Received 6 July 2010 Coherent Structures (LCSs) and invariants of the Cauchy–Green strain tensor field. Motivated by physical Received in revised form observations of trajectory patterns, we define hyperbolic LCSs as material surfaces (i.e., codimension- 12 November 2010 one invariant manifolds in the extended phase space) that extremize an appropriate finite-time normal Accepted 17 November 2010 repulsion or attraction measure over all nearby material surfaces. We also define weak LCSs (WLCSs) Available online 10 December 2010 Communicated by V. Rom-Kedar as stationary solutions of the above variational problem. Solving these variational problems, we obtain computable sufficient and necessary criteria for WLCSs and LCSs that link them rigorously to the Keywords: Cauchy–Green strain tensor field. We also prove a condition for the robustness of an LCS under Lagrangian Coherent Structures perturbations such as numerical errors or data imperfection. On several examples, we show how Invariant manifolds these results resolve earlier inconsistencies in the theory of LCS. Finally, we introduce the notion of a Mixing Constrained LCS (CLCS) that extremizes normal repulsion or attraction under constraints. This construct allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node. © 2010 Elsevier B.V. All rights reserved.

1. Introduction (1) An LCS should be a material surface, i.e., a codimension- one invariant surface in the extended phase space of a dynamical 1.1. Background system. This is because (a) an LCS must have sufficiently high dimension to have visible impact and act as a transport barrier and This paper is concerned with the development of a self- (b) an LCS must move with the flow to act as an observable core of consistent theory of coherent trajectory patterns in dynamical evolving Lagrangian patterns. systems defined over a finite time-interval. Following Haller (2) An LCS should exhibit locally the strongest attraction, repulsion and Yuan [1], we use the term Lagrangian Coherent Structures or shearing in the flow. This is essential to distinguish the LCS from (or LCSs, for short) to describe the core surfaces around which such all nearby material surfaces that will have the same stability type, trajectory patterns form. as implied by the continuous dependence of the flow on initial As an example, Fig. 1 shows the formation of passive tracer conditions over finite times. patterns in a quasi-geostrophic turbulence simulation described Based on (1)–(2), a purely physical definition of an observable in [1]. We seek to locate the dynamically evolving LCSs that form LCS can be given as follows (cf. [1]): the skeleton of these patterns. Beyond offering conceptual help in interpreting and forecasting complex time-dependent data sets, Definition 1 (Physical Definition of Hyperbolic LCS). A hyperbolic LCSs are natural targets through which to control ensembles of LCS over a finite time-interval I = [α, β] is a locally strongest re- trajectories. pelling or attracting material surface over I (cf. Fig. 2). As proposed in [1], repelling LCSs are the core structures generating stretching, attracting LCSs act as centerpieces of folding, This definition does not favor any particular diagnostic quantity, and shear LCS delineate swirling and jet-type tracer patterns. In such as finite-time or finite-size Lyapunov exponents, relative or order to act as organizing centers for Lagrangian patterns, LCSs are absolute dispersion, vorticity, strain, measures of hyperbolicity, expected to have two key properties: etc. Instead, it describes the main physical property of LCSs that enables us to observe them as cores of Lagrangian patterns. Ideally, a mathematical definition of an LCS should capture the ∗ essence of the above physical definition, and lead to computable Corresponding address: Department of Mechanical Engineering, Macdonald Bldg. Rm. 270 B, McGill University, 817 Sherbrooke Ave. West, Montreal, QC H3A mathematical criteria for the LCS. As we shall see below, however, 2K6, Canada. Tel.: +1 (514) 398 6296. such a mathematical definition and the corresponding criteria have E-mail address: [email protected]. been missing in the literature.

a 7 b 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -101234567 -101234567 c 7 d 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 -1 -1 -101234567 -101234567 e 7 f 7 6 6 5 5 4 4 Fig. 2. The geometry of Definition 1: an attracting LCS is locally the strongest

3 3 attracting material surface over the time interval [t0, t1] among all nearby 2 2 (i.e., sufficiently C 1-close) material surfaces. A sphere of nearby initial conditions 1 1 released at time t0 will then spread out in a fashion that the LCS will serve as its 0 0 centerpiece at time t . A similar sketch is shown for a repelling LCS. -1 -1 1 -101234567 -101234567 n where x0 ∈ R is the phase space variable, λn(x0) denotes t +T t +T Fig. 1. An initially square set of fluid trajectories evolve into a complex material the largest eigenvalue of C(x ) = [∇F 0 (x )]∗∇F 0 (x ), the pattern in a two-dimensional turbulence simulation. The snapshots (a)–(f) are taken 0 t0 0 t0 0 t +T at different time instances. For more information, see [1]. Cauchy–Green strain tensor computed from the flow map F 0 t0 between time t0 and t0 +T ; ξn(x0) is the eigenvector corresponding In particular, while LCSs have de facto become identified with to the largest eigenvalue of C(x0). Condition (1) turns out to be local maximizing curves (ridges) of the Finite-Time Lyapunov equivalent to Exponent (FTLE) field (see, e.g., [2–5], and the recent review −1 by Peacock and Dabiri [6]), simple counterexamples reveal ∇C (x0)[ξn(x0), ξn(x0), ξn(x0)] = 0, (2) conceptual problems with such an identification (see Section 2.3). − where ∇C 1(x ) is a three-tensor, the gradient of C−1, evaluated Computational results on geophysical data sets also show that 0 on the vector ξ . several FTLE ridges in real-life data sets do not repel or attract n Theorem 7 also states that a material surface satisfying (1) at nearby trajectories (see, e.g., [7,8]). time t must be orthogonal to the x vector field in order to be a The present paper addresses this theoretical gap by providing 0 ξn( 0) a mathematical version of the above physical LCS definition, and repelling LCS. This condition is non-restrictive for a long-lived LCS by deriving exact computable criteria for LCSs in n-dimensional because all√ repelling material surfaces turn out to align at a rate −bT ≈ dynamical systems defined over a finite time-interval. Our focus e λn−1/λn with directions normal to ξn(x0) (Theorem 4 is hyperbolic (repelling or attracting) LCSs; a similar treatment of and formula (30)). For small T , however, the orthogonality condi- shear LCSs will appear elsewhere. tion may not hold on any material surface, which underscores a fundamental limitation to identifying cores of Lagrangian patterns from short-term observations. 1.2. Summary of the main results Finally, Theorem 7 requires a matrix L(x0, t0, T ), defined in Our analysis is based on a new notion of finite-time hyperbol- (31), to be positive definite on the zero set (1) in order for the icity of material surfaces. This hyperbolicity concept is expressed underlying WLCS to be an LCS. A necessary condition for the through the normal repulsion rate and the normal repulsion ra- positive definiteness of L is tio that are finite-time analogues of the Lyapunov-type numbers − ∇2C 1(x )[ξ (x ), ξ (x ), ξ (x ), ξ (x )] > 0, (3) introduced by Fenichel [9] for normally hyperbolic invariant man- 0 n 0 n 0 n 0 n 0 2 −1 −1 ifolds. Unlike Fenichel’s numbers, however, the normal repulsion with the four-tensor ∇ C (x0), the second derivative of C , rate and ratio are smooth quantities that are computable for a given evaluated on the strongest strain eigenvector field ξn(x0) (Propo- material surface and flow. sition 8). We employ a variational approach to locate LCSs as material The LCSs we identify are robust under perturbations as long as surfaces that pointwise extremize the normal repulsion rate 1 2 among all C -close material surfaces. We also introduce the notion ⟨ξn, ∇ λnξn⟩ + ⟨∇λn, ∇ξnξn⟩ ̸= 0 (4) of a Weak LCS (WLCS), which is a stationary surface – but not holds along them (Theorem 11). The admissible perturbations to necessarily an extremum surface – for our variational problem. As the underlying dynamical system need not be pointwise small as we show, the use of WLCS resolves notable counterexamples to the long as they translate to small perturbations to the flow map. For identification of LCSs with FTLE ridges. example, large amplitude but short-lived localized perturbations Solving the variational problem leads to a necessary and to the vector field governing the dynamics are admissible. sufficient LCS criterion that involves invariants of the inverse Cauchy–Green strain tensor (Theorem 7). Specifically, WLCSs at These general results allow us to examine the relevance of FTLE for LCS detection in rigorous terms. FTLE ridges turn out to mark time t0 must be hypersurfaces in the phase space satisfying the equation the presence of LCSs under four conditions. First, along FTLE ridges, λ must be larger than one and of multiplicity one. Second, FTLE   n ∇λn(x0), ξn(x0) = 0, (1) ridges have to be normal to the ξn(x0) field. Third, along FTLE 576 G. Haller / Physica D 240 (2011) 574–598 ridges, the gradient of ξn(x0) in directions parallel to ξn(x0) must be small enough. Fourth, the FTLE ridge must be steep enough (cf. Proposition 14). For LCSs marked by such FTLE ridges, the robustness criterion (4) takes the more specific form 2 ⟨ξn, ∇ λnξn⟩ + ⟨∇λn, ∇ξnξn⟩ < 0, as we show in Proposition 15. This again implies that FTLE ridges 2 that are steep (i.e., ⟨ξn, ∇ λnξn⟩ ≪ 0), nearly flat (i.e., |∇λn| ≈ 0), and lie in regions of moderately nonlinear strain (i.e., |∇ξnξn| ≤ 1) are the most robust under perturbations. The present approach also allows for the treatment of a Constrained LCS (or CLCS), which is a solution of the above maximum repulsion problem under constraints. In this paper, we explore two such constraints: (1) The constraint that the LCS be Fig. 3. Material surface M(t) generated in the extended phase space by the flow an invariant manifold in phase space, not just in extended phase map from a surface M(t0) of initial conditions. space. (2) The constraint that the LCS be a level surface of a first integral. the dot denoting differentiation with respect to the time variable The CLCS approach enables us to identify unique attracting and t. This paper is primarily motivated by applications to fluid repelling LCSs in linear flows that have so far defied LCS extraction mechanics, in which case we have n = 2 (planar flows) or n = 3 techniques. CLCSs also turn out to be useful in extracting unique (three-dimensional flows). The finite length of the time interval weak unstable manifolds from finite-time data sets, even though [α, β] reflects temporal limitations to the available experimental such manifolds are nonunique in the classic theory of invariant or numerical flow data. manifolds. At time t, a trajectory of (5) system is denoted by x(t, t0, x0), We believe that these results establish the first rigorous starting from the initial condition x at time t . The flow map Ft (x ) 0 0 t0 0 link between a Lagrangian diagnostic tool, the Cauchy–Green maps the initial position x0 of the trajectory into its position at stains tensor, and invariant coherent structures in a finite-time time t: dynamical system. Notably, however, the recent work of Froyland Ft : U → U, et al. [10] provides a rigorous link between properties of the t0 Perron–Frobenius operator and almost invariant coherent sets of x0 → x(t, t0, x0). non-autonomous dynamical systems defined over infinite times. Classic results from the theory of differential equations guarantee that the flow map is as many times differentiable in x0 as is v(x, t) 1.3. Organization of the paper in x (see [11]). We recall that the Lagrangian approach to the analysis of the In Section 2, we fix our notation and review discrepancies dynamical system (5) focuses on the trajectories of the system. By between observable LCSs and their commonly assumed FTLE contrast, the Eulerian approach to analyzing (5) is concerned with signature. the properties of the underlying vector field v(x, t). Central to the In Section 3, we develop the notion of finite-time hyperbolicity Lagrangian view is the overall qualitative behavior of ensembles for material surfaces and show that normals of finite-time of trajectories x(t, t0, x0). The most important such ensembles are hyperbolic material surfaces align exponentially fast with the codimension-one sets of trajectories, because they locally divide largest strain eigenvector of the Cauchy–Green strain tensor. In the extended phase space into two regions between which no Section 4, we define Weak LCSs and LCSs as repelling material transport is possible. surfaces that are pointwise stationary surfaces and extrema, Specifically, a material surface M(t) is the t = const. slice of an respectively, of a variational principle for the repulsion rate. We invariant manifold M of system (5) in the extended phase space then solve this variational problem and obtain sufficient and U × [α, β], generated by the advection of an n − 1-dimensional necessary conditions for WLCSs and LCSs. surface of initial conditions M(t ) by the flow map Ft : 0 t0 Section 5 discusses the robustness of LCSs obtained from our M(t) = Ft (M(t )), dim M(t ) = n − 1, theory with respect to perturbations to the underlying dynamical t0 0 0 system. Applying our general results, we examine the relevance of M(t0) ⊂ U, α ≤ t0 ≤ t ≤ β, (6) FTLE ridges in LCS detection in Section 6. In Section 7, we discuss implications for the numerical detection of LCSs. with the corresponding geometry sketched in Fig. 3. Since Ft is a diffeomorphism, the material surface M(t) is as In Section 8, we review the counterexamples of Section 2.3 t0 and show how they are resolved by our main result, Theorem 7. smooth as the initial surface M(t0), and has the same dimension. Section 9 discusses constrained LCS problems with applications to linear flows and weak unstable manifolds. We present our 2.2. Finite-Time Lyapunov Exponents (FTLE) conclusions and directions for future work in Section 10. Since Definition 1 involves repulsion and attraction, it is 2. LCS and FTLE plausible to explore how the maximal finite-time Lyapunov exponent, a local measure of the largest particle separation rate in 2.1. Set-up and notation system (5), could be used to characterize LCSs.

We recall that an infinitesimal perturbation ξ0 to the trajectory Consider a dynamical system of the form x(t, t , x ) at time t evolves into the vector ∇Ft (x )ξ at 0 0 0 t0 0 0 time t under the linearized flow. The largest singular value of x˙ = v x t x ∈ U ⊂ n t ∈ [ ] (5) ( , ), R , α, β , the deformation gradient ∇Ft (x ), therefore, gives the largest t0 0 with a smooth vector field v(x, t) defined on the n-dimensional possible infinitesimal stretching along x(t, t0, x0) over the time bounded, open domain U over a time interval [α, β], and with interval [t0, t]. G. Haller / Physica D 240 (2011) 574–598 577

We introduce the Cauchy–Green strain tensor

t +T t +T ∗ t +T C 0 = [∇F 0 ] ∇F 0 (7) t0 t0 t0 , with the star denoting the transpose. We will use the notation

ξ1(x0, t0, T ), . . . , ξn(x0, t0, T ) for an orthonormal eigenbasis of t +T C 0 x , with the corresponding eigenvalues t0 ( 0)

