Parameter estimation of delay dynamical system from a scalar time series under external noise

We introduce an adaptive learning rules for estimating all unknown parameters of delay dynamical system using a scalar time series. Sufficient condition for synchronization is derived using Krasovskii-Lyapunov theory. This scheme is highly applicable in secure communication since multiple messages can be transmitted through multiple parameter modulations. One of the advantage of this method is that parameter estimation is also possible even when only one time series of the transmitter is available. We present numerical examples for Mackey-Glass system with periodic delay time which are used to illustrate the validity of this scheme. The corresponding numerical results and the effect of external noise are also studied.