Sometimes, through either ignorance of the variables in the system state vector or technological limitations on sensor technologies, experimentalists only get to see a onedimensional time series y(t)=Φ(x(t))y(t)=Φ(x(t)), where Φ:RN→RΦ:RN→R is a smooth observation or measurement function.
Note that when the evolution of the system states are known (i.e., we have knowledge of ΨΨ Equation 1), the system states can be recovered over time from the time series measurements by the Kalman filter [3].
But without this knowledge, can information about the system state x(t)x(t) be retained in this time series data y(t)y(t)?
Remarkably, Takens' theorem [5] answers this in the positive.
Takens first defined the delay coordinate map with MM delays F(Φ,GTs):M→RMF(Φ,GTs):M→RM by stacking MM previous entries of the time series y(t)y(t) sampled uniformly in time with sampling time TsTs up into a vector:
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(6)When it is clear, we will drop the subscripts (Φ,GTs)(Φ,GTs) from F(Φ,GTs)F(Φ,GTs).
FF is thus a mapping from the ambient space M⊂RNM⊂RN where the system state resides to a reconstruction spaceRMRM formed with the time series measurement.
When a dynamical system is confined to a manifold MM, Takens showed that given certain conditions on the time(Ts)(Ts) map, GTsGTs, the delay coordinate map FF is an embedding of MM onto the reconstruction space for almost every choice of measurement function ΦΦ.
The following theorem, first described in [5], gives the full details:
Theorem 3.1 (Takens' Embedding Theorem)
Let MM be a compact manifold of dimension KK and suppose we have a dynamical system defined by Equation 1 that is confined on this manifold.
Let M>2KM>2K and suppose:
 the periodic points of GTsGTs with periods less than or equal to 2K2K are finite in number, and
 GTsGTs has distinct eigenvalues on any such periodic points.
Then the observation functions ΦΦ for which the delay coordinate map FF Equation 6 is an embedding form an open and dense subset of C2M,RC2M,R.
Figure 1 illustrates the process of forming a delay coordinate map FF with M=3M=3 delays and portrays how FF is consequently an embedding of the system manifold MM. In the theorem Ck(Ω1,Ω2)Ck(Ω1,Ω2) refers the space of all functions f:Ω1→Ω2f:Ω1→Ω2 whose kkth derivative is continuous.
By an embedding, we mean that the operator FF is a onetoone immersion.
First, FF being onetoone means that distinct system states are not mapped to the same point in the reconstruction space.
Second, an immersion means that the differential operator at any point xx in the state space, DxFDxF, is itself a onetoone map.
The theorem make clear the meaning of almost every; basically the “good” observation functions form an open and dense subset of C2M,RC2M,R.
Let us give an example of why the extra conditions on periodic points are necessary.
Suppose MM is a limit cycle (or more simply a circle).
The conditions of the theorem for this manifold dictates that all periodic points of MM, at least of period 1, be finite in number.
Now suppose for an unfortunate choice of TsTs, we have GTs(x)=xGTs(x)=x for all x∈Mx∈M, meaning that the set of all periodic points of period 1 is the whole manifold itself.
Then no matter what observation function or however many delays MM we pick, the delay coordinate map will have the form F(x)=[Φ(x),Φ(x),⋯,Φ(x)]TF(x)=[Φ(x),Φ(x),⋯,Φ(x)]T for any x∈Mx∈M.
This implies that the limit cycle will be mapped onto a line in the reconstruction space, thus violating the onetoone property.
Summing up, this theorem says that if the conditions about periodic points on the dynamical systems are fulfilled, then for any function Φ'∈C2M,RΦ'∈C2M,R, there will be a function ΦΦ in an arbitrarily small neighborhood around Φ'Φ' such that the delay coordinate map F(Φ,GTs)F(Φ,GTs) is an embedding.
This means that for a large class of observation functions ΦΦ, FF preserves the topology of MM, therefore information about MM can be retained in the time series (measurements) output!
Indeed by preserving the topology of the manifold in the reconstruction space, many properties of the manifold and the dynamical system are retained, including its dimensionality and its lyapunov exponents  just to name a few.