LCS Tutorial: The finite-time Lyapunov exponent

3 The Finite-Time Lyapunov Exponent

The finite-time Lyapunov exponent, FTLE, which we will denote by $σTt (x)$ , is a scalar value which characterizes the amount of stretching about the trajectory of point $x ∈ D$ over the time interval [t, t + T]. For most flows of practical importance, the FTLE varies as a function of space and time. Therefore, one typically is interested in the FTLE field, in particular how it varies over space and time. The FTLE is not an instantaneous separation rate like the Okubo-Weiss criterion, Okubo (1970) and Weiss (1991), but rather measures the average, or integrated, separation between trajectories. This distinction is important because in time-dependent flows, the instantaneous velocity field often is not very revealing about actual trajectories, that is, instantaneous streamlines can quickly diverge from actual particle trajectories. However the FTLE accounts for the integrated effect of the flow because it is derived from particle trajectories, and thus is more indicative of the actual transport behavior.

To make the expression for the FTLE less obscure, let us derive it from considering the stretching between two neighboring particles. Consider an arbitrary point $x ∈ D$ at time t₀. When advected by the flow,

t0+T x ↦→ φ t (x) 0

after a time interval T. Since the flow has a continuous dependence on initial conditions, we know that an arbitrary point near x at time t₀ will behave similarly as x when advected in the flow, at least locally in time. However, as time evolves, the distance between this neighboring point and the point x will almost certainly change. Therefore, let us consider more closely the evolution of a point close to x, which we will write as $y = x + δx(t0)$ , where we assume $δx(t0)$ is infinitesimal and, for now, arbitrarily oriented. After a time interval T, this perturbation becomes

d φt0+T (x) δx(t +T ) = φt0+T (y) - φt0+T (x) = ----t0--------δx(t )+ 𝒪( ∥ δx(t )∥2) , 0 t0 t0 dx 0 0

(6)

where the second equality comes from taking the Taylor series expansion of the flow about point x. Above we employ the Landau notation, cf. Marsden and Hoffman (1993). That is, we write $f (x) = 𝒪(g)$ for a positive function $g(x)$ if and only if $f (x) ∕ g(x)$ remains bounded for all $x ∈ ℝ$ . Since $δx(t ) 0$ is infinitesimal, we can assume that the $2 𝒪( ∥δx(t0)∥ )$ term is negligible. Therefore, the magnitude of the perturbation is given by (using the standard vector L₂-norm)

┌ -------------------------------------------------------------- ┌ --------------------------------------------------------------- │ 〈 〉 │ 〈 * 〉 │ t0+T t0+T │ t0+T t0+T ∘ d-φ-t0-----(x)-- d-φ-t0-----(x)-- ∘ d-φ--t0----(x)-- d-φ-t0-----(x)-- ∥ δx(t0 + T )∥ = δx(t0), δx(t0) = δx(t0), δx(t0) dx dx dx dx

(7)

where we use the convention that M* denotes the adjoint (transpose) of M. The symmetric matrix

* d φt0+T (x) d φt0+T (x) -----t0--------- ----t0---------- Δ = dx dx

(8)

is a finite-time version of the (right) Cauchy-Green deformation tensor. Note that although $Δ$ is technically a function of t₀, T, and x, we try to avoid unnecessary notational clutter by simply writing $Δ$ instead of, say, $Δ(x; t0,T )$ .

Suppose we are interested in the maximum stretching that occurs between the points x and y. Notice that this occurs when $δx(t ) 0$ is chosen such that it is aligned with the eigenvector associated with the maximum eigenvalue of $Δ$ . That is, if $λ (Δ) max$ is the maximum eigenvalue of $Δ$ , thought of as an operator, then

∘ ----------------------------------------- ∘ --------------∥ ----- ∥ 〈 〉 max ∥ δx(t0 + T )∥ = δx(t0), λmax( Δ) δx(t0) = λmax( Δ) ∥ δx(t0) ∥ δx(t0)

(9)

where $--- δx(t0)$ is aligned with the eigenvector associated with $λmax( Δ)$ . If we define

∘ -------------- T --1-- σ t0(x) = ln λmax( Δ) , ∣T ∣

(10)

then Eq. (9) can be rewritten as

∥ ----- ∥ σTt (x)∣T ∣ ∥ ∥ max ∥ δx(t0 + T )∥ = e 0 δx(t0) . δx(t0)

(11)

Equation (10) represents the finite-time Lyapunov exponent at the point $x ∈ D$ at time t₀ with a finite integration time T.

Some remarks are in order:

Remark 1

The FTLE,

T σt0(x)

, is a function of the state variable x at time t₀, but if we vary t₀, then it is also a function of time.

Remark 2

Throughout this tutorial,

σTt0(x)

is often referred to as just

σ

when the extra notation can be dropped without causing ambiguity.

Remark 3

Even though

∘ ---------- λmax( Δ)

is the factor by which a perturbation is maximally stretched, perturbations often grow exponentially in time near LCS, which implies that the scaling/normalization introduced in (10) is better suited for locating such structures. This is especially true when considering large T and for systems defined on large or unbounded domains, as

∘ ---------- λmax( Δ)

can become numerically unstable.

Remark 4

It is interesting to note that,

∥ ∥ ∥ t0+T ∥ 1 ∥ d φ t (x) ∥ σT (x) = ----- ln ∥ ------0---------∥ , t0 ∣T ∣ ∥ dx ∥ 2

(12)

where the subscript in $∥ ⋅ ∥2$ explicitly denotes that the L₂-norm is being used, and in this case it's the matrix L₂-norm. To see this, we recall that by definition the matrix L₂-norm of an arbitrary matrix M is given by

∥M ∥2 = max ∥M x ∥2 , ∥x ∥2 = 1

where on the right hand side we are using the standard vector L₂-norm. But, following the reasoning above which led to Eq. (9), we have

∘ --------------------- * ∥M ∥2 = λmax (M M ) .

Remark 5

In fluid mechanics, the Eulerian perspective is typically defined as viewing the fluid at fixed points in the domain, perhaps at varying instances in time. When viewing the vector field of a dynamical system, this is typically the standard perspective. On the other hand, the Lagrangian perspective views the flow in terms of particle trajectories. While the FTLE field is technically a Eulerian field, it is thought of as a Lagrangian quantity since it is derived from particle trajectories.

Remark 6

Notice that even if the initial perturbation

δx(t0)

is not aligned with the eigenvector associated with

λmax( Δ)

, the perturbation will typically align very quickly with this direction. The reason for this is that if

δx(t0)

has a component in the

λmax( Δ)

eigenvector direction, then this component will quickly dominate because it is aligned with the most unstable direction.

Remark 7

In the definition of

σ

in (10), we used |T| instead of T because it is often the case that we are interested in computing

σ

for T > 0 and T < 0, to produce LCS akin to stable and unstable manifolds, as was mentioned towards the end of Sec. 2.

Remark 8

The finite-time Lyapunov exponent (FTLE) is sometimes referred to as the Direct Lyapunov Exponent (DLE), a name apparently due to Haller (2001). In this tutorial, we try to stick to the convention of calling it the finite-time Lyapunov exponent, however, we might occasionally refer to the FTLE as the DLE, but know that the two are equivalent.

Comments, Questions or Concerns can be sent to shawn@cds.caltech.edu