An analytic estimate of the maximum Lyapunov exponent in products of tridiagonal random matrices

In this work we present a theoretical and numerical study of the behaviour of the maximum Lyapunov exponent in products of random tridiagonal matrices in the limit of small coupling and small fluctuations. Such a problem is d irectly motivated by the investigation of coupled-map lattices in a regime- where the chaotic properties are quite robust and yet a complete understand ing has still not been reached. We derive some approximate analytic express ions by introducing a suitable continuous-time formulation of the evolution equation. As st first result, we show that the perturbation of the Lyapuno v exponent due to the coupling depends only on a single scaling parameter w hich, in the case of strictly positive multipliers, is the ratio of the cou pling strength with the variance of local multipliers. An explicit expressi on for the Lyapunov exponent is obtained by mapping the original problem on to a chain of nonlinear Langevin equations, which are eventually reduced to a single stochastic-equation. The probability distribution of this dynamic al equation provides an excellent description for the behaviour of the Lyap unov exponent. Finally, multipliers with random signs are also considered, finding: that the Lyapunov exponent again depends on a single scaling param eter, which, however, has a different expression.