Tool to recover scalar time-delay systems from experimental time series.

We propose a method that is able to analyze chaotic time series, gained from experimental data. The method allows to identify scalar time-delay systems. If the dynamics of the system under investigation is governed by a scalar time-delay differential equation of the form dy(t) dt = h(y(t), y(t−τ0)), the delay time τ0 and the function h can be recovered. There are no restrictions to the dimensionality of the chaotic attractor. The method turns out to be insensitive to noise. We successfully apply the method to various time series taken from a computer experiment and two different electronic oscillators. Time series analysis of chaotic systems has gained much interest in recent years. Especially, embedding of time series in a reconstructed phase space with the help of time-delayed coordinates was widely used to estimate fractal dimensions of chaotic attractors [1, 2] and Lyapunov exponents [3]. It is the advantage of embedding techniques that the time series of only one variable has to be analyzed, even if the investigated system is multi-dimensional. Furthermore, it can be applied, in principle, to any dynamical system. Unfortunately, the embedding techniques only yield information, if the dimensionality of the chaotic attractor under investigation is low. Another drawback is the fact that it does not give any information about the structure of the dynamical system, in the sense, that one is able to identify the underlying insta-bilities. In the following, we propose a method which is taylor-suited to identify scalar systems with a time-delay induced instability. We will show that the differential equation can be recovered from the time series , if the investigated dynamics obeys a scalar time-delay differential equation. There are no restrictions to the dimensionality of the chaotic attractor. Additionally, the method has the advantage to be insensitive to noise. We consider the time evolution of scalar time-delay differential equations ˙ y(t) = h(y(t), y(t − τ 0)), (1)