Recovery of scalar time-delay systems from time series

We present an analysis method that allows one to recover the differential equation of scalar time-delay systems having the form dy(t)/dt = f(y(t - tau(0))) - y(t) if only their time series are available. There exists a projection of an extremal section from the infinite-dimensional phase space to the (y(t - tau(0)),y(t))-plane, which has a fractal dimension less than or equal to one. This criterion can be used to extract the delay time tau(0) from the time series. Furthermore, the function f(Y(t - tau(0))) and, therefore, the complete time-evolution equation are obtained through a fitting procedure. The method is able to identify dynamical systems, the instability of which is time-delay induced. The method is successfully applied to experimental time series taken from two different types of electronic oscillators.