Functional Differential Equations on Manifolds

Abstract

In this chapter we deal, mainly, with the current status of the global and geometric theory of functional differential equations (FDE) on a finite dimensional manifold. Retarded functional differential equations (RFDE) and neutral functional differential equations (NFDE) (in particular retarded differential delay equations and neutral differential delay equations) will be considered. As we will show, for a generic initial condition, the dependence of solutions with time differs, enormously, from one case to the other. In RFDE with finite delays the solution (as a curve in the phase space) starts continuous and its smoothness increases with time; on the other hand, in NFDE, in general, the solution stays only continuous for all time. Also, the semiflow operator of a smooth RFDE is not necessarily one to one even when we restrict its action to the set of all (smooth) global bounded solutions; this corresponds to the existence of collisions, a phenomenon which never occurs in the category of smooth finite dimensional vector fields (see Example 3.2.18 and Chapter 7). A vector field on a finite dimensional manifold is a special case of a RFDE and any RFDE is a particular case of a NFDE. The global and geometric theory of flows of vector fields acting on a (finite dimensional) manifold is very well developed, much more than the corresponding theory of semiflows acting on infinite dimensional Banach manifolds, even when the semiflows are defined by RFDE. Analogously, the global and geometric theory for RFDE is much more developed than the corresponding one for NFDE and so, many interesting open questions appear naturally.