Formal power series method for nonlinear time delay systems with analytic initial data

The state of a class of differential delay systems with a single delay and analytic initial data is computed in a purely combinatorial method by means of composition between formal power series. Time axis is divided into durations of the time delay and for each time interval, the generating series of the Fliess operator, which is the input-output map of the given system is determined. The state from the previous interval and the external inputs are viewed as the input arguments to the map. The current state is computed by composing the generating series of the input-output map with the generating series of the state from the previous interval and the formal power series representing the analytic external input. Sufficient conditions for convergence are also provided.