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Notes on Predictability in the large: an extension of the concept of Lyapunov exponent by Aurell et al. (1997, J. Phys. A)

In this paper, the authors introduce the definition of finite-size Lyapunov exponents (FSLE), a measure of the predictability of systems with a broad range of spatial and temporal scales for which the classic definition of Lyapunov exponents is not suited (the classic definition quantifies dispersion only for asymptotic processes):

“In this paper we shall address the problem of predictability in systems with many characteristic times, e.g. the case of fully developed turbulence where a hierarchy of different eddy turnover times do exist, or when the threshold delta_i [the initial perturbation] is not small. In these cases the predictability time Tp is determined by the details of the nonlinear mechanism responsible for the growth of the error. In particular, Tp may have no relation with the maximum Lyapunov exponent governed by the linearized equations for the infinitesimal error. In general, in this case the predictability time strongly depends on the details of the system.”

“The natural starting point in looking for such a quantity is the time it takes for a perturbation to grow from an initial size delta_i to a tolerance delta_f. We call this the (delta_i,delta_f) predictability time and denote it by T(delta_i,delta_f). Generally speaking, the predictability time will fluctuate. The natural definition of the finite-size Lyapunov exponent is, therefore, an average of some function of the predictability time, such that if both delta_i and delta_f are in the infinitesimal range, we will recover the usual Lyapunov exponent, and an obvious choice is then

lambda = <1/T> ln(delta_f/delta_i)”

“In contrast to infinitesimal perturbations, for finite perturbations the threshold delta_f is typically not to be taken much larger than the perturbation delta_i. What is interesting, and what makes finite-size Lyapunov exponents different from Lyapunov exponents for infinitesimal perturbations, is the dependence on delta_i.”

Question: Could we use FSLE to detect chaos on finite-length time series? I did not find anything on the internet on this possibility.

Question: Can you get a clearer idea of the difference between the definitions of FSLE and Lyapunov exponents? One difference is that a Lyapunov exponent is a measure of the exponentially-increasing expansion over an infinite time and thus measures dispersion only if the process is asymptotic, while FLSE is a measure over a specific period and is a function of that period and can be applied to non-asymptotic processes such as in the case of turbulence. As Lehahn et al. (2007) put it, FLSE equals the Lyapunov exponent when delta_i goes to zero and the time T goes to infinity. Any other difference?