In the study of
dynamical systems
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
, 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.,
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
s), but it does not preserve the
geometric shape
A shape is a graphical representation of an object's form or its external boundary, outline, or external surface. It is distinct from other object properties, such as color, texture, or material type.
In geometry, ''shape'' excludes informat ...
of structures in
phase space
The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
.
Takens' theorem is the 1981 delay
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup.
When some object X is said to be embedded in another object Y ...
theorem of
Floris Takens. It provides the conditions under which a smooth
attractor
In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain c ...
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.
Delay embedding theorems are simpler to state for
discrete-time dynamical systems.
The state space of the dynamical system is a -dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
. The dynamics is given by a
smooth map
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
:
Assume that the dynamics has a
strange attractor
In the mathematics, mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor va ...
with
box counting dimension . Using ideas from
Whitney's embedding theorem, can be embedded in -dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
with
:
That is, there is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
that maps into
such that the
derivative
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
of has full
rank
A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial.
People Formal ranks
* Academic rank
* Corporate title
* Diplomatic rank
* Hierarchy ...
.
A delay embedding theorem uses an ''observation function'' to construct the embedding function. An observation function
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
:
is an
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup.
When some object X is said to be embedded in another object Y ...
of the strange attractor in
Simplified version
Suppose the
-dimensional
state vector
evolves according to an unknown but continuous
and (crucially) deterministic dynamic. Suppose, too, that the
one-dimensional observable
is a smooth function of
, and “coupled”
to all the components of
. Now at any time we can look not just at
the present measurement
, but also at observations made at times
removed from us by multiples of some lag
, etc. If we use
lags, we have a
-dimensional vector. One might expect that, as the
number of lags is increased, the motion in the lagged space will become
more and more predictable, and perhaps in the limit
would become
deterministic. In fact, the dynamics of the lagged vectors become
deterministic at a finite dimension; not only that, but the deterministic
dynamics 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
, and the minimal ''embedding dimension'' is often less.
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.
See also
*
Whitney embedding theorem
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:
*The strong Whitney embedding theorem states that any smooth real - dimensional manifold (required also to be Hausdorf ...
*
Nonlinear dimensionality reduction
Nonlinear dimensionality reduction, also known as manifold learning, is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which cannot be adequately captured by linear de ...
References
Further reading
*
*
*
*
*
*
*
*
*
* {{cite journal
, journal = Remote Sensing of Environment
, year = 2015
, title = Estimating determinism rates to detect patterns in geospatial datasets
, pages = 11–20
, author = R. A. Rios, L. Parrott, H. Lange and R. F. de Mello
, volume = 156
, doi = 10.1016/j.rse.2014.09.019, bibcode = 2015RSEnv.156...11R
External links
Scientio's ChaosKit product uses embedding to create analyses and predictions. Access is provided online via a web service and graphic interface.
Empirical Dynamic Modelling tools pyEDM and rEDM use embedding for analyses, prediction, and causal inference.
Theorems in dynamical systems