  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.
