SCALING-LAW FOR THE MAXIMAL LYAPUNOV EXPONENT

We study the scaling law for epsilon --> 0 of the maximal Lyapunov exponent for coupled chaotic map lattices and for products of random Jacobi matrices. To this purpose we develop approximate analytical treatments of the random matrix problem inspired by the theory of directed polymers in a random medium: a type of mean-field method and a tree approximation which introduces correlations. The theoretical results suggest a leading \log epsilon\-1 increase in the maximal Lyapunov exponent near epsilon = 0, which is confirmed by numerical simulations, also for coupled map lattices. A dynamical mechanism responsible for this behaviour is investigated for a 2 x 2 random matrix model. The theory also predicts a phase transition at a critical value of the coupling epsilon(c), which is not observed in numerical simulations and might be an artifact of the approximation.