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Attractor reconstruction (changes)

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Contents

Idea

According to Scholarpedia:

Attractor reconstruction refers to methods for inference of geometrical and topological information about a dynamical attractor from observations.

Details

Wikipedia has an article on 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.

In the recent book by Broer and Takens they claim that the reconstruction theorem was discovered independently by D. Aeyels and Takens.

Applications

Both as a basis for nonlinear time series in order to distinguish randomness from deterministic chaos caused by bifurcations. It has been refined to work with both continuous and discrete maps in many dimensions. Also used to predict when nonlinear becomes chaotic or to find tipping points in climate, ecology and many other sciences

References