Recovery of the time-evolution equation of time-delay systems from time series

We present a method for time series analysis of both scalar and nonscalar time-delay systems. If the dynamics of the system investigated is governed by a time-delay-induced instability, the method allows one to determine the delay time. In a second step, the time-delay differential equation can be recovered from the time series. The method is a generalization of our recently proposed method suitable for time series analysis of scalar time-delay systems. The dynamics is not required to be settled on its attractor, and this also makes transient motion accessible to the analysis. If the motion actually takes place on a chaotic attractor, the applicability of the method does not depend on the dimensionality of the chaotic attractor-one main advantage over all time series analysis methods known until now. For a demonstration, we analyze time series, which are obtained with the help of the numerical integration of a two-dimensional time-delay differential equation. After having determined the delay time, we recover the nonscalar time-delay differential equation from the time series, in agreement with the ''original'' time-delay equation. Finally, possible applications of our analysis method in such different fields as medicine, hydrodynamics, laser physics, and chemistry are discussed.