Coupled map lattices as models of deterministic and stochastic differential delay equations

We discuss the probabilistic properties of a class of differential delay equations (DDE’s) by first reducing the equations to coupled map lattices, 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 white noise. This scheme provides an explicit description of the evolution of phase space densities in DDE’s, and yields an evolution equation that approximates the analog for delay equations of the generalized Liouville and Fokker-Planck equations. It is shown that in many cases of interest, for both stochastic and deterministic delay equations, the phase space densities reach a limit cycle in the asymptotic regime. This statistical cycling is observed numerically in continuous time systems with delay and discussed in light of our analytical description of the transfer operators.