Analytic estimate of the maximum Lyapunov exponent in coupled-map lattices

In this work we present a theoretical and numerical study of the behaviour of the maximum Lyapunov exponent for a generic coupled-map-lattice in the weak-coupling regime. We explain the observed results by introducing a suitable continuous-time formulation of the tangent dynamics. The first general result is that the deviation of the Lyapunov exponent from the uncoupled-limit limit is function of a single scaling parameter which, in the case of strictly positive multipliers, is the ratio of the coupling strength with the variance of local multipliers. Moreover, we find an approximate analytic expression for the Lyapunov exponent by mapping the problem onto the evolution of a chain of nonlinear Langevin equations, which are eventually reduced to a single stochastic equation. The probability distribution of this dynamical equation provides an excellent description for the behaviour of the Lyapunov exponent. Furthermore, multipliers with random signs are considered as well, finding that the Lyapunov exponent still depends on a single scaling parameter, which however has a different expression.