Reconstruction of unidirectionally coupled time-delayed systems of first order from time series of the driven system

Time-delayed systems, including coupled ones, became popular models of different physical and biological objects. Often One or few variables of such models cannot be directly measured, these variables are called hidden variables. However, reconstruction of models from experimental signals in presence of hidden variables can be very suitable for model verification and indirect measurement. Current study considers reconstruction of parameters of both systems and hidden variable of the driving system from time series of the driven system in ensemble of two unidirectionally coupled first order time-delayed systems.
Initial condition approach was used; this method considers initial conditions for hidden variables as additional unknown parameters. The method was adapted for time-delayed systems: vector of initial conditions was used instead of a single initial condition.
Time series of the driven system, parameters of nonlinear function of both systems and the coupling coefficient were shown to be reconstructable from a time series of driven system in periodical regime, if starting guesses for the hidden variable were set using a priori information about the model. The space of starting guesses for parameters was studied; the probability of successful reconstruction was shown to be approximately 1/4 for starting guesses for parameters distant from their true values in 50 % of their absolute values.
The fundamental possibility of reconstruction of system with one time delay in presence of hidden variables from scalar time series was shown.