0 < λ1(x0, t0, T ) ≤ · · · ≤ λn−1(x0, t0, T ) ≤ λn(x0, t0, T ) (8) that satisfy

t +T C 0 x ξ x t T = x t T ξ x t T i = 1 n Fig. 4. A nonlinear strain flow with a repelling LCS that is not a ridge of the forward t0 ( 0) i( 0, 0, ) λi( 0, 0, ) i( 0, 0, ), ,..., . FTLE field. The Finite-Time Lyapunov Exponent (FTLE) is defined as Appendix B, the forward FTLE field is simply the constant field   t0+T 1 t0+T 1 Λ (x ) = log ∇F (x ) = log λ (x , t , T ), (9) t +T t0 0  t0 0  n 0 0 Λ 0 x ≡ 1 (11) T 2T t0 ( 0) , t +T where ‖∇F 0 x ‖ denotes the operator norm of the deforma- which has no ridges for any T > 0. t0 ( 0) t +T The even simpler linear, area-preserving strain flow tion gradient ∇F 0 . This norm is equal to the square root of t0 λn(x0, t0, T ), the maximum eigenvalue of the Cauchy–Green strain x˙ = x, t +T tensor. When T 0, we will refer to Λ 0 x as the forward FTLE; > t0 ( 0) y˙ = −y, (12) for T < 0, we refer to the same quantity as backward FTLE. has a phase portrait topologically equivalent to Fig. 4, but neither Formula (9) shows that for finite T , the FTLE and λn are directly its repelling LCS nor its attracting LCS can be detected as ridges of related. In what follows, we shall use λ in our analysis. n the forward or backward FTLE field. Specifically, both the forward t ±T FTLE and the backward FTLE fields are constant (Λ 0 x ≡ 1) t0 ( 0) 2.3. Relationship between FTLE and LCS: views and counterexamples and hence admit no ridges. An area-preserving version of (10) is given by the system In [2,3], we suggested that at time t0, a repelling LCS over 2 [t0, t] should appear as a local maximizing curve, or ridge, of the x˙ = x(1 + 3y ), (13) Finite-Time Lyapunov Exponent (FTLE) field computed over initial y˙ = −y − y3. conditions at t0. Similarly, an attracting LCS [t0, t] should be a ridge of the backward-time FTLE field. Because of the coupling between the two equations, an analytic LCSs indeed appear to create ridges in the FTLE field in several calculation of FTLE is more cumbersome. Still, a numerical applications (see [6] for a recent review). In [3], however, we computation of the forward FTLE field shows that system (13) = presented an example where an FTLE ridge indicates maxima of admits no FTLE ridges, even though the x 0 axis is an observable shear, as opposed to a repelling LCS. repelling LCS. Shadden et al. [4] (see also Lekien et al. [5]) propose an 2.3.2. Example 2: LCS that is a trough of the FTLE field alternative view by defining a repelling LCS at t0 as a ridge of the FTLE field computed over an interval [t0, t]. As they note, under Consider the saddle flow this definition, LCSs are no longer guaranteed to be Lagrangian. x˙ = x, Indeed, when the vector field v(x, t) has general time dependence 3 t0+T ˙ over a time interval [t t ], ridges of the scalar field Λ will y = −y − y , (14) 0, 1 t0 t +T t typically not evolve into ridges of Λ 1 under the flow map F 1 . To studied in Example 1 above. From Fig. 4, it is apparent that the t1 t0 quantify the degree of non-invariance of FTLE ridges, [4,5] present y = 0 axis is an attracting Lagrangian structure. Yet, as we show in an elegant expression for the leading-order material flux through Appendix B, the y = 0 axis is a trough of the backward-time FTLE FTLE ridges. field. Below we use four simple examples to demonstrate that Again, an area-preserving version of the above example is given observable LCSs need not be FTLE ridges, and FTLE ridges need not by (13). A numerical computation of the backward FTLE field mark observable LCSs. We show that this mismatch also arises in reveals the same type of trough along the y = 0 axis as the one area-preserving versions of our examples, and hence is relevant found analytically in system (14) (see Fig. 5). even for incompressible fluid flows. Finally, in Example 4, we highlight limitations to the validity of the above-mentioned ridge 2.3.3. Example 3: FTLE ridge that is not a repelling LCS flux formula (cf. Example 11 and Appendix C). Consider the two-dimensional area-preserving dynamical system 2.3.1. Example 1: LCSs in systems with no FTLE ridges x˙ = 2 + tanh y, (15) With the notation x = (x, y), consider the two-dimensional y˙ = 0, nonlinear saddle flow whose phase portrait is just the parallel shear flow sketched in x˙ = x, Fig. 6. As we show in more detail in Appendix B, the x axis of y˙ = −y − y3, (10) this example is a ridge of the forward, as well as the backward, FTLE field. As Fig. 6 shows, however, all trajectories preserve their for which the y axis is a readily observable repelling LCS, i.e., distance from the x axis for all times, thus the x axis is neither a the stable manifold shown in Fig. 4. By contrast, as we show in repelling nor an attracting LCS. 578 G. Haller / Physica D 240 (2011) 574–598

Fig. 5. The backward FTLE field admits a trough along the x axis, which is an Fig. 8. FTLE ridge at x = 0 for the area-preserving system (18) for t − t0 = 1.78. observable attracting LCS of example (13). The integration time was selected as This ridge remains at the same location for all t0, but no such fixed LCS exists in the t − t0 = −0.7450. flow.

Note, however, that Mˆ (t) is far from being Lagrangian: the flow of (16) crosses it orthogonally with speed x˙ = 1. Thus the area flux per unit length through Mˆ (t) is equal to one. This is at odds with the flux formula proposed in [4], which predicts an order ˆ O(1/|t − t0|) flux through M(t) as t increases (cf. Appendix B). An area-preserving version of Example 4 is given by

x˙ = 1 + tanh2 x, (18) 2 tanh x y˙ = − y. cosh2 x A numerical computation of the FTLE field for (18) again reveals the development of a ridge along the y axis for large enough integration Fig. 6. A parallel shear flow with an FTLE ridge that is not a repelling LCS, but a times and for any t0, even though no fixed LCS of the form (17) shear LCS, as shown by the deformation of a set of initial conditions. exists (cf. Fig. 8). The unit flux through Mˆ (t) is again erroneously predicted by formula (107) as O(1/|t − t0|) for growing t, as opposed to the correct value ϕ((0, y0), t) ≡ 1.

2.4. Summary

The above examples highlight the following points: 1. Observable LCS are not necessarily ridges of the FTLE field. Specifically, Fig. 9a, the usual motivating picture used in the literature for equating FTLE ridges with LCSs is not universally applicable, as illustrated by the examples in Fig. 9b and c. 2. Ridges of the FTLE field are not necessarily observable LCSs. Specifically, FTLE ridges may be indicators of large shear (see Example 3), or indicators of locally large stretching without Fig. 7. A flow with a forward FTLE ridge that has no LCS in its proximity. any underlying coherent structure (see Example 11). 3. Ridges of the FTLE field may in fact be far from any Lagrangian 2.3.4. Example 4: FTLE ridge that is far from any LCS structure (see Examples 4 and 11). Consider the dynamical system 4. The flux formula proposed in [4] for FTLE ridges is not generally applicable: its higher-order neglected terms may be as large or 2 x˙ = 1 + tanh x, (16) larger than its explicit leading-order terms, even as T → ∞ y˙ = −y, (cf. Examples 4 and 11, and Appendix C). whose phase portrait is shown in Fig. 7, along with the deformation In the following, we describe a mathematical theory that of a set of initial conditions under the flow. As we show in resolves these inconsistencies and gives a unified view on LCSs. Appendix B, the FTLE field computed for this flow admits a ridge along the y axis for any choice of t and large enough t > t . 0 0 3. Hyperbolicity of material surfaces Defining an LCS as a time-evolving surface which, at any time t, + coincides with a ridge of the FTLE field Λt T , would render the y t In our setting, LCSs are only assumed to exist over a finite axis a repelling LCS of the form time-interval [α, β]. To describe their attracting and repelling    ˆ 0 2 properties, therefore, we cannot rely on classic notions of M(t) ≡ ∈ R : y0 ∈ R . (17) y0 hyperbolicity that are inherently asymptotic in time (see, e.g., [9]). G. Haller / Physica D 240 (2011) 574–598 579

∈ space Nxt M(t), and hence a unit vector n0(x0, t0) Nx0 M(t0) will typically not be mapped into the normal space Nxt M(t) by the linearized flow. Rather, its image, ∇Ft (x )n , will be a vector of gen- t0 0 0 eral orientation, as shown in Fig. 10. ∈ Let nt Nxt M(t) denote a smoothly varying family of unit normal vectors along xt . To assess the growth of perturbations in the direction normal to M(t), we need to consider the projection of ∇Ft (x )n onto n , given by the repulsion rate t0 0 0 t ρt (x , n ) = ⟨n , ∇Ft (x )n ⟩. (19) t0 0 0 t t0 0 0 a If this projection is larger than one, the normal component of an infinitesimal normal perturbation to M(t0) grows by the end of the time interval [t , t]. Similarly, ρt (x , n ) < 1 indicates that the 0 t0 0 0 normal component of normal perturbations to M(t0) decreases by time t. We also introduce the repulsion ratio, a measure of the ratio between normal and tangential growth rates along M(t): t ⟨nt , ∇F (x0)n0⟩ t = t0 νt (x0, n0) min . 0 |e |=1 t 0 |∇Ft (x0)e0| e ∈T t 0 0 x0 M( 0) If νt (x , n ) > 1 holds, then infinitesimal normal growth along t0 0 0 bcM(t) dominates the largest tangential growth along M(t) over the time interval [t0, t]. In this case, the material surface M(t) is indeed observed as the locally dominant repelling structure. Fig. 9. (a) Heuristic motivation for the assertion that FTLE ridges mark repelling LCSs. It is at odds with the following two examples: (b) Linear saddle flow without The following proposition gives computable expressions for = ρt (x , n ) and νt (x , n ) in terms of the Cauchy–Green strain an FTLE ridge that has an observable repelling LCS at x 0. (c) Nonlinear saddle t0 0 0 t0 0 0 flow with an FTLE trough and an observable repelling LCS at x = 0. tensor. Proposition 2. The quantities ρt and νt can be computed and The concept of uniform finite-time hyperbolicity for individual t0 t0 estimated as follows: trajectories was apparently first introduced in [12], then elaborated on in [13,1,22,2,3,14]. Extensions of these results to higher- (i) dimensional and non-volume-preserving systems are given by Duc t 1 ρ (x0, n0) = , and Siegmund [15], Berger et al. [16,17] and Berger [18]. t0  ⟨n [Ct x ]−1n ⟩ Haller and Yuan [12] extended the concept of uniform finite- 0, t0 ( 0) 0 time hyperbolicity to material surfaces. They require the normal t ρ (x0, n0) component of normal perturbations to a material surface to grow t t0 ν (x0, n0) = min . t0 | |=  at a uniform rate over all subintervals of a finite time-interval e0 1 t e ∈T M(t ) ⟨e0, Ct (x0)e0⟩ [α, β]. Numerical simulations in the same paper and experimental 0 x0 0 0 results in [19] show, however, that this notion of finite-time (ii) hyperbolicity captures only a small subset of observed Lagrangian  t +T  λ (x , t , T ) ≤ ρ 0 (x , n ) ≤ λ (x , t , T ), coherent structures. 1 0 0 t0 0 0 n 0 0 In our discussion below, we relax this requirement of uniform   λ1(x0, t0, T ) t +T λn(x0, t0, T ) hyperbolicity to a weaker notion of finite-time hyperbolicity ≤ 0 x n ≤ νt0 ( 0, 0) . that ultimately enables us to locate LCSs more efficiently. This λn(x0, t0, T ) λ1(x0, t0, T ) notion of finite-time hyperbolicity resembles that used by Lin and Proof. Note that for any e ∈ T M(t ), we have ⟨e , n ⟩ = 0, and Young [20] in a different context. 0 x0 0 0 0 hence ∗ t ∗ 0 = ⟨e , n ⟩ = ⟨e ,(∇Ft ) (∇F 0 ) n ⟩ 3.1. Repulsion rate and repulsion ratio 0 0 0 t0 t 0

t t0 ∗ n = ⟨∇F e0,(∇F ) n0⟩, (20) Consider a compact material surfaceM(t) ⊂ R , as defined by t0 t formula (6). We want to express the physically observed repelling t t t ∗ 0 ∗ = 0 t ∗ where we have used the identity (∇Ft ) (∇Ft ) (∇Ft ∇Ft ) or attracting nature of this surface over a time interval [t0, t] in 0 0 = I. Formula (20) along with ∇Ft e ∈ T M(t) implies that mathematical terms. t0 0 xt t0 ∗ t0 ∗ To this end, at an arbitrary point x0 ∈ M(t0), we consider the nt = [(∇Ft ) n0]/|(∇Ft ) n0|. As a result, we have (n − 1)-dimensional tangent space Tx M(t0) of M(t0), as well as 0  t0 ∗  the one-dimensional normal space N M(t ), as shown in Fig. 10. t t (∇Ft ) n0 t x0 0 = ⟨ ∇ ⟩ = ∇ ρt (x0, n0) nt , Ft (x0)n0 t , Ft (x0)n0 The tangent space Tx M(t0) is carried forward along the trajectory 0 0 0 ∗ 0 0 |(∇Ft ) n0| def. = ; = t t t xt x(t, t0 x0) Ft (x0) by the linearized flow map ∇Ft (x0) 0 t 0 0 ⟨n0, ∇F (xt )∇F (x0)n0⟩ into the tangent space = t t0  t t0 t0 ∗ T M(t) = ∇F (x )T M(t ) ⟨n0,(∇Ft )(∇Ft ) n0⟩ xt t0 0 x0 0 1 by the invariance of M(t) under the flow map. In particular, a unit =  , tangent vector e0 ∈ Tx M(t0) is mapped by the linearized flow t −1 0 ⟨n0, [C (x0)] n0⟩ into ∇Ft (x )e , a tangent vector to M(t) at the point x . By con- t0 t0 0 0 t trast, ∇Ft (x )N M(t ) is generally not contained in the normal as claimed. t0 0 x0 0 580 G. Haller / Physica D 240 (2011) 574–598

Fig. 10. Geometry of the linearized flow map along the material surface M(t).

We also have Theorem 4 (Alignment Property of Hyperbolic Material Surfaces). t Assume that for all T ∈ [T−, T+], we have a normally repelling ⟨nt , ∇F (x0)n0⟩ t = t0 [ + ] ⊂ νt (x0, n0) min material surface M(t) over t0, t0 T I in the sense 0 |e |=1 t 0 |∇Ft (x0)e0| e ∈T t 0 of Definition 3, with the constants a, b > 0 selected uniformly in T . 0 x0 M( 0) t Then the eigenspace spanned by the first n − 1 eigenvalues of the ρ (x0) t +T t0 Cauchy–Green strain tensor C 0 x converges exponentially fast in = min , t0 ( 0) | |=  e0 1 t T to the tangent space of t at the point x . Specifically, we have e ∈T t ⟨e0, C (x0)e0⟩ M( 0) 0 0 x0 M( 0) t0 dist [T M(t ), span{ξ (x , t , T ), . . . , ξ (x , t , T )}] completing the proof of statement (i) of the Proposition. The √x0 0 1 0 0 n−1 0 0 ≤ − −bT estimates in statement (ii) follow directly from the definition of n 1e , √ t t − ρ (x0, n0) and ν (x0, n0). bT t0 t0 | sin α(n0(x0, t0, T ), ξn(x0, t0, T ))| ≤ n − 1e ,

for all T ∈ [T−, T+]. A similar statement holds for normally attracting 3.2. Finite-time hyperbolic material surfaces and their alignment material surfaces in backward time. property Proof. Consider a point x0 ∈ M(t0) and let e1(x0, t0, T ), . . . , en−1 (x , t , T ) be an orthonormal basis in the tangent space T (t ). We are now in a position to define normal attraction and 0 0 x0 M 0 repulsion for a material surface over a finite time-interval. For any basis vector ei(x0, t0, T ) and for the unit normal vector ∈ n0 Nx0 M(t0), we have the representation

Definition 3 (Finite-Time Hyperbolic Material Surface). A material n ⊂ [ + ] ⊂ − surface M(t) U is normally repelling over t0, t0 T I, if ei(x0, t0, T ) = aij(x0, t0, T )ξj(x0, t0, T ), there exist constants a, b > 0 such that for all points x0 ∈ M(t0) j=1 (22) and unit normals n ∈ N M(t ), we have n 0 x0 0 − n0 = bj(x0, t0, T )ξj(x0, t0, T ). t0+T aT x n ≥ e (21) j=1 ρt0 ( 0, 0) , t +T 0 x n ≥ ebT Note that ξ , e and n are all unit vectors, therefore we have νt0 ( 0, 0) . j i 0

Similarly, we call M(t) normally attracting over [t0, t0 + T ] ⊂ I if  n   n  − 2 − 2 it is normally repelling over [t0, t0 + T ] in backward time. Finally,  a  = 1,  b  = 1. (23)  ij  j  we call M(t) hyperbolic over [t0, t0 + T ] if it is normally repelling  j=1   j=1  or normally attracting over [t0, t0 + T ]. The repelling property of M(t) over [t0, t0 + T ] (as assumed in The first condition in (21) requires all small normal perturba- the statement of the Theorem uniformly for T ∈ [T−, T+]) implies = + tions to M(t0) to have strictly grown by time t t0 T ; the second  −1  1  t0+T  −2aT condition requires that by time t = t0 + T , any growth along M(t) = n , C (x ) n ≤ e , t +T 0 t0 0 0 [ 0 x ]2 is strictly smaller than growth normal to M(t). Since Definition 3 ρt0 ( 0) is only concerned with growth between the times t0 and t0 +T , the 1  t +T −  t0+T 0 1 exponential lower bounds in (21) will always exist as long as = n0, [Ct (x0)] n0 ρt0 2 0  t0+T  t0+T ν (x ) and are uniformly bounded from below by a constant larger t0 0 νt0  t +T  − than one. × max e C 0 x e ≤ e 2bT (24) 0, t0 ( 0) 0 . We now derive a local geometric relationship between a |e0|=1 e ∈T M(t ) normally repelling LCS and the largest eigenvector of the Cauchy– 0 x0 0 Green strain tensor. To state our result, for two codimension-one Note that n planes E, F ⊂ R with respective unit normals nE and nF , we 2  −1  n introduce the distance  t +T  − bj n C 0 x n = 0, t0 ( 0) 0 ,  2 λj dist [E, F] = 1 − ⟨nE , nF ⟩ , j=1 (25)   n which is equal to | sin α(nE , nF )|, with α(nE , nF ) denoting the angle t0+T − 2 ei, C (x0)ei = λja , between n and n . t0 ij E F j=1 G. Haller / Physica D 240 (2011) 574–598 581

Fig. 12. The geometry of defining an LCS as an extremum surface for the normal repulsion rate.

