COUPLED MAP LATTICES AS MODELS OF DETERMINISTIC AND STOCHASTIC DIFFERENTIAL-DELAY EQUATIONS

We discuss the probabilistic properties of a class of differential del ay equations (DDE's) by first reducing the equations to coupled map la ttices, and then considering the spectral properties of the associated transfer operators. The analysis is carried out for the deterministic case and a stochastic case perturbed by additive or multiplicative wh ite noise. This scheme provides an explicit description of the evoluti on of phase space densities in DDE's, and yields an evolution equation that approximates the analog for delay equations of the generalized L iouville and Fokker-Planck equations. It is shown that in many cases o f interest, for both stochastic and deterministic delay equations, the phase space densities reach a limit cycle in the asymptotic regime. T his statistical cycling is observed numerically in continuous time sys tems with delay and discussed in light of our analytical description o f the transfer operators.