  Takens' Embedding Theorem asserts that when the states of a hidden dynamical
system are confined to a low-dimensional attractor, complete information about
the states can be preserved in the observed time-series output through the
delay coordinate map. However, the conditions for the theorem to hold ignore
the effects of noise and time-series analysis in practice requires a careful
empirical determination of the sampling time and number of delays resulting in
a number of delay coordinates larger than the minimum prescribed by Takens'
theorem. In this paper, we use tools and ideas in Compressed Sensing to provide
a first theoretical justification for the choice of the number of delays in
noisy conditions. In particular, we show that under certain conditions on the
dynamical system, measurement function, number of delays and sampling time, the
delay-coordinate map can be a stable embedding of the dynamical system's
attractor.
