Reconstruction of systems with delayed feedback: I. Theory

High-dimensional chaos displayed by multi-component systems with a single t ime-delayed feedback is shown to be accessible to time series analysis of a scalar variable only. The mapping of the original dynamics onto scalar tim e-delay systems defined on sufficiently high dimensional spaces is thorough ly discussed. The dimension of the "embedding" space turns out to be indepe ndent of the delay time and thus of the dimensionality of the attractor dyn amics. As a consequence, the procedure described in the present paper turns out to be definitely advantageous with respect to the standard embedding t echnique in the case of high-dimensional chaos, when the latter is practica lly unapplicable. The mapping is not exact when delayed maps are used to re produce the dynamics of time-continuous systems, but the errors can be kept under control. In this context, the approximation of delay-differential eq uations is discussed with reference to different classes of maps. Appropria te tools to estimate the a priori unknown delay time and the number of hidd en components are introduced. The generalized Mackey-Glass system is invest igated in detail as a testing ground for the theoretical considerations.