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 discretetime dynamical systems. The state space of the dynamical system is a νdimensional manifold M. The dynamics is given by a smooth map
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 kdimensional Euclidean space with
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 twicedifferentiable 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
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: 712–716. doi:10.1103/PhysRevLett.45.712.
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