Scaling Law and Asymptotic Distribution of Lyapunov Exponents in Conservative Dynamical Systems with Many Degrees of Freedom

摘要

The infinite product of 2N x 2N conservative random matrices which mimics the chaotic behavior of hamiltonian systems with N+1 degrees of freedom made of weakly nearest neighbor coupled oscillators is studied. The maximum Lyapunov exponent (L1) exhibits a power law behavior as a function of the coupling constant Sigma: L1 = Sigma sup Beta (Beta = 1/2 or 2/3, depending on the probability distribution of the matrix elements). These power laws do not depend on N and when increasing N, L1 quickly tends to an asymptotic value which only depends on Sigma and the kind of probability distribution chosen for building up the matrices. The spectrum of the Lyapunov exponents is calculated, showing that it has a thermodynamic limit of large N. This suggests the existence of a Kolmogorov entropy per degree of freedom proportional to the asymptotic value.