Takens' theorem
Encyclopedia
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
Assume that the dynamics f has a strange attractor A with box counting dimension dA. Using ideas from Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space
with
That is, there is a diffeomorphism
φ that maps A into Rk 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
is an embedding of the strange attractor A.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a delay embedding theorem gives the conditions under which a chaotic
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
dynamical system
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
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
Floris Takens
Floris Takens was a Dutch mathematician known for contributions to the theory of chaotic dynamical systems.Together with David Ruelle he predicted that fluid turbulence could develop through a strange attractor, a term they coined, as opposed to the then-prevailing theory of accretion of modes....
. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic
Baire space
In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
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
Assume that the dynamics f has a strange attractor A with box counting dimension dA. Using ideas from Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
with
That is, there is a diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
φ that maps A into Rk such that the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of φ has full rank
Rank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
.
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
Baire space
In mathematics, a Baire space is a topological space which, intuitively speaking, is very large and has "enough" points for certain limit processes. It is named in honor of René-Louis Baire who introduced the concept.- Motivation :...
, 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.
External links
- Attractor Reconstruction (scholarpedia)
- http://www.scientio.com/Products/ChaosKit Scientio's ChaosKit product uses embedding to create analyses and predictions. Access is provided online via a web service and graphic interface.