Finite-Time Lyapunov Exponent

asymptotic Lyapunov exponent $\lambda$ requires $t\to \infty$, useless its for a finite-horizon prediction or transient dynamics. the finite time Lyapunov exponent computes same quantity over window length $T$, yielding a field over phase space.

For a flow ${\varphi }^t(x_0)$, the maximal FTLE is:

$${\sigma }^T(x_0)=\frac{1}{|T|}\ln \sqrt{{\lambda }_{\max }\left({\left[D{\varphi }^T(x_0)\right]}^{\top }D{\varphi }^T(x_0)\right)}$$

i.e. $\frac{1}{|T|}$ times log of largest singular value of the flow map Jacobian. this quantity represents maximum growth rate of any infinitesimal perturbation to $x_0$ over time $T$.

• $T\to \infty$: ${\sigma }^T\to {\lambda }_{\max }$ (global exponent)
• $T\to 0$: ${\sigma }^T\to {\lambda }_{\text{loc}}$ (Local Lyapunov Exponent)
• Backward integration ($T<0$) reveals attracting structure instead of repelling

---

Lagrangian Coherent Structures

Primary use case is that ridges in the FTLE field identify something called Lagrangian Coherent Structures (LCS), think material surfaces acting as transport barriers.

• Ridges of forward-time FTLE ($T>0$): repelling LCS, finite-time analogues of unstable manifolds
• Ridges of backward-time FTLE ($T<0$): attracting LCS, analogues of stable manifolds

Trajectories don’t cross LCS on timescale $T$, which is why they matter for geophysical fluid dynamics — ocean eddies, atmospheric mixing barriers, etc. The Eulerian velocity field alone doesn’t reveal them; you need to integrate trajectories and look at the FTLE field.