Takens' theorem

In mathematics, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes, but it does not preserve the geometric shape of structures in phase space.

Takens' theorem is the 1981 delay embedding theorem of Floris Takens. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function. Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other classes of functions.

Delay embedding theorems are simpler to state for discrete-time dynamical systems. The state space of the dynamical system is a ν-dimensional manifold M. The dynamics is given by a smooth map

$f: M \to M.$

Assume that the dynamics f has a strange attractor A with box counting dimension d_A. Using ideas from Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space with

$k > 2 d_A.\$

That is, there is a diffeomorphism φ that maps A into R^k such that the derivative of φ has full rank.

A delay embedding theorem uses an observation function to construct the embedding function. An observation function α must be twice-differentiable and associate a real number to any point of the attractor A. It must also be typical, so its derivative is of full rank and has no special symmetries in its components. The delay embedding theorem states that the function

$\phi_T(x)=\left(\alpha(x), \alpha\left(f(x)\right), \dots, \alpha\left(f^{k-1}(x)\right)\right)$

is an embedding of the strange attractor A.

References

Further reading

• N. Packard, J. Crutchfield, D. Farmer and R. Shaw (1980). "Geometry from a time series". Physical Review Letters 45 (9): 712–716. Bibcode 1980PhRvL..45..712P. doi:10.1103/PhysRevLett.45.712. 
• F. Takens (1981). "Detecting strange attractors in turbulence". In D. A. Rand and L.-S. Young. 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. Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898. Springer-Verlag. pp. 230–242. 
• G. Sugihara and R.M. May (1990). "Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series". Nature 344 (6268): 734–741. Bibcode 1990Natur.344..734S. doi:10.1038/344734a0. PMID 2330029. 
• Tim Sauer, James A. Yorke, and Martin Casdagli (1991). "Embedology". Journal of Statistical Physics 65 (3–4): 579–616. Bibcode 1991JSP....65..579S. doi:10.1007/BF01053745. 
• G. Sugihara (1994). "Nonlinear forecasting for the classification of natural time series". Phil. Trans. R. Soc. Lond. A 348 (1688): 477–495. Bibcode 1994RSPTA.348..477S. doi:10.1098/rsta.1994.0106. 
• P.A. Dixon, M.J. Milicich, and G. Sugihara (1999). "Episodic fluctuations in larval supply". Science 283 (5407): 1528–1530. Bibcode 1999Sci...283.1528D. doi:10.1126/science.283.5407.1528. PMID 10066174. 
• G. Sugihara, M. Casdagli, E. Habjan, D. Hess, P. Dixon and G. Holland (1999). "Residual delay maps unveil global patterns of atmospheric nonlinearity and produce improved local forecasts". PNAS 96 (25): 210–215. Bibcode 1999PNAS...9614210S. doi:10.1073/pnas.96.25.14210. PMC 24416. PMID 10588685. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=24416. 
• C. Hsieh; Glaser, SM; Lucas, AJ; Sugihara, G (2005). "Distinguishing random environmental fluctuations from ecological catastrophes for the North Pacific Ocean". Nature 435 (7040): 336–340. Bibcode 2005Natur.435..336H. doi:10.1038/nature03553. PMID 15902256. 

External links

• Attractor Reconstruction (scholarpedia)
• [1] Scientio's ChaosKit product uses embedding to create analyses and predictions. Access is provided online via a web service and graphic interface.