Liapounov exponents for the Kuramoto-Sivashinsky model

A one-dimensional version of the Kuramoto-Sivashinsky (1977, 1978) model was used to examine the transition to stochasticity in deterministic systems. The number of nonnegative values for the Liapunov exponent (LE), a measure of the divergence of neighboring trajectories, is the amount of chaos in the system. The LEs are calculated as the differences between the dimensions of parallelepipeds which undergo either expansion or contraction, i.e., the Haussdorff dimension of the invariant measure on the attractor of the dynamic system. A finite difference second order Crack-Nicolson/Adams Bashforth scheme was used in simulations of varying LE values in order to characterize the evolution of large scale turbulence phenomena.