The geometry of Theorem 4 is shown in Fig. 11. An inspection of the proof also reveals that along a material line satisfying the assumptions of Theorem 4, we can select Fig. 11. Geometry of the linearized flow map along the material surface M(t).

1 1 λn ∈ [ ] a < log λn, b < log , (29) therefore the first inequality in (24) yields, for any T T−, T+ , 2T 2T λn−1 the relation in which case the rate of alignment can be approximated as n 2 − bj − ≤ e 2aT , (26)  −bT λn−1 j=1 λj e ≈ . (30) λn which in turn, by (8) and (23), implies Therefore, the larger the spectral gap between λn and λn−1, the n b2 n b2 faster the normal of a repelling material surface aligns with ξ . −2aT − j − j 1 n e ≥ ≥ = , As we shall see later, on the most influential repelling material j=1 λj j=1 λn λn surfaces (i.e., on LCSs), the vector field ξn(x0, t0, T ) aligns exactly or, equivalently, with the surface normals.

2aT λn ≥ e , T ∈ [T−, T+]. (27) 4. Existence of hyperbolic Lagrangian coherent structures

Next, the second inequality in (24) can be re-written as 4.1. Weak LCS and LCS     n b2 n−1 − j − 2 2 a λk + a λn Consider now material surfaces that are small and smooth λ ik in j=1 j k=1 deformations of a normally repelling material surface M(t). All −2bT ≤ e , i = 1,..., n − 1, T ∈ [T−, T+], such deformed material surfaces will be normally repelling by the continuity of the inequalities (21) in their arguments. which implies For a repelling material surface M(t) to be a repelling LCS, we [ + ] n n 2 will require M(t) to be pointwise more repelling over t0, t0 T − − bj λn − a2 = a2 b2 ≤ a2 ≤ e 2bT , than any other nearby material surface. Specifically, at any point in in j in ∈ ∈ j=1 j=1 λj x0 M(t0), perturbations to M(t0) along its normal n0 t +T N t will be required to yield 0 xˆ nˆ values that are x0 M( 0) ρt0 ( 0, 0) or, equivalently, t +T strictly smaller than 0 x n (see Fig. 12). ρt0 ( 0, 0) −bT ain ≤ e , i = 1,..., n − 1, T ∈ [T−, T+]. We start by defining a weaker notion of an LCS, a repelling Weak Lagrangian Coherent Structure: ∈ Consequently, for any tangent vector ei Tx0 M(t0), we have Definition 5 (Hyperbolic Weak LCS). Assume that M(t) ⊂ U is   −bT ei, ξ ≤ e , T ∈ [T−, T+], (28)  n  a normally repelling material surface over [t0, t0 + T ]. We call [ + ] and hence M(t) a repelling Weak LCS (WLCS) over t0, t0 T if its normal repulsion rate admits stationary values along M(t0) among all    1 dist Tx0 M(t0), span ξ1,..., ξn−1 locally C -close material surfaces. We call M(t) an attracting WLCS  over [t0, t0 +T ] if it is a repelling WLCS over [t0, t0 +T ] in backward  n−1  √ time. Finally, we call M(t) a hyperbolic WLCS over [t0, t0 + T ] if it 2 − 2 −bT = 1 − ⟨n0, ξ ⟩ =  ⟨ei, ξ ⟩ ≤ n − 1e , n n is a repelling or attracting WLCS over [t0, t0 + T ]. i=1 Specifically, at each point x0 of a WLCS, the repulsion rate field as claimed. Also, we have t +T 0 xˆ nˆ has a zero derivative in the direction of n x t .A ρt0 ( 0, 0) 0( 0, 0)  WLCS may not be a locally unique core of trajectory patterns. For | sin n | = 1 − cos2 n α( 0, ξn) α( 0, ξn) instance, nearby attracting WLCSs obtained by normal translations  may all converge to each other at the same rate, and hence 2 = 1 − ⟨n0, ξn⟩ √ may all be viewed as cores of an emerging trajectory pattern − ≤ n − 1e bT , (see, e.g., Example 5). The following stronger definition of an LCS guarantees that it is which concludes the proof of the Theorem. observed as a unique core of a coherent trajectory pattern. 582 G. Haller / Physica D 240 (2011) 574–598

Definition 6 (Hyperbolic LCS). Assume that M(t) ⊂ U is a nor- for an appropriate choice of x0(s), and for some smooth function s t : V × → . mally repelling material surface over [t0, t0 + T ]. We call M(t) a α( , 0) R R To derive an expression for the unit normal n ∈ N M (t ), repelling LCS over [t0, t0 + T ] if its normal repulsion rate admits ε xε ε 0 we differentiate (34) with respect to s and substitute (33) into the a pointwise nondegenerate maximum along M(t0) among all lo- i cally C 1-close material surfaces. We call M(t) an attracting LCS over result to obtain [t0, t0 + T ] if it is a repelling LCS over [t0, t0 + T ] in backward time. ∂xε ∂α ∂n0 = ei + ε n0 + εα ∈ Tx Mε(t0), Finally, we call M(t) a hyperbolic LCS over [t0, t0 + T ] if it is a re- ε ∂si ∂si ∂si pelling or attracting LCS over [t0, t0 + T ]. i = 1,..., n − 1. (35) By a nondegenerate maximum, we mean a local maximum for t +T Seeking the unit normal nε ∈ Nx Mε(t) in the form 0 x n with a nondegenerate second derivative with respect ε ρt0 ( 0, 0) to changes normal to M(t0), as shown in Fig. 12. 2 3 nε(s, t0) = n0(x0, t0) + εβ(s, t0) + ε γ(s, t0) + O(ε ), (36) we substitute the expressions (35) and (36) into the two 4.2. Sufficient and necessary criteria for hyperbolic WLCS and LCS constraints   The following result provides computable sufficient and ∂xε ⟨n , n ⟩ = 1, , n = 0, i = 1,..., n − 1, necessary conditions for both weak LCS and LCS. In stating our main ε ε ε ∂si result, we will use the matrix (Eq. (31)) given in Box I whose first diagonal term, the second derivative of the inverse Cauchy–Green and compare equal powers of ε to deduce tensor, will be seen to equal − −n 1 ∂α 1 β = − ei, 2 −1 2 ∂s ∇ C [ξn, ξn, ξn, ξn] = − ⟨ξn, ∇ λnξn⟩ i=1 i λ2 (37) n − − 1 n 1 ∂α 2 n 1 ∂n  n−1 − − 0 λ − λ γ = − n0 − α , β ei, − n q  2 2 s s + 2 ξq, ∇ξnξn . (32) i=1 ∂ i i=1 ∂ i λqλn q=1   ∂n0 where we have used , n0 = 0. Eqs. (36) and (37) provide an ∂si Theorem 7 (Sufficient and Necessary Conditions for Hyperbolic expression for the unit normal to Mε(t) at the point xε. WLCS and LCS). Consider a compact material surface M(t) ⊂ U over We now derive a necessary condition for the normal repulsion t +T measure 0 x n to attain a pointwise extremum along t the interval [t0, t0 + T ]. Then ρt0 ( 0, 0) M( 0) relative to all other material surfaces M (t ) that are locally [ + ] ε 0 (i) M(t) is a repelling WLCS over t0, t0 T if and only if all the O(ε) C 1-close to M(t ). First note that Definition 5 is equivalent ∈ 0 following hold for all x0 M(t0): to the requirement 1. λn−1(x0, t0, T ) ̸= λn(x0, t0, T ) > 1, ⊥ 2. ξn(x0, t0, T ) Tx0 M(t0), ∂ t0+T ρ (x (s, t ), n (s, t ))| = = 0, (38) ⟨ ⟩ = t0 ε 0 ε 0 ε 0 3. ∇λn(x0, t0, T ), ξn(x0, t0, T ) 0. ∂ε (ii) M(t) is a repelling LCS over [t , t + T ] if and only if 0 0 which is to be satisfied for all x ∈ M (t ), and for any choice of 1. M(t) is a repelling WLCS over [t , t + T ], ε ε 0 0 0 the smooth function α(s, t ) in (34). 2. The matrix L(x , t , T ) is positive definite for all x ∈ M(t ). 0 0 0 0 0 We compute (38) using tensor notation with summation Proof. We first show that the conditions of the theorem are implied over repeated indices. We use the notation necessary, then argue that they are also sufficient. 1 n 1 n xε(s) = (xε,..., xε), nε(s) = (nε,..., nε), A. Conditions (i)–(ii) are necessary n (s) = (n1,..., nn), e (s) = (e1,..., en), We start by formulating the extremum property of the 0 0 0 p p p t +T = − repulsion rate 0 along t in precise terms. For a small p 1,..., n 1, ρt0 M( ) number ε > 0, we consider nearby material surfaces Mε(t) such t0+T 1 1 ρt (xε(s), nε(s)) = , (39) that Mε(t0) is O(ε) C -close to M(t0). On the compact time interval 0  −1 i j 1 C (xε(s)) n (s)nε(s) [t0, t0 + T ], Mε(t) will then remain O(ε) C -close to M(t) by ij ε t +T the smoothness of the flow map F 0 . If M(t) turns out to be a t0 with indices i, j, k, l varying over the integers 1,..., n, and with [ + ] repelling surface over t0, t0 T , then Mε(t) will be a repelling the indices p, q varying over 1,..., n−1. We denote differentiation [ + ] material surface over t0, t0 T , for ε > 0 small, by the continuity with respect to ε by prime. There will be no summation implied over of the inequalities (21) in their arguments. repeated occurrences of the index n. t +T In order to compute 0 at points of t , we need to derive ρt0 Mε( 0) In this notation, condition (38) can be written as a local representation of points and unit normals of Mε(t0) in the ′ 1 3 −1 i j k ′ −1 i ′ j vicinity of x0. For this, we consider a local parametrization of the ρ |ε=0 = − [ρ (C n n (x ) + 2C (n ) n )]ε=0 2 ij,k ε ε ε ij ε ε (n − 1)-dimensional manifold M(t0) by a parameter vector s = (s ,..., s ) ∈ V ⊂ n−1, so that for some orthonormal basis 1 1 n−1 R = − 3 [ −1 i j k − −1 i j ] ρ (x0, n0) αCij,k (x0)n0n0n0 2α,p Cij (x0)epn0 , (40) e1,..., en−1 in the tangent space Tx0 M(t0), we have 2 where we have used (36) and (37). From this equation we conclude ∂x0(s) ′ = ei, i = 1,..., n − 1. (33) that ρ | = = 0 can only hold for all choices of the smooth function ∂s ε 0 i α, if we require 1 ∈ By the O(ε) C -closeness of Mε(t0) to M(t0), a point xε − j C 1(x )ni n nk = 0, (41) Mε(t0) can be uniquely represented as ij,k 0 0 0 0 = + −1 i j = = − xε(s, t0) x0(s) εα(s, t0)n0(x0(s), t0) (34) Cij (x0)epn0 0, p 1,..., n 1. (42) G. Haller / Physica D 240 (2011) 574–598 583

 − −  2 −1   λn λ1   λn λn−1 ∇ C ξn, ξn, ξn, ξn 2 ξ1, ∇ξnξn ··· 2 ⟨ξn−1, ∇ξnξn⟩  λ1λn λn−1λn   λn − λ1 2λn − λ1   2 ⟨ξ , ∇ξ ξ ⟩ ··· 0   1 n n  L =  λ1λn λ1λn  (31)  . . . .   . . .. .   . . .   λn − λn−1 2λn − λn−1  2 ⟨ξn−1, ∇ξnξn⟩ 0 ··· λn−1λn λn−1λn

Box I.

−1 ⊥ Note that (42) implies C n0 Tx0 M(t0), or, equivalently, To evaluate condition (50), we differentiate (39) twice with −1 −1 C n0 ‖ n0; in other words, n0 must be an eigenvector of C . respect to ε to obtain For M(t0) to be normally repelling (cf. Definition 3), n0 must in [ ] ′′ 3  ′2 fact coincide with ξn(x0, t0, T ) and the corresponding eigenvalue ρ | = = ρ ε 0 ρ λn(x0, t0, T ) must be multiplicity one, otherwise growth normal ε=0  3 ′ to M(t) would not strictly dominate growth tangent to M(t) over ρ  − ′ − ′  − C 1ni nj xk + 2C 1 ni  nj [t0, t0 + T ], and hence M(t) would not be normally repelling by ij,k ε ε( ε) ij ε ε 2 ε=0 Definition 3. We conclude that for condition (42) to hold, we must √ necessarily have λ 3 = − n [ −1 i j k ′ + −1 i ′ j ]′ Cij,k nεnε(xε) 2Cij (nε) nε ε=0, (51) n0 = ξn, λn−1 ̸= λn > 1. (43) 2 ′ This in turn implies that condition (41) is equivalent to where we have used the fact that ρ |ε=0 = 0 holds at the extremum location by (38). −1 i j k = Cij,k (x0)ξn(x0, t0, T )ξn(x0, t0, T )ξn (x0, t0, T ) 0. (44) Evaluating (51) further, we note that [ −1 i j k ′ + −1 i ′ j ]′ To analyze the expression (44) further, we consider the Cij,k nεnε(xε) 2Cij (nε) nε ε=0 eigenvalue problem = [ −1 i j k ′ l ′ + −1 i ′ j k ′ Cij,klnεnε(xε) (xε) 4Cij,k (nε) nε(xε) 1 −1 j = i + −1 i ′′ j + −1 i ′ j ′] Cij ξn ξn, 2Cij (nε) nε 2Cij (nε) (nε) ε=0 λn − − = α2C 1 ξ i ξ j ξ kξ l − 4αα, C 1ei ξ j ξ k and differentiate it with respect to xk to obtain ij,kl n n n n p ij,k p n n   −1 1 i k k i j − − j λn,k 1 + 2C − α,p α,p ξ − αξ β e ξ C 1ξ j + C 1ξ = − ξ i + ξ i . ij 2 n ,p p n ij,k n ij n,k λ2 n λ n,k n n + −1 i j 2α,p α,q Cij epeq Multiplying both sides by ξ i and using the identity n = 2 −1 i j k l − −1 i j k α Cij,klξnξnξn ξn 4αα,p Cij,k ξpξnξn ξ j ξ j = 0, (45) α α 2α α, α, n,k n − ,p ,p − k k i i + p p ξ,pβ ξpξ 2 j j λn λn λp (which follows from ξnξn = 1), we obtain = 2 −1 i j k l − −1 i j k α Cij,klξnξnξn ξn 4αα,p Cij,k ξpξnξn − λn,k C 1ξ i ξ j = − , (46) [ ] ij,k n n 2 2 1 λn + α,p α,p − , (52) λp λn which implies where we have used (49). ⟨ ⟩ = k = − 2 −1 i j k ∇λn, ξn λn,kξn λnCij,k ξnξnξn . (47) Eqs. (51)–(52) show that (50) holds for all choices of α(s) if and only if the matrix (Eq. (53)) given in Box II is positive definite. Therefore, by (47), the necessary condition (44) is equivalent to To compute L(x0, t0, T ) in a coordinate-free form, we differen- i i = k ⟨∇λn, ξn⟩ = 0. (48) tiate the identity ξqξn δqn with respect to x to obtain Note that (43) and (48) together complete the proof of necessity ξ i ξ i + ξ i ξ i = 0, q = 1,..., n − 1. (54) for the conditions in statement (i) of the theorem. Also note that q,k n q n,k { }n−1 by (43), the orthonormal basis ei i=1 can be chosen as Next, for any q = 1,..., n − 1, we differentiate the identity −1 i j = k Cij ξqξn 0 with respect to x to obtain ep(s) = ξp(x0(s), t0, T ), p = 1,..., n − 1. (49) −1 i j + −1 i j + −1 i j = To prove the necessity of conditions (ii)/1 and (ii)/2 for the Cij,k ξqξn Cij ξq,kξn Cij ξqξn,k 0, existence of an LCS, first observe that (ii)/1 is clearly a necessary q = 1,..., n − 1. condition because an LCS is necessarily a WLCS. Next note that for k an LCS, beyond the stationarity condition (38), we must necessarily Multiplying this last equation by ξn and using the definition of ξn have the nondegenerate local maximum condition and ξq gives

2 1 1 ∂ + −1 i j k i i k i i k t0 T C ξ ξ ξ + ξ ξ ξ + ξ ξ ξ = 0, ρt (xε(s, t0), nε(s, t0))|ε=0 < 0 (50) ij,k q n n q,k n n q n,k n ∂ε2 0 λn λq by Definition 6. q = 1,..., n − 1, (55) 584 G. Haller / Physica D 240 (2011) 574–598

−1 i j k l − −1 i j k − −1 i j k · · · − −1 i j k  Cij,klξnξnξn ξn 2Cij,k ξ1ξnξn 2Cij,k ξ2ξnξn 2Cij,k ξn−1ξnξn  2 1 − −1 i j k − ···  2Cij,k ξ1ξnξn 0 0   λ1 λn   2 1   − −1 i j k − ···  =  2C ξ2ξnξn 0 0  L  ij,k λ λ  (53)  2 n   ......   . . . . .   2 1  − −1 i j k − 2Cij,k ξn−1ξnξn 0 0 0 λn−1 λn

