Exponential Dichotomies and Homoclinic Orbits in Functional Differential Equations*

Suppose an autonomous functional differential equation has an orbit r which is homochnic to a hyperbolic equilibrium point. The purpose of this paper is to give a procedure for determining the behavior of the solutions near r of a functional differential equation which is a nonautonomous periodic perturbation of the original one. The procedure uses exponential dichotomies and the Fredholm alternative. It is also shown that any smooth function p(f) defined on the reals which approaches zero monotonically as t + k co is the solution of a scalar functional differential equation and generates an orbit homoclinic to zero. Examples illustrating the results are also given.