This article is from the Nonlinear Science FAQ, by James D. Meiss firstname.lastname@example.org with numerous contributions by others.
(Thanks to Jim Crutchfield for contributing to this answer)
This is the application of dynamical systems techniques to a data series,
usually obtained by "measuring" the value of a single observable as a function
of time. The major tool in a dynamicist's toolkit is "delay coordinate
embedding" which creates a phase space portrait from a single data series. It
seems remarkable at first, but one can reconstruct a picture equivalent
(topologically) to the full Lorenz attractor (see [2.12])in three-dimensional
space by measuring only one of its coordinates, say x(t), and plotting the
delay coordinates (x(t), x(t+h), x(t+2h)) for a fixed h.
It is important to emphasize that the idea of using derivatives or delay
coordinates in time series modeling is nothing new. It goes back at least to
the work of Yule, who in 1927 used an autoregressive (AR) model to make a
predictive model for the sunspot cycle. AR models are nothing more than delay
coordinates used with a linear model. Delays, derivatives, principal
components, and a variety of other methods of reconstruction have been widely
used in time series analysis since the early 50's, and are described in
several hundred books. The new aspects raised by dynamical systems theory are
(i) the implied geometric view of temporal behavior and (ii) the existence of
"geometric invariants", such as dimension and Lyapunov exponents. The central
question was not whether delay coordinates are useful for time series
analysis, but rather whether reconstruction methods preserve the geometry and
the geometric invariants of dynamical systems. (Packard, Crutchfield, Farmer &
G.U. Yule, Phil. Trans. R. Soc. London A 226 (1927) p. 267.
N.H. Packard, J.P. Crutchfield, J.D. Farmer, and R.S. Shaw, "Geometry
from a time series", Phys. Rev. Lett. 45, no. 9 (1980) 712.
F. Takens, "Detecting strange attractors in fluid turbulence", in: Dynamical
Systems and Turbulence, eds. D. Rand and L.-S. Young
(Springer, Berlin, 1981)
Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., and Tsimring, L.Sh.T.
"The analysis of observed chaotic data in physical systems",
Rev. Modern Physics 65 (1993) 1331-1392.
D. Kaplan and L. Glass (1995). Understanding Nonlinear Dynamics,
E. Peters (1994) Fractal Market Analysis : Applying Chaos Theory to
Investment and Economics, Wiley