Systems with delayed feedback can possess chaotic attractors with extremely high dimension, even if only a few physical degrees of freedom are involved. We propose a state space reconstruction from time series data of a scalar observable, coming along with a novel method to identify and model such systems, if a single variable is fed back. Making use of special properties of the feedback structure, we can understand the structure of the system by constructing equivalent equations of motion in spaces with dimensions which can be much smaller than the dimension of the chaotic attractor. We verify our method using both numerical and experimental data.

We propose a memory form of exchange-correlation potential y XC ͑r, t͒ for time-dependent interacting many-particle systems. Unlike previous memory-XC potentials, our y XC is not limited to the linear response regime. The proposed form of y XC is a generalized local-density approximation chosen so as to satisfy the nonlinear harmonic potential theorem and Newton's third law. For the case of the inhomogeneous electron gas, we give an explicit prescription for y XC based solely on an existing parametrization of the linear XC response kernel f hom XC ͑n, v͒ of the uniform gas. Application to quantum wells seems promising. [S0031-9007(97)

ZusammenfassungWe 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 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, which 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 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. P.A.C.S.: 05.45.+b * published in Phys. Rev. E 56 (1997) 5083

High-dimensional chaos displayed by multi-component systems with a single time-delayed feedback is shown to be accessible to time series analysis of a scalar variable only. The mapping of the original dynamics onto scalar time-delay systems defined on sufficiently high dimensional spaces is thoroughly discussed. The dimension of the "embedding" space turns out to be independent of the delay time and thus of the dimensionality of the attractor dynamics. As a consequence, the procedure described in the present paper turns out to be definitely advantageous with respect to the standard "embedding" technique in the case of high-dimensional chaos, when the latter is practically unapplicable. The mapping is not exact when delayed maps are used to reproduce the dynamics of time-continuous systems, but the errors can be kept under control. In this context, the approximation of delay-differential equations is discussed with reference to different classes of maps. Appropriate tools to estimate the a priori unknown delay time and the number of hidden components are introduced. The generalized Mackey-Glass system is investigated in detail as a testing ground for the theoretical considerations.