Extreme value theory for synchronization of coupled map lattices

Abstract

We show that the probability of the appearance of synchronization in chaotic coupled map lattices is related to the distribution of the maximum of a certain observable evaluated along almost all orbits. We show that such a distribution belongs to the family of extreme value laws, whose parameters, namely the extremal index, allow us to get a detailed description of the probability of synchronization. Theoretical results are supported by robust numerical computations that allow us to go beyond the theoretical framework provided and are potentially applicable to physically relevant systems.

On the Computation of the Extremal Index for Time Series

“…The natural question is therefore to ask when condition (15) is verified. For a large class of dynamical systems, in particular verifying the assumptions of theorem 4.2.7 in [13], it can be shown that condition (15) holds with k = 0 if the target point z is not periodic, and with k = p if z is periodic of prime period p (see Proposition 4.2.13 in [13]).…”

“…(14) or Eq. (7) have been computed up to now and in the framework of dynamical systems only around periodic points, or around the diagonal in coupled systems (see [15]) where at most only one of them is different from zero. We will present later on further examples of explicit computations of the q k .…”

“…We will present later on further examples of explicit computations of the q k . In particular section 4.1.2 will give an example of a stationary random dynamical system for which all the q k are different from zero and from each other, which implies that condition (15) is violated at all orders. In this respect our Eq.…”

“…where h is the density of µ, which, with our assumptions, is bounded from below by the constant C. Since ||(L S − L S )h|| L 1 (G×m) ≤ Ce −un ||h|| B , we can apply the perturbation theory mentioned in section 2, (see [32], [15], [13] ch. 7), and show that P (M n ≤ u n ) converges to the Gumbel law e −θτ .…”

“…In the attempt to describe the synchronisation of coupled map lattices, Faranda et al [15] introduced the neighborhood of the diagonal in the n-dimensional hypercube and defined accordingly the extremal index related to the first time the maps on the lattices become close together. Faranda et al [15] then used that approach to define a new type of observable, first in [20] and successively generalized in [8], which brought to a different interpretation of the extremal index, in terms of Lyapunov exponents instead of periodic behaviors. Let us consider the k−fold (k > 1) direct product (X, µ, T ) k with the direct product map T k = T • · · · • T acting on the product space X k and the product measure µ k = µ • · · · • µ.…”