Box II. where q indicates that there is no summation implied over q. B. Conditions (i)–(ii) are also sufficient Substituting (54) into (55), we obtain We first note that applying Proposition 2 and using assumptions (i)/1 and (i)/2 for any point x0 ∈ M(t0) and unit normal n0 ≡ λn − λq ξ (x , t , T ) ∈ N M(t ), we obtain −1 i j k = − i i k = − n 0 0 x0 0 Cij,k ξqξnξn ξqξn,kξn , q 1,..., n 1. (56) λnλq t +T 1 ρ 0 (x , n ) = t0 0 0  l  −1  Next, we differentiate Eq. (46) with respect to x to obtain  t +T  ξ C 0 x ξ n, t0 ( 0) n = − −1 i j − 2 −1 i j − 2 −1 i j λn,kl 2λnλn,lCij,k ξnξn λnCij,klξnξn 2λnCij,k ξn,lξn. (57)  = λn(x0, t0, T ) > 1, k l l t Multiplying (57) by ξ ξ and using the fact that λn,lξ = 0 by n n n t +T ρt (x0) ν 0 (x , n ) = min 0 condition (i)/3, we obtain t0 0 0  |e0|=1 t e ∈T M(t ) ⟨e0, Ct (x0)e0⟩ k l = − 2 −1 i j k l − 2 −1 i j k l 0√x0 0 0 λn,klξn ξn λnCij,klξnξnξn ξn 2λnCij,k ξn,lξnξn ξn, λn(x0, t0, T ) = √ > 1. (61) or, equivalently, λn−1(x0, t0, T ) Since t is assumed compact, (61) implies the existence of − 1 − M( 0) C 1 ξ i ξ j ξ kξ l = − λ ξ kξ l − 2C 1ξ i ξ l ξ j ξ k. (58) ij,kl n n n n 2 n,kl n n ij,k n,l n n n constants a, b > 0 such that Definition 3 holds for M(t0), and λn hence M(t0) is a repelling material surface over [t0, t0 + T ]. To evaluate the second term on the right-hand side of (58), we Given this, condition (38) ensures that M(t0) is a stationary t +T ∇ ∗ = ∗ surface for the repulsion rate ρ 0 (x , n ). We have seen that if left-multiply the identity ( ξn) ξn 0 by ξn to conclude that t0 0 0 conditions (43) hold, M(t0) is indeed a stationary surface which

∇ξnξn ⊥ ξn, (59) proves the sufficiency of conditions (ii)/1–(ii)/3 for M(t0) to be a WLCS. We have also seen that the positive definiteness of and hence L(x0, t0, T ) guarantees that the strict extremum condition (50) n−1 holds, and hence M(t0) is an LCS under conditions (ii)/1 and (ii)/2 − of the theorem. This completes the proof that conditions (i) and (ii) ∇ξnξn = ⟨ξq, ∇ξnξn⟩ξq. q=1 of Theorem 7 are also sufficient. A consequence of Theorem 7 is the following set of necessary Using this identity and (56), we can write conditions for hyperbolic LCS: −1 i l j k = −1⟨ ∇ ⟩ i j k Cij,k ξn,lξnξnξn Cij,k ξq, ξnξn ξqξnξn Proposition 8 (Necessary Conditions for Hyperbolic LCS). Consider = ⟨ ∇ ⟩ −1 i j k ξq, ξnξn Cij,k ξqξnξn a compact material surface M(t) ⊂ U which is a repelling LCS λ − λ over the interval [t0, t0 + T ]. Then the following must hold for = − n q ⟨ ∇ ⟩ i i k ξq, ξnξn ξqξn,kξn all x0 ∈ M(t0), the corresponding inverse Cauchy–Green strain λnλq −1 def. t0+T −1 tensor C (x0) = [Ct (x0)] , and its largest strain eigenvector n−1 λ − λ 0 = − − n q ⟨ ∇ ⟩2 def. ξq, ξnξn , ξn(x0) = ξn(x0, t0, T ): q=1 λnλq (1) λn−1(x0, t0, T ) ̸= λn(x0, t0, T ) > 1, ⊥ thus (58) can be re-written as (2) ξn(x0, t0, T ) Tx0 M(t0), (3) − 1 − C 1 ξ i ξ j ξ kξ l = − ⟨ξ , ∇2λ ξ ⟩ ∇C 1(x )[ξ (x ), ξ (x ), ξ (x )] = 0, ij,kl n n n n 2 n n n 0 n 0 n 0 n 0 λn or, equivalently, n−1 − − λn λq 2 + 2 ⟨ξ , ∇ξ ξ ⟩ , (60) ⟨∇λn(x0, t0, T ), ξn(x0)⟩ = 0. λ λ q n n q=1 n q 2 −1 (4) ∇ C (x0)[ξn(x0), ξn(x0), ξn(x0), ξn(x0)] > 0. as we claimed in (32). Proof. Conditions (1) and (2) are just re-statements of the Substituting the expressions (56) and (60) into the expression corresponding conditions in Theorem 7. Condition (3) is equivalent (53) for L(x0, t0, T ), we obtain the coordinate-free form (31) for to condition (i)/3 of Theorem 7 by formula (47). Finally, condition L(x0, t0, T ). This completes the proof that condition (ii) is necessary (4) of the proposition as a necessary condition follows from the for M(t0) to be an LCS. application of Sylvester’s theorem to the matrix L(x0, t0, T ) defined G. Haller / Physica D 240 (2011) 574–598 585 in (31). Specifically, the positivity of the first diagonal entry of Theorem 11 (Sufficient Condition for Robustness of Hyperbolic L(x0, t0, T ) is a necessary condition for the positive definiteness of LCS). Assume that M(t0) ⊂ U is a compact repelling LCS over L(x0, t0, T ). [t0, t0 + T ] such that Another consequence of note arises from an inspection of the 2 ⟨ξn, ∇ λnξn⟩ + ⟨∇λn, ∇ξnξn⟩ ̸= 0 (63) proof of Theorem 7. holds at each point x0 ∈ M(t0). Then, for any small enough smooth t +T perturbation of the flow map F 0 , there exists a unique repelling Proposition 9 (Nonexistence of a Least Repelling Material Sur- t0 face). There exists no normally repelling material surface along which QLCS that perturbs smoothly from M(t0). t +T the repulsion rate 0 admits a pointwise nondegenerate local min- ρt0 Proof. We first note that properties (1) and (3) in Definition 10 are imum among all C 1-close material surfaces. defined by open conditions that are smooth with respect to small t +T changes in F 0 . It remains to show that for small perturbations t0 Proof. Assume the contrary, i.e., assume that M(t) is such that at t +T to F 0 , the zero set defined by condition (2) of Definition 10 some x ∈ M(t ), we have t0 0 0 smoothly persists. For a small parameter δ ≥ 0, let ∂ t0+T ρ (x (s, t ), n (s, t ))| = = 0, (62) t0 ε 0 ε 0 ε 0 + + + ∂ε ˆt0 T t0 T t0 T Ft = Ft + δ8t 2 0 0 0 ∂ t0+T ρ (x (s, t ), n (s, t ))| = > 0, t0+T 2 t0 ε 0 ε 0 ε 0 denote a smooth perturbation of F . We seek to solve the ∂ε t0 equation where we used the framework and formula (38) of the proof of Theorem 7. ˆ ˆ ⟨∇λn(x0, t0, T ; δ), ξn(x0, t0, T ; δ)⟩ = 0, (64) Again, the first extremum condition in (62) is equivalent to for the perturbed flow map. By assumption, any x0 ∈ M(t0) is a λn−1 ̸= λn > 1, ξn = n0, ⟨∇λn, ξn⟩ = 0, solution to this equation for δ = 0. Eq. (64) has a unique solution 1 that is O(δ)C -close to M(t0) for δ > 0 small enough, if from and the second condition in (62) would require the matrix i Eq. (64) we can locally express one coordinate x0 in the form L(x0, t0, T ) defined in (31) to be negative definite. For L(x0, t0, T ) i = 1 i−1 i+1 n to be negative definite, all its diagonal elements would need to be x0 f (x0,..., x0 , x0 ,... x0, δ), strictly negative. This is, however, not possible, because λq < λn for some smooth function f . By the implicit function theorem, this holds for all q = 1,..., n−1 on a repelling material surface by the t +T is possible if (i) the left-hand side of (64) is differentiable in x and requirement ν 0 > 1. 0 t0 δ, which is the case, and (ii) at least one coordinate of the gradient of the left-hand side of (64) is nonzero at δ = 0, i.e., 5. Robustness of hyperbolic LCS 2 ∗ ∇ λnξn + (∇ξn) ∇λn ̸= 0.

A major question about any coherent structure identification Since M(t0) is assumed to be an LCS, the above gradient is parallel to the unit vector , and hence the gradient is nonzero precisely scheme is its robustness. If small data imperfections, numerical ξn when its inner product with is nonzero, as assumed in (63). errors, or other sources of noise can significantly alter the identified ξn structures, then the structures are of little practical use. We close by noting that LCSs are also expected to be robust The only non-robust feature of an LCS turns out to be the exact with respect to stochastic perturbations of the flow map, as long alignment of its normals with the ξn field. For material surfaces as individual realizations of the perturbation are smooth, and the that are normally repelling for long enough T , however, the mean of the perturbed flow satisfies the robustness condition (63). alignment error becomes numerically undetectable by Theorem 4. We will explore stochastic persistence of LCSs in more detail in This motivates the following relaxed definition of a quasi-LCS another publication. (QLCS) for the purposes of establishing robustness. 6. When does an FTLE ridge indicate a hyperbolic LCS? Definition 10 (Hyperbolic Quasi-LCS). Assume that M(t) ⊂ U is a normally repelling material surface over [t0, t0 + T ]. We call First, we give a definition of an FTLE ridge that is somewhat M(t) a repelling quasi-LCS (QLCS) over [t0, t0 + T ] if for all points weaker than the second-derivative ridge definition we used in x0 ∈ M(t0): analyzing the examples of Section 2.3 (cf. Appendix B).

1. λ − (x , t , T ) ̸= λ (x , t , T ) > 1. n 1 0 0 n 0 0 Definition 12 (FTLE Ridge). For a fixed time interval [t0, t0 + T ], ⟨ ⟩ = 2. ∇λn(x0, t0, T ), ξn(x0, t0, T ) 0. we call a compact hypersurface M(t0) ⊂ U an FTLE ridge if for all 3. L(x0, t0, T ) is positive definite. x0 ∈ M(t0), we have

We call M(t) an attracting QLCS over [t0, t0 +T ] if it is a repelling ∇ ∈ λn(x0, t0, T ) Tx0 M(t0), (65) QLCS over [t0, t0 + T ] in backward time. Finally, we call M(t) a 2 ⟨ξn(x0, t0, T ), ∇ λn(x0, t0, T )ξn(x0, t0, T )⟩ < 0. hyperbolic QLCS over [t0, t0 + T ] if it is a repelling or an attracting QLCS over [t0, t0 + T ]. This definition formulates the ridge conditions in terms of the Note that a hyperbolic QLCS is only a candidate for a hyperbolic λn field, which is equivalent to similar conditions on the FTLE field t +T LCS: one has to verify that the time interval T is long enough so Λ 0 by formula (9). t0 that ξn(x0, t0, T ) is practically aligned with the normal of the QLCS modulo exponentially small errors in T . We have the following Theorem 13 (Sufficient and Necessary Condition for a Hyperbolic robustness result for QLCS. LCS Based on an FTLE Ridge). Assume that M(t0) ⊂ U is a 586 G. Haller / Physica D 240 (2011) 574–598 compact FTLE ridge. Then M(t) = F t [M(t )] is a repelling LCS over where we have used (67). Therefore, the second condition in (66) is t0 0 [t0, t0 + T ] if and only if its points x0 ∈ M(t0) satisfy the following also satisfied if assumption (4) of the proposition is satisfied. conditions: Assumptions (3)–(4) of Proposition 14 hold, for example, if the

1. λn−1(x0, t0, T ) ̸= λn(x0, t0, T ) > 1, Cauchy–Green strain tensor is globally diagonal, and hence the ⊥ derivative ∇ξ vanishes identically (see Examples 1, 2, and 4). 2. ξn(x0, t0, T ) Tx0 M(t0), n 3. The matrix L(x0, t0, T ) is positive definite. We close this section by pointing out that the robustness condition (63) can be made more specific for FTLE ridges. Proof. Applying Theorem 7, we first note that assumptions (1) and (2) above are just re-statements of conditions (i)/1 and (i)/2 of Proposition 15 (Sufficient Condition for Robustness of an LCS that theorem. Next note that the first condition in (65) along with Marked by an FTLE Ridge). Assume that M(t0) ⊂ U is a compact FTLE assumption (2) of this proposition implies that assumption (i)/3 of ridge whose points x0 ∈ M(t0) satisfy all the assumptions of Propo- Theorem 7 is also satisfied, and hence M(t0) is a repelling WLCS. sition 14. Assume further that It remains to note that assumption (3) is identical to assumption 2 (ii)/2 of Theorem 7, and hence the statement of Theorem 13 ⟨ξn, ∇ λnξn⟩ + ⟨∇λn, ∇ξnξn⟩ < 0 (68) follows. holds at each point x0 ∈ M(t0). Then, for any small enough smooth t +T The following sufficient condition for an FTLE-ridge-based LCS perturbation of the flow map F 0 , there exists a unique nearby re- t0 gives more geometric insight into the types of FTLE ridges on pelling QLCS that perturbs smoothly from M(t0). which condition (3) of Theorem 13 is known to hold. (At the same 2 Proof. By Definition 12, we have ⟨ξn, ∇ λnξn⟩ < 0 along an time, preliminary calculations show that this sufficient condition FTLE ridge, and hence if that ridge is guaranteed to be an LCS by may reveal significantly fewer LCS than a direct application of Proposition 14, then (63) follows from (68), as claimed. Proposition 14.) 7. On the numerical detection of LCS Proposition 14 (Sufficient Condition for a Hyperbolic LCS Based on ⊂ an FTLE Ridge). Assume that M(t0) U is a compact FTLE ridge 7.1. Alignment of the LCS normal with the largest strain eigenvector whose points x0 ∈ M(t0) satisfy all the following conditions:

