Takens's theorem explained

In the study of dynamical systems, 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 that system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms), 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.

It is the most commonly used method for attractor reconstruction.[1]

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

f:M\toM.

A\subM

with box counting dimension . Using ideas from Whitney's embedding theorem, can be embedded in -dimensional Euclidean space with

k>2dA.

That is, there is a diffeomorphism that maps into

\Rk

such that the derivative of has full rank.

A delay embedding theorem uses an observation function to construct the embedding function. An observation function

\alpha:M\to\R

must be twice-differentiable and associate a real number to any point of the attractor . 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

\varphiT(x)=l(\alpha(x),\alpha(f(x)),...,\alpha(fk-1(x))r)

is an embedding of the strange attractor in

\Rk.

Simplified version

Suppose the

d

-dimensional state vector

xt

evolves according to an unknown but continuousand (crucially) deterministic dynamic. Suppose, too, that theone-dimensional observable

y

is a smooth function of

x

, and “coupled”to all the components of

x

. Now at any time we can look not just atthe present measurement

y(t)

, but also at observations made at timesremoved from us by multiples of some lag

\tau:yt+\tau,yt+2\tau

, etc. If we use

k

lags, we have a

k

-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

k\toinfty

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

2d+1

, and the minimal embedding dimension is often less.[2] [3]

Choice of delay

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]

See also

Further reading

External links

Notes and References

  1. Sauer . Timothy D. . 2006-10-24 . Attractor reconstruction . Scholarpedia . en . 1 . 10 . 1727 . 10.4249/scholarpedia.1727 . 1941-6016 . free . 2006SchpJ...1.1727S .
  2. Book: Shalizi. Cosma R.. Deisboeck. ThomasS. Kresh. J.Yasha. Complex Systems Science in Biomedicine. limited. 2006. Springer US. 978-0-387-30241-6. 33–114. Methods and Techniques of Complex Systems Science: An Overview. 10.1007/978-0-387-33532-2_2. Topics in Biomedical Engineering International Book Series. nlin/0307015. 11972113.
  3. Barański . Krzysztof . Gutman . Yonatan . Śpiewak . Adam . 2020-09-01 . A probabilistic Takens theorem . Nonlinearity . 33 . 9 . 4940–4966 . 10.1088/1361-6544/ab8fb8 . 1811.05959 . 2020Nonli..33.4940B . 119137065 . 0951-7715.
  4. Book: Strogatz, Steven . Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering . 2015 . 978-0-8133-4910-7 . Second . Boulder, CO . 12.4 Chemical chaos and attractor reconstruction . 842877119.
  5. Fraser . Andrew M. . Swinney . Harry L. . 1986-02-01 . Independent coordinates for strange attractors from mutual information . subscription . Physical Review A . 33 . 2 . 1134–1140 . 10.1103/PhysRevA.33.1134. 9896728 . 1986PhRvA..33.1134F .

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Takens's theorem".

Except where otherwise indicated, Everything.Explained.Today is © Copyright 2009-2024, A B Cryer, All Rights Reserved. Cookie policy.