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A Brief Survey on the Numerical Dynamics for Functional Differential Equations

This is a survey on discretizing delay equations from a geometric-qualitative view-point. Concepts like compact attractors, hyperbolic periodic orbits, the saddle structure around hyperbolic equilibria, center-unstable manifolds of equilibria, inertial manifolds, structural stability, and Kamke monotonicity are considered. Error estimates for smooth and nonsmooth initial data in various Cj topologies are provided. The emphasis is put on Runge–Kutta methods with natural interpolants. The paper ends with a collection of the related results on retarded functional differential equations with bounded delay.