Partial averaging of functional differential equations

Develops a framework for averaging functional differential equations (FDEs) with two time scales. Averaging is performed on the fast time system, while slow time is 'frozen.' This creates an averaged equation which is slowly time-varying, hence the terminology of partial averaging. We show that solutions of the original FDE and its corresponding partially averaged equation remain close on arbitrarily long but finite time intervals. Next, assuming that the partially averaged system has an exponentially stable equilibrium point and that we restrict our interest to initial conditions that lie in the domain of exponential stability, the finite-time averaging results are extended to infinite time. In the special case of pointwise delays, exponential stability of the averaged system can be related to the frozen-time eigenvalues of its linearization.