(1) λn−1(x0, t0, T ) ̸= λn(x0, t0, T ) > 1, As we remarked earlier, requiring the zero set of ⟨∇λn, ξn⟩ ⊥ to be orthogonal to the vector field x t T appears to be a (2) ξn(x0, t0, T ) Tx0 M(t0), ξn( 0, 0, ) (3) |∇ξnξn| ≤ 1, restrictive condition, yet it emerges as a necessary requirement for ⟨ ∇2 ⟩ − |∇ | ∑n−1[ − ] the existence of an LCS by Theorem 7. (4) ξn, λnξn < 2λn ξnξn q=1 λn/λq 1 . In view of the alignment result in Theorem 4, this orthogonality Then M(t) = F t [M(t )] is a repelling LCS over [t , t + T ]. t0 0 0 0 requirement is no longer surprising. Specifically, for a material Proof. We only need to show that under assumptions (3)–(4) of surface that is normally repelling over a long enough time interval, the present proposition, assumption (3) of Theorem 13 is satisfied, the strain eigenvectors ξn(x0, t0, T ) align with the normals of the surface up to errors that are exponentially small in the time i.e., L(x0, t0, T ) is positive definite. To see this, recall the classic criterion from linear algebra that a symmetric matrix with positive interval length T (cf. Fig. 11). If, for a given T , the normals to the zero set of ⟨∇ ⟩ are not yet fully aligned with the field, then there diagonal elements and with row-diagonal dominance is positive λn, ξn ξn is not yet a locally strongest repelling material surface emerging definite. Applying this criterion to L(x , t , T ) translates to the 0 0 over the time interval [t , t + T ]. This means that there exist sufficient conditions 0 0 small deformations of the zero set that produces slightly stronger n−1 λ − λ repelling surfaces. −⟨ ∇2 ⟩ + 2 − n q ⟨ ∇ ⟩2 ξn, λnξn 2λn ξq, ξnξn > 0, λnλq q=1 7.2. Algorithmic steps in LCS extraction n−1 − λn − λq 2λ2 [|⟨ξ , ∇ξ ξ ⟩| − ⟨ξ , ∇ξ ξ ⟩2] Theorem 7 suggests the following procedure for locating n λ λ q n n q n n q=1 q n hyperbolic LCSs: 2 For the total time interval [t0, t0 + T ] of interest, < −⟨ξn, ∇ λnξn⟩, Step 1 Compute the two largest strain eigenvalue fields, λn(x0, λq t0, T ) and λn−1(x0, t0, T ), as well as the largest strain |⟨ξq, ∇ξnξn⟩| < 1 + , q = 1,..., n − 1. (66) 2(λn − λq) eigenvector field ξn(x0, t0, T ). Step 2 Locate the solution set Z of the nonlinear equation Since for all q < n, we have λn > λq by assumption (1) of the ⟨∇λn(x0, t0, T ), ξ (x0, t0, T )⟩ = 0. proposition, the first condition in (66) is always satisfied on an FTLE n Step 3 Identify repelling WLCSs at time t0 as the subset ZWLCS ridge (cf. the second inequality in (65)). ⊂ Z on which (i) λn−1 ̸= λn > 1 and (ii) ξ (x0, t0, T ) ⊥ Under assumption (3) of the proposition, we have n Tx0 ZWLCS. Step 4 Identify repelling LCSs as the n − 1-dimensional surface |⟨ξq, ∇ξnξn⟩| < 1, q = 1,..., n − 1, (67) ZLCS ⊂ ZWLCS on which L(x0, t0, T ) (cf. (31)) is positive which ensures that the last condition in (66) is satisfied. definite. Finally, note that Step 5 Repeat the above steps starting from t0 + T back to t0 in backward time to obtain attracting WLCSs and LCSs at time n−1 − λn − λq t + T . 2λ2 [|⟨ξ , ∇ξ ξ ⟩| − ⟨ξ , ∇ξ ξ ⟩2] 0 n q n n q n n Step 6 Verify that the quantity ⟨ξ ,(∇2λ )ξ ⟩ + ⟨∇λ ,(∇ξ )ξ ⟩ is = λqλn n n n n n n i 1 bounded away from zero to ensure the robustness of the n−1 [ ] − λn LCSs. ≤ 2λn − 1 |⟨ξ , ∇ξ ξ ⟩| q n n As we have discussed, the angle between ξ and a local normal i=1 λq n to a repelling LCS becomes exponentially small for large enough T . n−1 [ ] − λn As a result, an approximate alignment between ξ and the normals ≤ 2λ |∇ξ ξ | − 1 , n n n n of Z in Step 3 above is sufficient for computational purposes. i=1 λq G. Haller / Physica D 240 (2011) 574–598 587

7.3. Numerical errors

The computation of the invariants of the Cauchy–Green strain tensor Ct is numerically sensitive along LCSs. This is due to t0 the (locally strongest) exponential separation of trajectories near repelling LCSs, which makes the accurate calculation of derivatives with respect to initial conditions challenging. In addition, Steps 2, 4, and 6 involve the calculation of second derivatives, further increasing the demand for a high-end numerical platform for the reliable extraction of LCSs. As a payoff, this computational investment leads to a sufficient and necessary mathematical criterion that eliminates false positives and missed structures. Fig. 13. Each advected horizontal line of this linear strain flow forms a material surface that is an attracting WLCS. Indeed, all blobs of initial conditions are visibly 7.4. Tracking LCS in time attracted at the same normal rate to these material surfaces while all WLCSs converge to each other. By the approach taken in this paper, LCSs are truly Lagrangian structures: they are material surfaces across which no phase space Since λ2 is a constant field, condition (ii)/2 of Theorem 7 is transport occurs. By Definition 6, the existence of such an LCS is violated, and hence system (70) admits no repelling LCSin the sense tied to a finite time-interval [t0, t0 + T ] over which it satisfies an Definition 6. extremum problem. As for WLCS, note that conditions (i)/1 and (i)/3 of Theorem 7 The same material surface is not guaranteed to satisfy a hold at any point x0 in the phase space. Then condition (i)/2 similar extremum problem over another time interval [t t + 1, 1 and (71) imply that all vertical material lines are repelling WLCSs. T ] in dynamical systems with general time dependence. It is Similarly, we obtain that all horizontal material lines are attracting natural to try and estimate the degree of non-invariance for LCS WLCSs by applying Theorem 7 to system (70) in backward time. computed over a rolling time interval [t, t + T ], but the resulting Indeed, each horizontal line is advected by this linear flow into a flux estimate, in general, is more complicated than previously moving material line that acts as a core Lagrangian structure on thought (see Appendix C) and requires the computation of higher- which trajectories accumulate, as shown in Fig. 13. order terms. Overall, therefore, the benefit of deriving a general flux formula over calculating the flux through an FTLE ridge numerically is unclear. 8.0.2. Example 6: linear–cubic saddle flow Applying the following steps will ensure that the extracted LCSs are fully Lagrangian, i.e., invariant, not just almost invariant: Consider again the linear–cubic saddle point flow Step A Find M(t ) by computing the conditions of Theorem 7 0 ˙ = (or one of the related propositions) over the maximum time x x, 3 interval [t0, t0 + T ] available. y˙ = −y − y , (72) Step B Locate later positions of the LCS M(t0) as already analyzed in Example 1. We have concluded that this system M(t) = Ft [M(t )]. (69) t0 0 has a trough in backward time along the x axis; therefore the x axis is not an attracting LCS by Theorem 7 (applied in backward time). Step B requires accurate numerical advection schemes, as the The Cauchy–Green strain tensor for this example is explicitly implementation of (69) involves the propagation of an unstable computable as surface under the flow map. Still, we gain conceptual clarity − and invariance using the above construction. Employing (69) also e2(t t0) 0  eliminates the emergence of ill-defined LCS (manifested by a fade- t (t−t0) C (x0) =  e  , t0 0 out of FTLE plots for larger t) as we approach the end of a finite-time 2 − 2 [ 1 + y e2(t t0) − y ]3 data set. ( 0) 0 √ as we show in Appendix B. In backward time (i.e., for t −log 5 < 8. Examples of hyperbolic LCS 0 t < t0; cf. Appendix B) we find for the above tensor that − 8.0.1. Example 5: linear saddle flow e4(t t0) λ2(y0, t0, t − t0) = > 0, [ + 2 2(t−t0) − 2]3 (1 y0)e y0 Consider again the linear strain flow   − 0 ˙ = λ = e2(t t0) < λ , ξ = , x x, (70) 1 2 2 1 y˙ = −y,   ∇λ2(0, t0, t − t0), ξ2(0, t0, t − t0) ≡ 0. analyzed in Example 1. The Cauchy–Green strain tensor for this example is simply Therefore, applying Theorem 7 in backward time, we conclude that the y = 0 axis is an attracting WLCS, as originally expected from  2T  + t0 T e 0 Fig. 4. C (x0) = − , t0 0 e 2T Since the strain eigenvector field is constant in x0, we have ∇ ≡ ⟨ ∇2 ⟩ with eigenvalues and eigenvectors ξn 0. Therefore, Theorem 7 requires ξ2, λ2ξ2 < 0 as a necessary condition for a repelling LCS for system (72) in backward −2T 2T λ1 = e , λ2 = e , (71) time. As we have seen in Appendix A, the x axis is a trough for the     ⟨ ∇2 ⟩ 0 1 backward FTLE field, and hence ξ2, λ2ξ2 > 0. We therefore ξ = , ξ = . 1 1 2 0 conclude that the x axis is not an attracting LCS. 588 G. Haller / Physica D 240 (2011) 574–598

8.0.3. Example 7: parallel shear flow with eigenvalues and unit eigenvectors satisfying

 2x 2 Consider again the parallel shear flow 1 + e 0 4x0 + λ ((0, y ), t ) = e 1 , x˙ = 2 + tanh y, 2 0 0  2x(t;t ,x )  + e 0 0 1 4x(t;t ,x ) y˙ = 0, e 0 0 +1 1 analyzed in Example 2, with its phase portrait sketched in Fig. 6. ξ ((0, y ) , t ) = , 2 0 0 0 Recall that this system has a second-derivative forward FTLE ridge   along the y = 0 axis, yet that axis is not a repelling LCS. − − 0 λ ((0, y ), t ) = e 2(t t0), ξ ((0, y ), t ) = . From the explicit solutions (112), we obtain that along the y0 = 1 0 0 1 0 0 1 0 axis, the Cauchy–Green strain tensor takes the form   Observe that t +T 1 1 + T C 0 ((x , 0)) = . t0 0 1 + T 1 + T 2 ξ2((0, y0), t0) = n0((0, y0), t0), For any t > t , the eigenvalues and unit eigenvectors of this tensor 0 and by our analysis in Appendix A, for large enough t = t + T , we can be written as 0 have 1 1  λ = 1 + T 2a2 + Ta T 2a2 + 4, 2 λ ((0, y ), t ) ̸= λ ((0, y ), t ) > 1, 2 √ 2 1 0 0 2 0 0  − 1 T 2a2 + 4 − 1 Ta    2 2 ∇λ2 ((0, y0), t0) , ξ2((0, y0), t0) = 0. (76)  √  + 1 2 2 + 1 2 2 +  1 2 T a 2 Ta T a 4  Also observe from Eq. (75) that the that strain eigenvector field, ξ =   , ∗ 2   ξ = (1, 0) , is constant at all points in the phase space. As a  1  2  √  result, we have ∇ξnξn ≡ 0 and hence assumptions (3)–(4) of + 1 2 2 + 1 2 2 + 1 2 T a 2 Ta T a 4 Proposition 14 are satisfied if the y0 axis is an FTLE ridge. 1 1 To apply Proposition 14, it remains to show that the y0 axis is 2 2  2 2 λ1 = 1 + T a − Ta T a + 4, an FTLE ridge in the sense of Definition 12. The first condition in 2 2 √ (65) is satisfied, and the second condition in (65) takes the specific  1 T 2a2 + 4 − 1 Ta  2 2 form  √  1 1  + 2 2 − 2 2 + 2  1 2 T a 2 Ta T a 4  ⟨ ∇ ⟩ =   ξ2((0, y0), t0), λ2((0, y0), t0)ξ2((0, y0), t0) < 0, (77) ξ1   .  1   √  which we have already verified for system (73) in Appendix A. + 1 2 2 − 1 2 2 + We conclude that by (76) and (77), assumptions (1)–(3) of 1 2 T a 2 Ta T a 4 Proposition 14 hold. Observe that for any T ∈ , we have R Therefore, by Proposition 14, for any choice of the initial time t   0 ± 0 and for large enough T , the material surface ξ ̸= n = , 2 0 ±1       ± = 0 ∈ 2 0 = t 0 with n denoting the two possible choices of a unit normal along M(t) R : Ft (78) 0 y y 0 y0 the y0 = 0 axis. As a result, condition (i)/2 of Theorem 7 fails, and = the y0 0 axis is therefore neither a WLCS nor an LCS. is a repelling LCS over the time interval [t0, t0 + T ]. Here the flow map Ft is defined implicitly by the formula (113) for the solutions t0 8.0.4. Example 8: LCS in Example 4 of system (16). The LCS defined in (78) is guaranteed to be locally the strongest Recall the dynamical system repelling material line, and should therefore be observable as the core of an expanding Lagrangian trajectory pattern. To verify this, x˙ = 1 + tanh2 x (73) we slightly modify (16) to the form y˙ = −y, x from Example 4, with its phase portrait shown in Fig. 7. We have x˙ = 1 + 40 tanh2 , (79) shown that defining LCSs as FTLE ridges would yield a repelling LCS 2 of the form y˙ = −0.1y,    ˆ 0 2 to enhance the extremum property of M(t) on shorter time scales M(t) ≡ ∈ R : y0 ∈ R . (74) y0 for the purposes of illustration. In Fig. 14, we show three snapshots of an array of trajectories We have pointed out, however, that Mˆ (t) is not even approxi- starting at time t = 0, then evolving into their later positions at mately Lagrangian: the area flux per unit length through this axis 0 time T = 1.05. Note that the FTLE field develops a ridge which is equal to one. gradually moves towards the x = 0 axis, becoming indistinguish- We now show how Definition 6 and Proposition 14 resolve the above contradiction. First, recall from Appendix A that the able from that axis by time T . Accordingly, the material surface Cauchy–Green strain tensor for this example is of the form M(t) defined in (78) emerges as the strongest normally repelling material surface over the time interval [0, T ], as confirmed by the  2x 2  1 + e 0 subplot (c). On shorter times scales, the FTLE ridge is still further 4x0 + e 1 0 away from the x = 0 axis; accordingly, the strongest normally re- t =  2x(t;t ,x )   Ct (x0)  e 0 0  , (75) 0 1 + ; pelling LCS is a vertical material line that is different from the x = 0  e4x(t t0,x0)+1  − − 0 e 2(t t0) axis, as shown in subplot (b). G. Haller / Physica D 240 (2011) 574–598 589

Particle positions at t= 0 Particle positions at t= 0.15 ab 2 2 1 1 y

y 0 0 -1 -1 -2 -2 0 1020304050 0 1020304050 x x Forward FTLE Forward FTLE 4 10 4 2 8 2 10

y 6 y 0 0 4 5 -2 2 -2 -4 -4 0 -4 -3 -2 -1 0 1 2 3 4 0 -4 -3 -2 -1 0 1 2 3 4 xx Particle positions at t= 1.05 c 2 1

y 0 -1 -2 0 1020304050 x Forward FTLE 4 3 2 2.5 y 0 2 1.5 -2 1 0.5 -4 -4 -3 -2 -1 0 1 2 3 4 x

Fig. 14. Three snapshots of the evolution of an array of trajectories for system (79). (a) Initial positions, with the x = 0 line highlighted, which approximates a repelling SLCS over [0, 1.06]. (b) Intermediate positions, with the FTLE plot revealing a strong repelling LCS over [0, 0.15], which is initially located along the vertical line x ≈ 0.6 at time t = 0. (c) Final particle positions at T = 1.05, confirming that the surface defined in Eq. (78) is indeed the strongest repelling material surface over the time interval [0, 1.05].

9. Constrained LCS The corresponding Cauchy–Green strain tensor field is defined as T = ∇ T ∗∇ T So far, we have identified hyperbolic LCSs as solutions of C (x0) ( F (x0)) F (x0), an extremum problem. Here we restrict this general extremum with eigenvalues and eigenvectors problem to a constrained extremum problem: we seek to find the most attracting or repelling material surface out of a prescribed 0 ≤ λ1(x0, T ) ≤ λ2(x0, T ), family of codimension-one surfaces. T C (x0)ξi(x0, T ) = λi(x0, T )ξi(x0, T ), i = 1, 2. An example where this approach is conceptually useful is the linear saddle flow x˙ = x, y˙ = −y. Out of all the attracting WLCS For any x with v(x) ̸= 0, we define the unit vector fields we have identified for this system in Example 5 (i.e., all horizontal v(x)  0 1 material lines), the x axis is the only one that is also an invariant n(x) = , = , (81) −1 0 manifold in the phase space of (70). Seeking the most normally |v(x)| repelling or attracting invariant manifold (as opposed to a general v(x) e(x) = , material line) in this flow is therefore a constrained LCS problem. |v(x)| Another example of interest is finding an LCS in a Hamiltonian system. Again, instead of seeking general material surfaces that which are pointwise normal and tangential, respectively, to all pointwise extremize the normal repulsion rate, one may restrict non-equilibrium trajectories of (80). the extremum search to material surfaces that are codimension- For any trajectory crossing a point x0 ∈ U with v(x0) ̸= 0, we one level sets of the Hamiltonian. define locally the trajectory-normal repulsion rate ρT (x0) over the In both of the above examples, the LCS constraint can be time interval [0, T ] as implemented by expressing the unit normal n in the definition of 0 T T t +T ρ (x ) = ⟨n(F (x )), ∇F (x )n(x )⟩, (82) the repulsion rate 0 (cf. (19)) as a function of x and t . Below T 0 0 0 0 ρt0 0 0 we discuss this approach for two classes of dynamical systems: where n(x0) is a smooth unit normal vector field to the trajectories planar autonomous systems and n-dimensional non-autonomous of (80). We also define the trajectory-normal repulsion ratio dynamical systems with a first integral. T T ⟨n(F (x0)), ∇F (x0)n(x0)⟩ ν (x ) = . (83) T 0 T 9.1. Constrained LCS in two-dimensional autonomous systems |∇F (x0)e(x0)|

Consider the two-dimensional autonomous system As before, ρT (x0) > 1 indicates strict normal growth of infinitesimal normal perturbations to the trajectory through x0 over 2 x˙ = v(x), x ∈ U ⊂ R , (80) the time interval [0, T ]. Similarly, νT (x0) > 1 implies that this normal growth dominates any potential infinitesimal growth in the with the one-parameter flow map tangent direction along the trajectory. T F : U → U, In analogy with Proposition 2, the above finite-time normal x0 → x(T , 0, x0). repulsion measures can be computed as follows: 590 G. Haller / Physica D 240 (2011) 574–598

