Rössler attractor reconstructed by Takens' theorem, using different delay lengths. Orbits around the attractor have a period between 5.2 and 6.2.
In the study ofdynamical systems, adelay embedding theorem gives the conditions under which achaotic dynamical system can be reconstructed from a sequence of observations of the state of that system. The reconstruction preserves the properties of the dynamical system that do not change under smoothcoordinate changes (i.e.,diffeomorphisms), but it does not preserve thegeometric shape of structures inphase space.
Takens' theorem is the 1981 delayembedding theorem ofFloris Takens. It provides the conditions under which a smoothattractor can be reconstructed from the observations made with ageneric function. Later results replaced the smooth attractor with a set of arbitrarybox counting dimension and the class of generic functions with other classes of functions.
It is the most commonly used method forattractor reconstruction.[1]
A delay embedding theorem uses anobservation function to construct the embedding function. An observation function must be twice-differentiable and associate a real number to any point of the attractorA. It must also betypical, so its derivative is of full rank and has no special symmetries in its components. The delay embedding theorem states that the function
Suppose the-dimensional state vector evolves according to an unknown but continuousand (crucially) deterministic dynamic. Suppose, too, that theone-dimensional observable is a smooth function of, and “coupled”to all the components of. Now at any time we can look not just atthe present measurement, but also at observations made at timesremoved from us by multiples of some lag, etc. If we use lags, we have a-dimensional vector. One might expect that, as thenumber of lags is increased, the motion in the lagged space will becomemore and more predictable, and perhaps in the limit would becomedeterministic. In fact, the dynamics of the lagged vectors becomedeterministic at a finite dimension; not only that, but the deterministicdynamics are completely equivalent to those of the original state space (precisely, they are related by a smooth, invertible change of coordinates,or diffeomorphism). In fact, the theorem says that determinism appears once you reach dimension, and the minimalembedding dimension is often less.[2][3]
Takens' theorem is usually used to reconstruct strange attractors out of experimental data, for which there is contamination by noise. As such, the choice of delay time becomes important. Whereas for data without noise, any choice of delay is valid, for noisy data, the attractor would be destroyed by noise for delays chosen badly.
The optimal delay is typically around one-tenth to one-half the mean orbital period around the attractor.[4][5]
^Strogatz, Steven (2015). "12.4 Chemical chaos and attractor reconstruction".Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (Second ed.). Boulder, CO.ISBN978-0-8133-4910-7.OCLC842877119.{{cite book}}: CS1 maint: location missing publisher (link)
F. Takens (1981). "Detecting strange attractors in turbulence". In D. A. Rand andL.-S. Young (ed.).Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898. Springer-Verlag. pp. 366–381.
R. Mañé (1981). "On the dimension of the compact invariant sets of certain nonlinear maps". In D. A. Rand and L.-S. Young (ed.).Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898. Springer-Verlag. pp. 230–242.
R. A. Rios, L. Parrott, H. Lange and R. F. de Mello (2015). "Estimating determinism rates to detect patterns in geospatial datasets".Remote Sensing of Environment.156:11–20.Bibcode:2015RSEnv.156...11R.doi:10.1016/j.rse.2014.09.019.{{cite journal}}: CS1 maint: multiple names: authors list (link)
