Scaling-law for the maximal Lyapunov exponent

The authors study the scaling law for epsilon to 0 of the maximal Lyapunov exponent for coupled chaotic map lattices and for products of random Jacobi matrices. To this purpose they 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 mod log epsilon mod ^-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*2 random matrix model. The theory also predicts a phase transition at a critical value of the coupling epsilon _e, which is not observed in numerical simulations and might be an artifact of the approximation.