Proof. The proof of the theorem follows closely the proof of Proposition 16. The quantities ρT (x0) and νT (x0) satisfy the expressions Theorem 7. The only difference is that in our search for WCLCS and CLCS, we restrict the class of C 1-close material surfaces to C 1-close  2 T trajectory segments. As a result, we have constraints on the normal |v(x0)| det C (x0) ρT (x0) = , nε(s, t0) and the perturbed trajectory point xε(s, t0) in the form ⟨v(x ), CT (x )v(x )⟩ 0 0 0 (84) v(xε(s, t0)) 2 T |v(x0)| det C (x0) nε (s, t0) = n(xε(s, t0)) = , ν (x ) = . |v(xε(s, t0))| T 0 T ⟨v(x0), C (x0)v(x0)⟩ xε(s, t0) = x0(s) + εα(s, t0)n(xε(s, t0))

2 ∂n(x0(s)) Proof. We obtain this result by following the proof of Proposi- = x0(s) + εα(s, t0)n(x0(s)) + ε α(s, t0) . tion 2, and noting that ∂x0

∗ − 1 The two extremum conditions, (38) and (50), used in the proof of [CT (x )] 1 = CT (x ), 0 T 0 Theorem 7 now simplify to det C (x0)

∂ t0+T ∂ which can be verified by direct calculation, using the symmetry of ρ (x (s, t ), n(x (s, t )))| = = ρ (x (s, t )) | = T t0 ε 0 ε 0 ε 0 T ε 0 ε 0 C and the definition of . ∂ε ∂ε = 0, We now reformulate our earlier general definitions of finite- 2 2 time normal repulsion and attraction for a trajectory segment of ∂ t0+T ∂ ρ (xε(s, t0), n(xε(s, t0)))|ε=0 = ρT (xε(s, t0)) |ε=0 the two-dimensional autonomous system (80). ∂ε2 t0 ∂ε2 < 0, (85) Definition 17 (Hyperbolic Trajectory Segment). A trajectory seg- or, equivalently, ment M ⊂U is normally repelling over [0, T ] if there exist constants   a, b > 0 such that for all points x0 ∈ M we have ∂ρT (x0) α , n(x0) = 0, aT ∂x ρT (x0) ≥ e , 0 bT νT (x0) ≥ e . and Similarly, we call M normally attracting over [0, T ] if it is repelling ∂ 2 ρ (x (s, t ))| = over [0, T ] in backward time. Finally, we call M hyperbolic over ∂ε2 T ε 0 ε 0 [0, T ] if it is normally repelling or normally attracting over [0, T ].   ∂ ∂  = α(s, t0) ρT (xε(s, t0)), n(xε(s, t0))  The following two definitions are analogues of our earlier ∂ε ∂x0 ε=0 definitions for weak LCS and LCS:  2  ∂ ρT (x0) = 2 s t n x n x α ( , 0) ( 0), 2 ( 0) Definition 18 (Hyperbolic Weak Constrained LCS). Assume that a ∂x0 trajectory segment M ⊂ U is normally repelling over [0, T ] ⊂ I.   2 ∂ρT (x0) ∗ We call M a repelling Weak Constrained LCS (WCLCS) over [0, T ] if + α (s, t0) , [∇n(x0)] n(x0) ∂x0 its normal repulsion rate admits a pointwise stationary value along  2  1 ∂ ρT (x0) M among all locally C -close trajectories. Similarly, we call M an = α2(s, t ) n(x ), n(x ) < 0, [ ] [ ] 0 0 2 0 attracting WCLCS over 0, T it is a repelling WCLCS over 0, T in ∂x0 backward time. Finally, we call M a hyperbolic WCLCS over [0, T ] it [ ]∗ = is a repelling or attracting WCLCS over [0, T ]. where we have used the relation ∇n(x0) n(x0) 0, obtained by differentiating ⟨n(x0), n(x0)⟩ = 1 with respect to x0. Definition 19 (Hyperbolic Constrained LCS). Assume that a trajec- We conclude that the extremum conditions (85) are equivalent tory segment M ⊂ U normally repelling over [0, T ]. We call M a to repelling Constrained LCS (CLCS) over [0, T ] if its normal repulsion   ∂ρT (x0) rate admits a nondegenerate maximum along M among all locally α , n(x0) = 0, (86) C 1-close trajectories. Similarly, we call M an attracting CLCS over ∂x0  2  [0, T ] it is a repelling CLCS over [0, T ] in backward time. Finally, 2 ∂ ρT (x0) α n(x0), n(x0) < 0. we call M a hyperbolic CLCS over [0, T ] it is a repelling or attracting ∂x2 CLCS over [0, T ]. 0 The first of these conditions in (86) (to be satisfied for any The following theorem is a reformulation of Theorem 7 for the choice of the smooth function α) is sufficient and necessary present two-dimensional, constrained LCS context. for ρT to admit stationary values along the trajectory segment M with respect to C 1-close trajectories. The first and second Theorem 20 (Sufficient and Necessary Condition for Hyperbolic conditions in (86) together are sufficient and necessary for ρ ⊂ T Weak CLCS and CLCS). Consider a compact trajectory segment M to admit nondegenerate local maxima along M with respect to U. Then C 1-close trajectories. Based on these considerations, the proof of (i) M is a repelling WCLCS over [0, T ] if and only if all the following the theorem is complete. hold for all x0 ∈ M: 1. ρT (x0) > 1, νT (x0) > 1, 9.1.1. Example 9: attracting CLCS in a linear saddle flow ⟨ ⟩ = 2. ∇ρT (x0), n(x0) 0. We again consider the linear flow (ii) M is a repelling CLCS over [0, T ] if and only if 1. M is a repelling WCLCS over [0, T ], x˙ = x, (87) 2 2. ⟨n(x0), ∇ ρT (x0)n(x0)⟩ < 0 for all x0 ∈ M. y˙ = −y, G. Haller / Physica D 240 (2011) 574–598 591 analyzed in Examples 1 and 5. Recall from Example 5 that any horizontal line 2 M(t0) = {(x, y) ∈ R : y = y0} gives rise to an attracting WLCS, M(t) = Ft [M(t )], over any finite t0 0 time-interval [0, T ]. So far, we have not found a way to distinguish the {y0 = 0} axis from all other WLCSs even though this axis is the fundamental attracting structure to which all WLCSs converge. Intuitively, however, we expect {y0 = 0} to be the locally strongest normally attracting trajectory (as opposed to material line) away from the origin. This is because the direction of strongest attraction is parallel to the y axis, and hence during any finite time-interval, the {y0 = 0} experiences the largest attraction in its normal direction. All other trajectories only experience a projection of that normal growth rate to their pointwise varying normal directions. Below we use Theorem 20 to verify this assertion. In our simplified notation for autonomous systems, the Cauchy–Green strain tensor of Example 5 can be re-written as Fig. 15. The graph of the normal repulsion rate ρT for the linear saddle flow x˙ = x, y˙ = −y. (The function ρT is only plotted over the domain where ρT > 1  2T  holds.) Note that away from the origin, any segment of the {y = 0} axis is a ridge T e 0 C (x0) = − . for ρT , and hence any such segment in a strong attracting CLCS. 0 e 2T

For any T < 0, the corresponding eigenvalues and eigenvectors are Note that the ρT surface in Fig. 15 loses its smoothness at the − origin. This is consistent with the fact that in any neighborhood λ (x , T ) = e2T < λ (x , T ) = e 2T ̸= 1, T < 0, (88) 1 0 2 0 of x0 = 0, there are no other trajectories of system (70) that are 1 0 locally C 1 close to the {y = 0} axis. As a result, the construction in ξ (x , T ) = , ξ (x , T ) = . 1 0 0 2 0 1 the proof of Theorem 20 does not apply. Recall, however, that on an empty set, all statements are true Using Proposition 16, we obtain by definition, and hence {y = 0} is the locally strongest attract-  ing trajectory segment among all C 1 close trajectories in any small x2 + y2 = = neighborhood of x0 0. As a result, Definition 19 deems the full ρT (x0) − | | | | , e 2 T x2 + e2 T y2 {y0 = 0} axis an attracting CLCS globally, including the neighbor- (89) hood of x0 = 0. x2 + y2 ν (x ) = T 0 −2|T | 2 2|T | 2 e x + e y 9.1.2. Example 10: nonexistence of CLCS in Example 4 for all T < 0. Consider again the dynamical system Substituting ξ from (88) into condition (i)/3 of Theorem 20, we 2 2 obtain that along any attracting weak CLCS of system (87), we must x˙ = 1 + tanh x, (91) have ∂yρT (x0) = 0, i.e., by (89), we must have y˙ = −y,

− | | | | 2 2 2 T − 2 T analyzed earlier in Examples 4 and 8. Note that for any x0 ∈ , we 2y0x0(e 2y0e ) R = 0. (90) have −2|T | 2 + 2|T | 2 2 (e x0 e y0) 1  0  { = } { = } e(x ) = , n(x ) = , This leaves us with the x0 0 and y0 0 axes as potential 0 0 0 −1 attracting WCLCS candidates, with the origin excluded from both   for technical reasons to avoid the blow-up in the definition (81). ∂x(t; x0) { = } T = 0 Note that condition (i)/2 Theorem 20 only holds for the y 0 ∇F (x0)  ∂x0  , − axis, which is therefore an attracting WCLCS by (88), (90) and (i) 0 e T of Theorem 20. This is nothing new yet compared with our earlier results on this example, which established that the {y = 0} axis is and hence from the original definition of ρT in (82), we have, for one of the infinitely many attracting WLCS in the flow. any T > 0, However, as opposed to λ (x , T ) defined in (88), the normal T T 2 0 ρT (x0) = ⟨n(F (x0)), ∇F (x0)n(x0)⟩, repulsion rate ρT (x0) is not a constant function, and hence {y0 = 0} −T may satisfy (ii)/2 of Theorem 20 and qualify as an attracting CLCS. = e < 1. (92) Indeed, We conclude that system (91) has no normally repelling trajectory 2 segments, and hence no repelling WCLCS or CLCS. Recall that the ⟨ξ2((x0, 0), T ), ∇ ρT ((x0, 0))ξ2((x0, 0), T )⟩ same system has a repelling LCS, which is a material line starting e|T | − e5|T | = 2 x 0 = 0 x ̸= 0 from the y axis at time zero. ∂y ρT (( 0, )) 2 < , 0 . x0

Therefore, with the exception of the origin x0 = 0, we have 9.1.3. Example 11: unique weak unstable manifold as repelling CLCS established that (ii)/1–(ii)/4 of Theorem 20 hold, and hence any Consider the autonomous system compact trajectory segment of the {y = 0} axis away from the [ ] x2 origin is an attracting CLCS over any finite time-interval 0, T . We x˙ = x + tanh , (93) show the graph of ρT for T = −1 in Fig. 15, confirming that the 4 {y = 0} axis is indeed a ridge of the ρ−1 field away from the origin. y˙ = x + 2y, 592 G. Haller / Physica D 240 (2011) 574–598

2 ab2 1.5 2.3 1.5

1 1

0.5 0.5 2.25

y 0 y 0

-0.5 -0.5 2.2 -1 -1 -1.5 -1.5 -2 2.15 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x x c 2 900 1.5 800 1 700

0.5 600

0 500 y

400 -0.5

300 -1 200 -1.5 100

-2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x

Fig. 16. LCS and CLCS analysis of (93): (a) Trajectories of the system. (b) FTLE field and directions of maximum strain computed over the interval [0, 3.19]. (c) Trajectory- normal repulsion rate ρT and trajectory-normal vectors at T = 3.19. whose origin is an unstable node with a weaker and a stronger 9.2. Constrained LCS in systems with first integrals instability (cf. Fig. 16a). The tanh(x2/4) term in the first equation of (93) introduces The results of the above section carry over with minor modifica- strong but highly localized stretching into the system on the two tions to n-dimensional systems with a conserved quantity, such as sides of the y axis. On the right-hand side of the y axis, the Hamiltonian systems. We consider an n-dimensional smooth dy- nonlinearity causes large trajectory separation and creates an FTLE namical system ridge (cf. Fig. 16b). This ridge, however, is not a hyperbolic LCS n x˙ = v(x, t), x ∈ U ⊂ R , n ≥ 2, t ∈ [α, β], (94) or WLCS by Theorem 7, since it is not normal to the directions ξ of maximal stretching. We also deduce from Fig. 16a that the with a conserved quantity H(x, t). Specifically, for the smooth 2 × [ ] → flux through the FTLE ridge is O(1) as T → ∞. Note that the function H: U α, β R, we assume flux formula (107) incorrectly predicts an O(1/T ) flux through the DH ∂H present FTLE ridge. = + ⟨∇H, v⟩ = 0. (95) Dt ∂t By contrast, the trajectory-normal growth rate field, ρT admits The level surfaces of H in the extended phase space U × [α, β], a ridge along one of the weak unstable manifolds of the origin. As defined as seen in Fig. 16c, the ridge is pointwise normal to trajectory normals, and hence is a repelling CLCS by Theorem 20. MC = {(x, t) ∈ U × [α, β]: H(x, t) = C}, On a more general note, we recall that the classic theory are therefore invariant sets in U × [α, β]. By the implicit function of invariant manifolds is unable to identify a unique weak theorem, MC is a codimension-one manifold in U if ∇H(x, t) ̸= 0 unstable manifold in system (93). Indeed, all we have guaranteed holds at each point (x, t) ∈ MC . In that case, the time t slice of MC by the classic theory is the existence of infinitely many weak is a material surface, denoted as unstable manifolds, all of which are tangent to the weak unstable MC (t) = {x ∈ U: H(x, t) = C}. (96) eigenvector of the origin. By contrast, the ρT (x0) field marks precisely one of these manifolds as the most influential one, i.e., the We will refer to MC (t) as an integral material surface to distinguish one that normally repels other such manifolds at the largest rate. it from other material surfaces that do not lie in a single level set Although not pursued here, a unique most attracting or repelling of the first integral H. center manifold could also be extracted in two-dimensional flows Some of the invariant level sets of H are dynamically using our CLCS approach. distinguished in that they repel or attract nearby trajectories G. Haller / Physica D 240 (2011) 574–598 593

1 at locally the highest rate in the flow. A simple example of maxima along MC (t0) among all locally C -close integral surfaces such distinguished level sets are open subsets of the stable and of H. Similarly, we call MC (t) an attracting CLCS over [0, T ] if it unstable manifolds of the classic pendulum equation containing is a repelling CLCS over [t0, t0 + T ] in backward time. Finally, we the origin. In this example, the system is Hamiltonian, and hence call MC (t) a hyperbolic CLCS over [t0, t0 + T ] if it is a repelling or the conserved quantity is the Hamiltonian, i.e., the energy of the attracting CLCS over [t0, t0 + T ]. pendulum. The main elements of our theory developed for material The following theorem is a direct extension of our two- surfaces carry over to integral material surfaces. The fundamental dimensional result on constrained LCSs to n-dimensional systems. difference is that we now seek the locally most repelling or attracting material surface among C 1-close integral material Theorem 24 (Sufficient and Necessary Conditions for Hyperbolic surfaces of the form (96), as opposed to all C 1-close material WCLCS and CLCS). Consider a compact integral material surface surfaces. MC (t) ⊂ U on which ∇H(x, t) is nonvanishing. Then t0+T The normal repulsion rate ρt (x0) for integral material 0 (i) MC (t) is a repelling WCLCS over [t0, t0 + T ] if and only if all the surfaces takes the form following hold for all x0 ∈ MC (t0): t +T t +T t0+T t0+T t0+T 0 0 x = ⟨n F x ∇F x n x ⟩ (97) 1. ρt (x0) > 1, νt (x0) > 1, ρt0 ( 0) t0+T ( t0 ( 0)), t0 ( 0) t0 ( 0) , 0 0 t +T 2. ⟨∇ρ 0 (x ), ∇H(x , t )⟩ = 0. and the normal repulsion ratio is defined as t0 0 0 0 (ii) MC (t) is a repelling CLCS over [t0, t0 + T ] if and only if t +T ρ 0 (x ) 1. M (t) is a repelling WCLCS over [t , t + T ], t0+T t0 0 C 0 0 ν (x ) = min . (98) t +T t0 0 t +T 2 0 |e |=1 0 2. ⟨∇H(x0, t0), ∇ ρt (x0)∇H(x0, t0)⟩ < 0 for all x0 ∈ 0 |∇F (x0)e0| 0 e ∈Tx M(t ) t0 0 0 0 MC (t0). t +T Again, 0 x 1 indicates a strict normal growth of infinites- ρt0 ( 0) > Proof. The proof of this theorem is identical to that of Theorem 7, 1 imal normal perturbations to the integral surface through x0 over with the class of C -close material surfaces restricted to the t +T the time interval [t t + T ]. Similarly, 0 x 1 implies that class of C 1-close integral surfaces, whose normals satisfy formula 0, 0 νt0 ( 0) > this normal growth dominates any growth in directions tangent to (100). the integral surface. As in Proposition 16, the above finite-time normal repulsion 9.2.1. Example 12: CLCS in a rotating saddle flow measures can be computed as follows: Consider the two-dimensional rotating saddle flow t +T t +T Proposition 21. The quantities ρ 0 (x ) and ν 0 (x ) satisfy the   t0 0 t0 0 sin 4t 2 + cos 4t x˙ = x (101) expressions −2 + cos 4t − sin 4t

