Universal scaling law for the largest Lyapunov exponent in coupled map lattices

We consider coupled map lattices of the type x(i)(n + 1) = (1 - epsilon)f(x(i)(n)) + (epsilon/2)[f(X(i-1)(n)) + f(x(i+1)(n))], where for concreteness we take f(x) = 1 - (mu/4)/1 - 2X/(p). With p > 1. We show that near epsilon = 0 (no coupling) and mu = 4 the envelope of the largest Lyapunov exponent of the full system obeys the scaling law <(Lambda)over bar> = Lambda(0) - [a epsilon + b(4 - mu)](1/p). We further argue that this law is universal in that it is independent of the details of f(x) insofar as f(x) has a single critical point x(c) in the interval [0, 1] and its lowest order power expansion about x(c) has the form /x - x(c)/(p). The dependence of <(Lambda)over bar> on the size of the lattice as well as on the range of the coupling is also discussed.