Multistability and probabilistic properties of differential delay equations

The dynamics of a class of nonlinear delay differential equations (D.D.E's) is studied. We focus attention on D.D.E's with a discrete delay used as models for production/destruction processes. The design of an electronic analog computer simulating an integrable D.D.E is presented. This computer is used to illustrate the presence of bistable solutions in the system. The multistability is investigated numerically with an analytic integration algorithm. Higher order multistability is reported, and the structure of basin boundaries in the space of initial functions is investigated. Pathological dependence of solution behavior on the initial function is shown to be present in large regions of parameter space. A D.D.E obtained as the singular perturbation of the one dimensional "hat map" is studied numerically. Several schemes to undertake a statistical analysis of the equation are presented. We first focus attention on the construction of densities along trajectories, and then on the construction of densities for ensembles of trajectories generated by ensembles of initial functions. A cycling of densities is observed in both cases, and compared to the asymptotic periodicity of the Frobenius-Perron operator for the hat map. Functional analytic techniques used for the analysis of stochastic wave propagation in continuous media and in quantum field theory are extended to the statistical study of D.D.E's, and provide a theoretical framework within which to study D.D.E dynamics in the spirit of ergodic theory and statistical mechanics