t +T |∇H(x0, t0)| ρ 0 (x ) = , (99) from [14]. This is an example of a non-autonomous dynamical t0 0   −1  system whose instantaneous phase portraits taken at constant t  t +T  ∇H x t C 0 x ∇H x t ( 0, 0), t0 ( 0) ( 0, 0) all suggest center-type behavior around the origin. Indeed, for any fixed t, the eigenvalues of the coefficient matrix of (101) are equal t +T ρ 0 (x ) to ±3i. t0+T t0 0 ν (x0) = min . The origin, however, turns out to be a saddle point with t0 | |=  e0 1 t0+T e ∈T M(t ) ⟨e C x e⟩ ± 0 x0 0 , t0 ( 0) Lyapunov exponents 1. Since the largest FTLE is constant in this linear example, the stable and unstable manifolds of (101) cannot Proof. The proof of this proposition is identical to that of be detected as ridges of the Lyapunov exponent field. Along with Proposition 2, once one observes that infinitely many parallel lines, the stable and unstable manifolds are found to be repelling and attracting WLCSs, respectively, from an = ∇ |∇ | nt0 (x0) H(x0, t0)/ H(x0, t0) (100) application of Theorem 7. System (101) admits a first integral given by is a unit normal to MC (t0) at the point x0 ∈ MC (t0). 1 Finite-time normal repulsion, attraction and hyperbolicity H(x, y, t) = (y2 − x2) cos 4t − xy sin 4t, for the material surface MC (t) are defined in Definition 3. 2 The following two definitions are generalizations of our CLCS which can be constructed in a rotating frame where the system definitions given in the two-dimensional context. becomes autonomous (cf. [14]). To illustrate Theorem 24, we note that Definition 22 (Hyperbolic Weak Constrained LCS). Assume that the   ⊂ −x integral material surface MC (t) U is normally repelling over ∇H(x, y, 0) = , (102) [t0, t0 + T ]. We call MC (t) a repelling Weak CLCS (WCLCS) over y [t , t + T ] if its normal repulsion rate admits pointwise stationary 0 0 which we use in formula (99) to locate the unstable manifold of values along M (t ) among all locally C 1-close integral surfaces of C 0 the origin as an attracting CLCS. Fig. 17 shows the results obtained H. Similarly, we call M (t) an attracting WCLCS over [t , t + T ] if + C 0 0 from computing ρ0 T with T = −0.1. it is a repelling WCLCS over [t , t + T ] in backward time. Finally, 0 0 0 The gradient field (102) is seen to be normal to the emerging we call M (t) a hyperbolic WCLCS over [t , t + T ] if it is a repelling + C 0 0 ridge of ρ0 T and hence (i)/2 of Theorem 24 also holds. As a result, or attracting WCLCS over [t , t + T ]. 0 0 0 the computation correctly identifies a repelling CLCS whose t = 0 Definition 23 (Hyperbolic Constrained LCS). Assume that the slice, ⊂ 2 integral material surface MC (t) U is normally repelling over M(0) = {(x, y) ∈ R : x = y}, [t0, t0 + T ]. We call MC (t) a repelling Constrained LCS (CLCS) over [0, T ] ⊂ I if its normal repulsion rate admits strict pointwise is the t = 0 slice of the unstable manifold of the origin. 594 G. Haller / Physica D 240 (2011) 574–598

The present work has focused on hyperbolic LCS. Similar variational formulations for the locally most shearing material surfaces are possible and will appear elsewhere. By contrast, defining and extracting an elliptic LCS (i.e., finite-time generalizations of KAM tori) appears more challenging. An initial numerical study of such surfaces appears in [21]. Finally, lower-dimensional LCS can be defined and analyzed in a manner similar to what we pursued for codimension-one LCS in this paper. Normal perturbations to a lower-dimensional material surface, however, span higher-dimensional normal spaces. As a result, several normal repulsion rates and ratios will need to be defined and extremized simultaneously.

Acknowledgements

I am grateful to Harry Dankowitz, Mohammad Farazmand, Jim Meiss, Ronny Peikert, Gilead Tadmor, and Lai-Sang Young for their insightful questions and comments that have led to improvements in this manuscript. I am also thankful to Wenbo − Fig. 17. The graph of 0.1 is plotted over the subset of the plane where condition ρ0 Tang for reviewing some of the proofs in detail. Finally, I would like = (i)/1 of Theorem 24 is satisfied. Note the emergence of a ridge along the x y line. to express my gratitude to Shawn Shadden for reading this work in Also shown is the gradient field (102). detail, catching a number of misprints, and providing a number of thoughtful suggestions. 10. Conclusions

We have developed a variational theory of hyperbolic La- Appendix A. An Eulerian upper estimate for FTLE grangian Coherent Structures (LCSs) in finite-time dynamical We derive here a simple upper bound for the FTLE based on systems of arbitrary dimension. Our objective was to identify global properties of the vector field v. observed cores of Lagrangian patterns rigorously without a priori favoring any particular Lagrangian diagnostic tool. Our analysis has Proposition 25 (Upper Bound of FTLE). The maximum of the largest yielded an exact relationship between observable LCSs and invari- singular value of ∇v over the domain U¯ × [α, β] is always an upper ants of the Cauchy–Green strain tensor. bound for all forward and backward FTLE computed on the same We have defined hyperbolic LCSs as locally the strongest domain. Specifically, we have normally repelling or attracting material surfaces (codimension-  one invariant manifolds in the extended phase space). We then Λt (x ) ≤ max λ [[∇v(x, t)]∗∇v(x, t)], t0 0 max employed a variational approach to derive sufficient and necessary x∈U¯ ,t∈[α,β] criteria for the existence of LCSs. We have also introduced the x ∈ U¯ , t, t ∈ [α, β], (103) notion of a Weak LCS (WLCS) that is a stationary surface (but not 0 0 necessarily an extremum) for the above variational principle. with U¯ denoting the closure of U. WLCSs turn out to be zero sets for a three-tensor, the inverse t Proof. The deformation gradient ∇F (x0) is the fundamental Cauchy–Green strain tensor, evaluated along directions of maximal t0 matrix solution of the equation of variations along x(t, t , x ), strain. Out of all WLCSs, the material surfaces on which an 0 0 i.e., it satisfies the initial value problem appropriate tensor field (defined in (31)) is positive definite turn out to be LCSs. A necessary condition for this to happen is the d t = t positivity of the second derivative of the Cauchy–Green strain ∇Ft (x0) ∇v(x(t, t0, x0), t)∇Ft (x0), (104) dt 0 0 tensor (a four-tensor) along directions of maximal strain. t ∇F 0 (x ) = I. Efficient implementation of our theory will require further t0 0 development in the computational tools used in LCS detection. Integrating both sides of the first line of (104) from t0 to t ≥ t0, and Specifically, both WLCS and LCS are now understood to be solutions using the initial condition from the second line of (104) gives of a nonlinear equation (cf. (1), or equivalently, (2)), which needs ∫ t to be solved automatically with high accuracy, even though its left- ∇Ft (x ) = I + ∇v(x(s, t , x ), s)∇Fs (x )ds. t0 0 0 0 t0 0 hand side is typically only known on a numerical or experimental t0 grid. Once the solution set is found, its subsets satisfying the Taking the operator norm of both sides and applying Gronwall’s conditions for WLCS and LCS need to be identified. While this inequality gives involves the evaluation of higher derivatives, the robustness criteria for LCS (cf. Section 5) also imply robustness with respect ∫ t ‖∇Ft (x )‖ ≤ exp ‖∇v(x(s, t , x ), s)‖ ds. to numerical errors. t0 0 0 0 t The variational approach used in this paper extremizes the 0 pointwise normal repulsion rate of material surfaces, as opposed to Taking the logarithm of both sides and dividing by t −t0, we obtain an integral of the repulsion rate. We have favored this formulation 1 ∫ t because (1) it ensures that the LCSs act as observed cores of t Λ (x0) ≤ ‖∇v(x(s, t0, x0), s)‖ ds, t > t0. (105) t0 − Lagrangian patterns at all of their points; (2) it enables us to solve t t0 t0 the variational problem without posing any boundary conditions Reversing the direction of time in the above derivation yields the for the LCS. Applications of the present ideas in controlling LCS will, similar result however, likely lead to a variational problem for the integral of the ∫ t0 repulsion rate along the LCS. In that case, the boundary conditions t 1 Λ (x0) ≤ ‖∇v(x(s, t0, x0), s)‖ ds, t < t0. (106) will be obtained from the desired spatial location of the LCS. t0 t0 − t t G. Haller / Physica D 240 (2011) 574–598 595

Using the relationship between the singular value of a matrix is given by and its operator norm (cf. (9)), we obtain from (105)–(106) the ⟨t, ∇Λt ⟩     estimate (103). The max on the right-hand side is finite due to t0 ∂n 1 ϕ(p, t) = t, − ∇v(p, t0)n + O , (107) the assumption that U is bounded, and hence its closure, U¯ , is ⟨n, 6t n⟩ ∂t |T | t0 0 compact. where n(p, t0) and t(p, t0) are the unit normal and unit tangent to the FTLE at the point p at time t0, respectively; ∇v(p, t0) is the Appendix B. Details for Examples 1–4 Jacobian of the vector field at the same location and time; and T is the length of time over which the FTLE field is computed starting In this appendix, we provide detailed calculations for Examples from time t0. A version of the flux formula (107) appears for 1–4 that illustrate inconsistencies with defining repelling LCSs as n-dimensional dynamical systems in [5]. ridges of the FTLE field. The first and most precise idea of such a As we show in Appendix C, however, formula (107) is not definition appears in [4] for two-dimensional flows, and in [5] for generally applicable without further assumptions, some of which n-dimensional flows, which we recall below for reference. turn out to be restrictive. Without such assumptions, the higher-  1  order terms denoted as O |T | in (107) may be of equal magnitude B.1. The definition of an LCS as an FTLE ridge or larger than the leading-order terms, even as T → ∞.

Since all three examples we discuss here are two-dimensional, B.3. Details for Example 1 we list below the main regularity and nondegeneracy assumptions of Shadden et al. [4] under which they define LCSs as FTLE ridges The x and y components of system (10) decouple, and hence the in two dimensions. Cauchy–Green strain tensor can be written in the general form The assumptions in [4] are as follows: [ ]2  ∂x(t, t0, x0) (A1) The vector field v(x, t) is at least class C 0 in t and class C 2 0 in x. t  ∂x0  C (x0) = t0  [ ]2 (A2) There exists a constant K such that  ∂y(t, t0, y0)  0 − ∂y0 ‖∇Ft ‖ ≤ eK(t t0) t0 − e2(t t0) 0  holds for all t ∈ [α, β]. [ ]2 =  ∂y(t, t0, y0)  . (108) (A3) For all t0, t0 + T ∈ [α, β] and for all x0 ∈ U, we have 0 ∂y0 t0+T t0+T log λmin[Ct (x0)] < 0 < log λmax[Ct (x0)]. 0 0 Differentiation of the second equation in (10) with respect to y0 Under these assumptions, Shadden et al. [4] use the following gives definition for an FTLE ridge: d ∂y(t) ∂y(t) = −[1 + 3y2(t)] , dt ∂y0 ∂y0 Definition 26 (Second-Derivative FTLE Ridge). A second-derivative which, after integration, leads to the estimate ridge of Λt is an injective curve c: [a, b] → U satisfying the t0 following conditions for all s ∈ a b :    t 2 ( , ) ∂y(t) − [1+3y (s)]ds −(t−t ) = e t0 ≤ e 0 (109) ′   SR1 The vectors c (s) and ∇Λt (c(s)) are parallel  ∂y0  t0 t 2 t SR2 For the Hessian 6 (x ) = ∇ Λ (x ) and for a unit normal for all t > t0. From (108), (109), and definition (9), we obtain t0 0 t0 0 n(s) to the curve c(s), we have the result (11) for the forward FTLE field. As a result, according to Definition 27, there would be no repelling LCS in (10). ⟨ t ⟩ = ⟨ t ⟩ n(s), 6t (c(s))n(s) min u, 6t (c(s))u < 0. 0 |u|=1 0 B.4. Details for Example 2 With the above definition at hand, the definition of a repelling For t ≤ t , system (10) admits the explicit solution LCS as an FTLE ridge can be stated as follows: 0

(t−t0) x(t) = x0e , (110) Definition 27 (Shadden et al. [4] and Lekien et al. [5]). At each y0 y(t) = . time t0, a repelling Lagrangian Coherent Structure (LCS) is a second-  2 − 2 t0+T + 2(t t0) − derivative ridge of the field Λ x . (y0 1)e y0 t0 ( 0) We note that the repelling nature of the LCS in the above Note that for nonzero y0, the y component of the above solution definition appears first in the more general setting of Lekien blows up in finite time. Restricting initial conditions to the set et al. [5], which, however, also covers the two-dimensional context   2 1 U = (x, y) ∈ R : |y| < , relevant for the examples below. We also note that Lekien et al. [5] 2 replaces assumption (A2) of [4] with the assumption that the domain U¯ is compact, and point out that the existence of a K > 0 however, we obtain that the solution (110)√will exist at least on the satisfying (A2) then follows (see our Proposition 25 for details). backward-time interval I =[t0, t0 − log 5]. For all (x0, y0) ∈ U and for all t ∈ I, the Cauchy–Green strain tensor in (108) can be explicitly computed as B.2. Flux through an FTLE ridge − e2(t t0) 0  t 4(t−t ) Shadden et al. [4] propose that at time t0, the area flux per unit C (x ) = e 0 , (111) t0 0   t0+T 0 length at a point p of a ridge of a two-dimensional FTLE field Λ [ + 2 2(t−t0) − 2]3 t0 (1 y0)e y0 596 G. Haller / Physica D 240 (2011) 574–598 with the two eigenvalues B.6. Details for Example 4

− e4(t t0) y t = 0 Solutions of system (16) are given by the system of equations λ2( 0, 0) 3 > ,  + 2 2(t−t0) − 2 (1 y0)e y0 1 1 1 1 2x 2x0 x − x0 + arctan(e ) − arctan(e ) = t − t0, 2(t−t0) λ1 = e < λ2. 2 2 2 2 − − y(t) = y e (t t0). (113) Hence 0 ∂ Implicit differentiation of the first equation with respect to x0 λ2(y0, t0) yields ∂y0 2x 2x0 − − y0 1 ∂x 1 1 e ∂x 1 e = − 6(e2(t t0) − 1)e4(t t0) , − + − = 0, 2(t−t0) + 2 2(t−t0) − 2 4 4x + 4x0 + (e y0e y0) 2 ∂x0 2 2 e 1 ∂x0 2 e 1 ∂ 2 or, equivalently, y t 2 λ2( 0, 0) ∂y 2x 0 1 + e 0 − − ∂x 4x0 + 2(t t0) − 2 2(t t0) + 2 e 1 2(t−t ) 4(t−t ) e 7y0e 7y0 (t; x0) = ; . (114) = − 0 − 0 e2x(t x0) 6(e 1)e . ∂x0 + 2(t−t0) + 2 2(t−t0) − 2 5 1 4x(t;x ) (e y0e y0) e 0 +1 Consequently, we have By inspection of the right-hand side of (16), we find that on any fixed trajectory, we have x˙ > 0 uniformly bounded away from zero d = ∞ = for all forward times. Therefore, we have limt→∞ x(t) on all λ2(0, t0) 0, → ∞ dy0 trajectories, and hence taking the t limit in (114), we obtain the function d2 2(t−t0) 4(t−t0) λ2(0, t0) = 6(1 − e )e > 0, t < t0. dy2 ∂x e2x0 0 lim (t; x ) = 1 + , (115) →∞ 0 4x t ∂x0 e 0 + 1 Therefore, λ2(y0, t0) has a minimum ridge (trough) at y0 = 0. which has a unique global maximum at x0 = 0. From system (16) and expression (114), the Cauchy–Green B.5. Details for Example 3 strain tensor can be written as

 2x 2  1 + e 0 System (15) is arbitrary many times differentiable, and hence 4x0 + e 1 0 t  2x(t;t ,x )   assumption (A1) holds. By (103), the FTLE obeys the global bound C (x0) = e 0 0 , t0  +  t 1 4x(t;t ,x ) Λ (x0) ≤ 1, thus  e 0 0 +1  t0 − − 0 e 2(t t0) t |t−t0| ‖∇F (x0)‖ ≤ e t0 with eigenvalues satisfying

