Local Lyapunov Exponent

The standard Lyapunov exponent $\lambda$ is a global, asymptotic average over an infinite orbit. The local Lyapunov exponent strips away the averaging to give the instantaneous divergence rate at a single point in phase space.

For a 1D map $f$:

$${\lambda }_{\text{loc}}(x)=\ln |f^{\prime }(x)|$$

No limit, no average — just the log-derivative at $x$. The global $\lambda$ is then the time-average of this along an orbit:

$$\lambda =\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{k=0}^{n-1}{\lambda }_{\text{loc}}(x_k)$$

For a flow $\dot{x}=F(x)$, the local exponent in perturbation direction $\hat{\delta }$ is the Rayleigh quotient of the Jacobian:

$${\lambda }_{\text{loc}}(x,\hat{\delta })={\hat{\delta }}^{\top }DF(x)\hat{\delta }$$

showing that local divergence is direction-dependent — some directions stretch while others compress at the same point.

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Why it matters

The global $\lambda$ washes out spatial structure. The local LLE reveals where stretching and folding actually happen, which matters for:

• Intermittency: orbits alternating between high-LLE (stretching) and low-LLE (compressing) regions can have $\lambda \approx 0$ globally while being locally explosive
• Finite-horizon predictability: forecast validity depends on the local LLE along your specific trajectory, not the attractor average
• FTLE: averaging ${\lambda }_{\text{loc}}$ over a finite window gives the FTLE, which interpolates between local structure ($T\to 0$) and the global exponent ($T\to \infty$)