Prolongations in Semidynamical Systems

Abstract

This chapter describes the prolongations in semidynamical systems. Semidynamical systems (sds) are continuous flows defined for all future time. Natural examples of sds are furnished by functional differential equations for which existence and uniqueness conditions hold. Though a substantial part of the theory of dynamical systems (ds) extends to sds, many new and interesting notions, for example, a start point and a singular point, arise in sds. Prolongations in ds have been studied and are useful in stability problems. However, prolongations, when defined as in ds, lose some of their basic properties. The chapter presents another system of prolongations, reviews which of the lost properties are restored, and also presents some results on prolongations and their limit sets. The definitions of D(x), J(x) being the same as in dynamical systems, the observation that sds are continuous maps defined for all nonnegative t tempts, it is believed that their properties in dynamical systems also hold in sds.