2 2 for all t, t ∈ and all x ∈ . Consequently, assumption (A2)  2x0  0 R 0 R 1 + e also holds. [ t ] = e4x0 +1 λmax Ct (x0)  ;  , 0 e2x(t t0,x0) The trajectories of (15) are given by 1 + ; e4x(t t0,x0)+1 t −2(t−t ) x(t; t0, x0) = x0 + (t − t0)(tanh y0 + 2), (112) λ [C (x )] = e 0 . min t0 0 y(t; t , x ) = y . 0 0 0 Note that the larger eigenvalue of Ct (x ) also satisfies t0 0 = ′ = − With the notation a(y) tanh (y0) and T t t0, we obtain from 2  e2x0  (112) the eigenvalues  t  = + lim λmax Ct (x0) 1 , t→∞ 0 e4x0 + 1 t +T 1 1  0 2 2 4 4 10x 8x 4x 2x λmax[Ct (x0)] = T a + T a + 4 + 1, ∂ −e 0 − e 0 + e 0 + e 0 0 2 2 [ t ] = lim λmax Ct (x0) 4 , x t→∞ 0 e4x0 + 1 3 t +T 1 1  ∂ 0 ( ) λ [C 0 (x )] = T 2a2 − T 4a4 + 4 + 1, min t0 0 ∂ 2 2 2 [ t ] lim λmax Ct (x0) ∂x2 t→∞ 0 which satisfy, for any x0, the relation 0 4 t +T t +T 2x0 4x0 8x0 10x0 log λ [C 0 (x )] < 0 < log λ [C 0 (x )]. = (2e + 4e − 8e − 10e ) min t0 0 max t0 0 (e4x0 + 1)3 This means that the nondegeneracy condition (A3) holds. e4x0 2x0 4x0 8x0 10x0 t0+T 1 t0+T − 48 (e + e − e − e ), (116) Next note that Λ = log λ [C (x )] has a nondegen- 4x 4 t0 2T max t0 0 (e 0 + 1) erate ridge along the y = 0 axis for any choice of t0 and T . To see + ∂ t0 T [ t ] ≡ this, it is enough to observe that λmax[C (x0)] has a nondegen- lim λmax Ct (x0) 0. t0 ∂y t→∞ 0 erate ridge along the same axis, because it reaches a strict maxi- 0 2 ′ 2 mum wherever a = [tanh (y0)] reaches a strict maximum, i.e., at Therefore, at any x0 = (0, y0), we have t +T y = 0. The corresponding ridge of Λ 0 has constant height, as 0 t0 9 t +T t 0 lim λmax[C ((0, y0))] = , (117) Λ does not depend on y . t0 t0 0 t→∞ 4 Based on the above, Definition 27 pronounces the x axis of ∂ t lim λmax[C ((0, y0))] = 0, system to be a repelling LCS for system (15). →∞ t0 ∂x0 t G. Haller / Physica D 240 (2011) 574–598 597 ∂ 2 [ t ] = − We conclude that for any t0 and large enough times t > t0, the lim λmax Ct ((0, y0)) 6, 2 t→∞ 0 y axis is a nondegenerate ridge of the FTLE field, and hence would ∂x0 be defined a repelling Lagrangian coherent structure at all times ∂ t ∂  t  lim λmax[C (x0)] = lim λmax C ((0, y0)) = 0, t0 by Definition 27. Note, however, that the y axis is far from being →∞ t0 →∞ t0 ∂y0 t ∂x0 t Lagrangian, as all trajectories cross it perpendicularly with velocity 2 2 ˙ = ∂ t ∂ t x 1. lim λmax[C (x0)] = lim λmax[C ((0, y0))] = 0. →∞ t0 2 →∞ t0 To evaluate the flux formula (107), recall that the above ridge ∂x0∂y0 t ∂y t 0 has constant height. As a result, ∇Λt vanishes along the ridge, and t0 From (116) and (117), we conclude that for large enough times the flux formula (107) gives the flux per unit length for this ridge t > t0: as (1) The scalar field λ [Ct (x )] develops a ridge approaching max t0 0   the unique global maximum of (115) as t increases. This ridge 1 ϕ(x0, t0) = O persists for all later times while both Lyapunov exponents tend to |t − t0| zero. as |t − t | → ∞. This implies lim| − |→∞ ϕ(x , t )| = 0, as (2) At any t > 0, the above ridge has constant height because 0 t t0 0 0 ridge t 1 t opposed to the correct value ϕ(x0, t0)|ridge ≡ 1. Λ = log λmax[C ((0, y0))] does not depend on y0. As a t0 2(t−t0) t0 result, ∇Λt vanishes along the ridge. t0 Appendix C. More on the flux through FTLE ridges (3) It follows from (114) that for any t > t0,

 2x 2 1 + e 0 Here we provide a simplified derivation of the phase space [ t ] = e4x0 +1 ≥ flux formula (107) through an FTLE ridge, originally obtained by log λmax Ct ((0, y0)) log   µmax 0 + e2x(t) Shadden et al. [4]. We also identify three additional assumptions 1 4x(t)+ e 1 that are not made explicitly in [4], but are necessary for formula  2x 2 1 + e 0 (107) to be correct. 4x0 + = log e 1 > 0. First, the derivation of (107) implicitly assumes that the  + 1  1 2 underlying dynamical system is known for all times, otherwise the term O(1/|T |) in (107) cannot be defined. Accordingly, we have to ≥ + Also, for any t t0 1, we have assume that def. log λ [Ct ((0, y ))] = −2(t − t ) ≤ µ = −2 < 0. (H1) System (5) is defined for all times, i.e., min t0 0 0 min We conclude that assumption (A3) is satisfied. [α, β] = [−∞, ∞]. (4) The vector field on the right-hand side of Eq. (16) is arbitrary many times differentiable, and hence assumption (A1) is satisfied. Under this assumption, we fix a time T > 0 and let R(t0) denote (5) The linearized flow map obeys the bound an n − 1-dimensional second-derivative ridge of the FTLE field t t0+T t Λ (x0)|t−t0| K|t−t | Λ (x ). We select a local parametrization of R(t ) in the form ‖∇F (x )‖ = e t0 ≤ e 0 , t0 0 0 t0 0 n−1 where K = 3.5 can be selected based on Proposition 25 and on the R(t0) = {x ∈ U: x = r(s, t0), s ∈ V ⊂ R }, observation n−1 with V denoting an open subset of R , and r: V × [α, β] →  ∗ max λmax[[∇v(x)] ∇v(x)] U denoting a smooth function. In the domain of this local x∈R2 parametrization, n(s, t0) will denote a smoothly varying unit    d 2 normal vector field to R(t0). 2 t +T = max 1 + tanh x < 3.5, (118) Denoting the n eigenvalues of the Hessian ∇2Λ 0 (x ) by dx t0 0 µ (x , t ) ≤ · · · ≤ µ (x , t ), which can be deduced by calculating the global maximum of min 0 0 max 0 0 2  + d 2  we recall that the following conditions must hold for R(t0) to be a 1 dx tanh x . We conclude that assumption (A2) is satisfied. (6) The limit of the Hessian of the FTLE field computed on the second-derivative ridge: ridge is given by (cf. (117)) t0+T ⟨∇Λt (r(s, t0)), n(s, t0)⟩ = 0, (120) lim 6t (x ) 0 t0 0 t→∞ 2 t0+T ∇ Λ (x0)n(s, t0) = µmin(x0, t0)n(s, t0), (121)  2  t0 1 ∂ t log λmax[C (x0)] 0 µ (x , t ) < 0. (122) = lim  2(t − t ) ∂x2 t0  min 0 0 t→∞ 0 0 0 0 For small enough ε > 0, there exists a nearby ridge R(t0 + ε) x0=(0,y0) that is a smooth deformation of R(t0). (This can be concluded from  ∂2  t   λmax C (x0) the smoothness of the underlying flow map, as well as from the ∂x2 t0  0 0 implicit function theorem using (120)–(122).) Specifically, we can = lim  t  →∞ λmax C ((0, y0))  t  t0  locally represent the points and unit normals of the ridge R(t0 +ε) 0 0 as x0=(0,y0) r(s, t + ε) = r(s, t ) + α(s, t ; ε)n(s, t )  −6  0 0 0 0 0 = + + 2 1 2 r(s, t0) εα1(s, t0)n(s, t0) O(ε ), = lim  3   . (119) →∞  2  t 2(t − t0) n(s, t + ε) = n(s, t ) + ∂ n(s, t )ε + O(ε2). (123) 0 0 0 0 t0 0 Since R(t + ε) is a second-derivative ridge, it should also Therefore, for the unit normal n = (1, 0) to the {x = 0} FTLE ridge, 0 satisfy the conditions (120)–(122). Specifically, substitution of the and for large enough times t > t , we have 0 expressions (123) into (120) gives −4 t + ⟨n, 6 (x0)n⟩ ≈ < 0. t0 T t0 ⟨∇Λ (r(s, t0 + ε)), n(s, t0 + ε)⟩ = 0. 3(t − t0) t0 598 G. Haller / Physica D 240 (2011) 574–598

Taylor expanding this expression in ε, comparing O(ε) terms in the If assumption (H2) holds, (125) and (128) imply expansion, and using (120)–(121) gives 1 + 1 + = ⟨ ⟩ + ⟨ ∇ t0 T ⟩ + ⟨∇ t0 T ⟩ ϕ v, n ∂t0 Λt , n Λt , ∂t0 n t +T µ 0 µ 0 ⟨∇Λ 0 r s t n s t ⟩ min min ∂t0 t0 ( ( , 0)), ( , 0) α1(s, t0) = − . (124)  µ (r(s, t ), t ) 1 2 t0+T ∗ t0+T min 0 0 = ⟨v n⟩ − ∇ Λ v + [∇v] ∇Λ , t0 t0 µmin Here ∂t0 refers to partial differentiation with respect to the + t0 T      explicit dependence of Λt and n on t0. This is to be contrasted 1 1 t +T 1 0 + O , n + ⟨∇Λ 0 , ∂ n⟩ + O with the vanishing total derivative t0 t0 |T | µmin |µminT | d   t0+T 1 + 1 ⟨∇Λ (r(s, t )), n(s, t )⟩ ≡ 0 t0 T t0 0 0 = ⟨∇Λ , ∂ n − [∇v]n⟩ + O , dt t0 t0 0 µmin |µminT | obtained by differentiating (120) with respect to t0. which is only equivalent to the flux formula (107) given in Shadden By definition, the local instantaneous volume flux per unit area et al. [4] if we assume through a point r(s, t ) ∈ R(t ) is given by 1 0 0 (H3) lim sup ∞ for all x ∈ t . |T |→∞ |µ (x ,t ,T )| < 0 R( 0) [ ]  min 0 0 dr(s, t0) For instance, in Example 4, formula (119) implies ϕ(r(s, t0), t0) = v(r(s, t0), t0) − , n(s, t0) , dt 0 1 1 lim sup = lim = ∞, i.e., by the ridge-normal projection of the velocity of a trajectory |T |→∞ |µmin(x0, t0, T )| |T |→∞ |µmin(x0, t0, T )| relative to the moving ridge. From the first equation in (123) and from (124), we obtain and hence assumption (H3) is not satisfied for Example 4. Accordingly, as we have seen, the flux formula (107) gives an dr(s, t ) incorrect value. 0 = α1(s, t0)n(s, t0) In general, unless assumptions (H1)–(H3) hold, the terms dt0  1  t0+T denoted as O by Shadden et al. [4] in formula (107) may be ⟨∇Λ r s t n s t ⟩ |T | ∂t0 t0 ( ( , 0)), ( , 0) = − n(s, t0), as large or larger than the leading-order terms, even as T → ∞. µmin(r(s, t0), t0) therefore we have References

ϕ(r(s, t0), t0) = ⟨v(r(s, t0), t0), n(s, t0)⟩ [1] G. Haller, G. Yuan, Lagrangian coherent structures and mixing in two- dimensional turbulence, Physica D 147 (2000) 352–370. t +T ⟨∇Λ 0 r s t n s t ⟩ [2] G. Haller, Distinguished material surfaces and coherent structures in three- ∂t0 t0 ( ( , 0)), ( , 0) + . (125) dimensional fluid flows, Physica D 149 (2001) 248–277. µmin(r(s, t0), t0) [3] G. Haller, Lagrangian coherent structures from approximate velocity data, Phys. Fluids A 14 (2002) 1851–1861. We note that the second term on the right-hand side of (125) [4] S.C. Shadden, F. Lekien, J.E. Marsden, Definition and properties of Lagrangian is identically zero for any autonomous dynamical system. Also coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D 212 (2005) 271–304. note that this term will remain small for slowly varying dynamical [5] F. Lekien, S.C. Shadden, J.E. Marsden, Lagrangian coherent structures in systems. n-dimensional systems, J. Math. Phys. 48 (2007) 065404. 1-19. Shadden et al. [4] prove that along a trajectory x(t) with a well- [6] T. Peacock, J. Dabiri, Introduction to focus issue: Lagrangian coherent structures, Chaos 20 (2010) 01750. defined Lyapunov exponent, we have [7] F. Lekien, A. Coulliette, A.J. Mariano, E.H. Ryan, L.K. Shay, G. Haller, J. Marsden,   Pollution release tied to invariant manifolds: a case study for the coast of d t +T 1 Florida, Physica D 210 (2005) 1–20. Λ 0 x t = t0 ( ( 0)) O , [8] W. Tang, M. Mathur, G. Haller, D.C. Hahn, F.H. Ruggiero, Lagrangian coherent dt0 |T | structures near a subtropical jet stream, J. Atmospheric Sci. 67 (2010) 2307–2319. which implies [9] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971) 193–225. t +T t +T [10] G. Froyland, S. Lloyd, N. Santitissadeekorn, Coherent sets for nonautonomous Λ 0 x = −⟨∇Λ 0 x v x t ⟩ + x t T (126) ∂t0 t0 ( 0) t0 ( 0), ( 0, 0) w( 0, 0, ), dynamical systems, Physica D 239 (2010) 1527–1541.   [11] V.I. Arnold, Ordinary Differential Equations, MIT Press, Cambridge, 1978. 1 [12] G. Haller, A. Poje, Finite-time transport in aperiodic flows, Physica D 119 (1998) w(x0, t0, T ) = O . (127) |T | 352–380. [13] G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos 10 (2000) 99–108. Next, Shadden et al. [4] differentiate (126) with respect to x0 [14] G. Haller, An objective definition of a vortex, J. Fluid Mech. 525 (2005) 1–26. and conclude [15] L.H. Duc, S. Siegmund, Hyperbolicity and invariant manifolds for planar nonautonomous systems on finite time intervals, Internat. J. Bifur. Chaos Appl.   Sci. Engrg. 18 (2008) 641–674. t +T t +T ∗ t +T 1 ∇ 0 = −[∇2 0 + [∇ ] ∇ 0 ] + [16] A. Berger, T.S. Doan, S. Siegmund, Non-autonomous finite-time dynamics, ∂t0 Λt Λt v v Λt O . (128) 0 0 0 |T | Discrete Contin. Dyn. Syst. Ser. B 9 (2008) 463–492. [17] A. Berger, T.S. Doan, S. Siegmund, A remark on finite-time hyperbolicity, PAMM This equation, however, is not correct without further assump- Proc. Appl. Math. Mech. 8 (2008) 10917–10918. [18] A. Berger, On finite-time hyperbolicity, Discrete Contin. Dyn. Syst. Ser. S (2010)  1  tions, because the spatial derivative of an O |T | term is (in press).   [19] M. Mathur, G. Haller, T. Peacock, J.E. Ruppert-Felsot, H.L. Swinney, Uncovering 1 the Lagrangian skeleton of turbulence, Phys. Rev. Lett. 98 (2007) 144502. not necessarily of order O | | . (An example is the function T [20] K. Lin, L.-S. Young, Shear-induced chaos, Nonlinearity 21 (2008) 899–915. 1 sin(T ⟨x0, x0⟩).) To deduce (128), we therefore have to assume [21] F.J. Beron-Vera, M.J. Olascoaga, M.G. Brown, H. Koçak, I.I. Rypina, Invariant- T tori-like Lagrangian coherent structures in geophysical flows, Chaos 20 (2010) (H2) lim sup T |∂ w(x , t , T )| < ∞ for all points x in a 017514/1-13. |T |→∞ x0 0 0 0 [22] G. Haller, Lagrangian structures and the rate of strain in a partition of two- neighborhood of the ridge R(t0). dimensional turbulence, Phys. Fluids 13 (2001) 3365–3385.