Finite size Lyapunov exponent: review on applications

Abstract

In dynamical systems, the growth of infinitesimal perturbations is well characterized by the Lyapunov exponents. In many situations of interest, however, important phenomena involve finite amplitude perturbations, which are ruled by nonlinear dynamics out of tangent space, and thus cannot be captured by the standard Lyapunov exponents. We review the application of the finite size Lyapunov exponent (FSLE) for the characterization of noninfinitesimal perturbations in a variety of systems. In particular, we illustrate their usage in the context of predictability of systems with multiple spatiotemporal scales of geophysical relevance, in the characterization of nonlinear instabilities, and in some aspects of transport in fluid flows. We also discuss the application of the FSLE to more general aspects such as chaos-noise detection and coarse-grained descriptions of signals.

Lyapunov analysis of multiscale dynamics: the slow bundle of the two-scale Lorenz 96 model

“…Here, we follow the excellent review [Cencini and Vulpiani(2013)], where applications to L96 were also discussed. Given a generic trajectory x(t), the idea is to define a series of thresholds δ n = δ 0 σ n , with σ > 1, and to measure the times τ (δ n ) needed by the norm of a finite perturbation ∆x(t) = x (t) − x(t) to grow from the amplitude δ n to δ n+1 .…”

“…(28). The FSLE in principle depends on the norm used to define the size of the perturbation [Cencini and Vulpiani(2013)]. However, by construction, for vanishing perturbations, the FSLE should coincide with the largest LE regardless of the norm lim δ→0 Λ(δ) = λ 1 .…”

“…Interestingly, it was observed that also the height of the second plateau seems to be roughly norm independent. This observation led to the conjecture that the nonlinear evolution of large perturbations may be controlled by the linear dynamics of an effective lower dimensional system, capturing the essence of the slow-variable dynamics [Cencini and Vulpiani(2013)].…”

Taming chaos to sample rare events: The effect of weak chaos

“…The growth of the divergence of two trajectories separated by a finite distance δ x (x) is not necessarily well described by exp(−λ t (x)t) (see Ref. [18] for more on finite-size Lyapunov exponents). This violation is more evident in trajectories x where λ t (x) < 0.…”

Power law error growth in multi-hierarchical chaotic systems—a dynamical mechanism for finite prediction horizon

“…This has given rise to the notion of scale dependent error growth: the conventional Lyapunov exponent should be replaced by a scale dependent quantity, e.g. a finite size Lyapunov exponent [12]. Indeed, in a study of scale dependent error growth in the Global Forecast System of the National Center for Environmental Prediction, Harlim et al [13] have shown that there is a scale dependent error growth rate which becomes very large if the errors become small (see figure 1 in [13]).…”