Numerical study of Lyapunov exponents for products of correlated random matrices

We numerically study Lyapunov spectra and the maximal Lyapunov exponent (MLE) in products of real symplectic correlated random matrices, each of which is generated by a modified Bernoulli map. We can systematically investigate the influence of the correlation on the Lyapunov exponents because the statistical properties of the sequence generated by the map, whose correlation function shows power-law decay, have been well investigated. It is shown that the form of the scaled Lyapunov spectra does not change much even if the correlation of the sequence increases in the stationary region, and in the nonstationary region the forms are quite different from those obtained in the delta-correlated purely random case. The fluctuation strength dependence of the MLE changes with increasing correlation, and a different scaling law from that of the delta-correlated case can be observed in the nonstationary region. Moreover, the statistical properties of the probability distribution of the local Lyapunov exponents are quite different from those obtained from delta-correlated random matrices. Slower convergence that does not obey the central-limit theorem is observed for increasing correlation.