  Delay-coordinate mapping is an effective and widely used technique for
reconstructing and analyzing the dynamics of a nonlinear system based on
time-series outputs. The efficacy of delay-coordinate mapping has long been
supported by Takens' embedding theorem, which guarantees that delay-coordinate
maps use the time-series output to provide a reconstruction of the hidden state
space that is a one-to-one embedding of the system's attractor. While this
topological guarantee ensures that distinct points in the reconstruction
correspond to distinct points in the original state space, it does not
characterize the quality of this embedding or illuminate how the specific
parameters affect the reconstruction. In this paper, we extend Takens' result
by establishing conditions under which delay-coordinate mapping is guaranteed
to provide a stable embedding of a system's attractor. Beyond only preserving
the attractor topology, a stable embedding preserves the attractor geometry by
ensuring that distances between points in the state space are approximately
preserved. In particular, we find that delay-coordinate mapping stably embeds
an attractor of a dynamical system if the stable rank of the system is large
enough to be proportional to the dimension of the attractor. The stable rank
reflects the relation between the sampling interval and the number of delays in
delay-coordinate mapping. Our theoretical findings give guidance to choosing
system parameters, echoing the trade-off between irrelevancy and redundancy
that has been heuristically investigated in the literature. Our initial result
is stated for attractors that are smooth submanifolds of Euclidean space, with
extensions provided for the case of strange attractors.
