Identification and characterization of systems with delayed feedback: (I) Theory and tools

Abstract

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.