Takens's Theorem

Takens's theorem shows connection between space and time, which is an important result in dynamical system. It also relates to ergodic theory and have applications in many fields. Here we introduce the proof of Takens's theorem^[2][4] and some applications^[1][3].

box counting dimension

The Minkowski-Bouigand dimension, also known as Minkowski dimension or box-counting dimension.

Denote \(N(\varepsilon)\) is the number of boxes of side length \(\varepsilon\) required to cover the set. Then the box-counting dimension is defined as \[ \dim_{\text{box}}(S):=\lim_{\varepsilon\to 0}\frac{\log N(\varepsilon)}{\log 1 / \varepsilon} \]

If \(S\) is a smooth space (a manifold) of integer dimension \(d\), then \(N(1 / n) \thickapprox Cn^{d}\), which corresponds with the normal definition of dimension.

When the above limit does not exist, upper and lower box-counting dimensions can be defined, and they are strongly related to the Hausdorff dimension.

some definitions

First we need to clarify what is an attractor.

Def 1 Let \(\phi(t,x)\) is a flow. A positively invariant set \(A\) from a flow \(\phi(t,x)\) is a set such that if \(\phi(t_0,x) \in A\) for some \(t_0\), then \(\phi(t,x)\in A\) for all \(t\geqslant t_0\).

Def 2 A stable set from a continuous dynamical system of flow \(\phi(t,x)\) is a set such that there exists a neighborhood \(B\) of \(S\) satisfying that if \(\phi(t_0,x)\in B\), then \(\phi(t,x)\in B\) for all \(t\geqslant t_0\).

Futhermore, if there exists a neighborhood \(B\) such that, for every neighborhood \(B' \subset B\), if \(\phi(t_0,x) \in B\), then there exists \(t_1\geqslant t_0\) such that \(\phi(t_0,x)\in B'\) for every \(t\geqslant t_1\), then \(S\) is also an asymptotically stable set.

Def 3 An attracting set of an ODE is a closed, positively invariant and asymptotically stable set. An attractor of an ODE is an attracting set which contains a dense orbit.

Def 4 let \(M\) be a manifold. The set of \(C^{r}\) functions from \(M\) to itself which are also diffeomorphisms (have \(C^{r}\) inverse) is called \(Diff^{r}(M)\).

proof of Takens' embedding theorem

Theorem 1 (Takens) Let \(M\) be a compact manifold of dimension \(m\). For pairs \((\phi, y)\), with \(\phi \in Diff^{2}(M)\), \(y \in C^{2}(M, \mathbb{R})\), it is a generic property that the map \(\Phi_{(\phi, y)}\colon M \rightarrow \mathbb{R}^{2m+1}\), defined by \[ \Phi_{(\phi, y)}(x) = (y(x), y(\phi(x)), \cdots , y(\phi^{2m}(x))) \]

is an embedding. Recall Whitney embedding theorem. This mean that \(\Phi_{(\phi,y)}(M)\) is diffeomorphic to \(M\) (in the sense of \(C^{2}\))

Here 'generic' means open and dense, and we use the \(C^{1}\) topology. We refer to the functions \(y\in C^{2}(M,\mathbb{R})\) as measurement functions.

Another version focusing on one particular \(\phi\) is that:

Theorem 2 Let \(M\) be as above. Let \(\phi \colon M\rightarrow M\) be a diffeomorphism, with the properties: - the periodic points of \(\phi\) with periods less than or equal to \(2m\) are finite in number; - if \(x\) is any periodic point with period \(k\leqslant 2m\) then the eigenvalues of the derivative of \(\phi^{k}\) at \(x\) are all distinct.

Then for generic \(y \in C^{2}(M, \mathbb{R})\), the map \(\Phi_{(\phi,y)}\colon M \rightarrow \mathbb{R}^{2m+1}\), defined as in Theorem 1, is an embedding.

Reference

1. Jia-Wen Hou, Huan-Fei Ma, et al. Harvesting random embedding for high-frequency change-point detection in temporal complex systems. National Science Review, 2022. ↩︎
2. J.P.Huke. Embedding Nonlinear Dynamical Systems: A Guide to Takens' Theorem. 2006. ↩︎
3. Huanfei Ma, Siyang Leng, et al. Randomly distributed embedding making short-term high-dimensional data predictable. PNAS, 2018. ↩︎
4. Jordi Penalva Vadell. Takens' Theorem: Proof and Applications. Advanced Physics and Applied Mathematics, 2018. ↩︎