Nonautonomous systems, cocycle attractors and variable time-step discretization

The concept of an attractor for autonomous systems is generally too restrictive in the nonautonomous context. An appropriate generalization is the cocycle attractor which consists of a family of equivariant sets. Here the cocycle description of a nonautonomous system and the concept of a cocycle attractor are reviewed in the context of nonautonomous ordinary differential equations and variable time-step numerical schemes for autonomous ordinary differential equations. In the latter case, theorems are stated for the existence and convergence of numerical cocycle attractors to an assumed attractor of an autonomous ordinary differential equations.