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

Abstract

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 directly motivated by the investigation of coupled-map lattices in a regime where the chaotic properties are quite robust and yet a complete understanding has still not been reached. We derive some approximate analytic expressions by introducing a suitable continuous-time formulation of the evolution equation. As a first result, we show that the perturbation of the Lyapunov exponent due to the coupling depends only on 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. An explicit expression for the Lyapunov exponent is obtained by mapping the original problem onto 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. Finally, multipliers with random signs are also considered, finding that the Lyapunov exponent again depends on a single scaling parameter, which, however, has a different expression.