Correlation Dimension

defined for an orbit of a map, rather than a set. wont require boxes.

Let:

$$S=\{{\vec{v}}_0,{\vec{v}}_1,\dots \}$$

be an orbit of the map $\vec{f}$ in ${\mathbb{R}}^n$ and $S_{\mathbb{N}}$ denote the first $\mathbb{N}$ points on the orbit.

For each $r>0$, define the correlation function $C(r)$ to be the proportion of pairs of orbit points withhin $r$ units of one another.

$$C(r)=\underset{n\to \infty }{\lim }\frac{\#\{pairs\{{\vec{w}}_1,{\vec{w}}_2\}:{\vec{w}}_1,{\vec{w}}_2\in S_N,|{\vec{w}}_1-{\vec{w}}_2|<r\}}{\#\{pairs\{{\vec{w}}_1,{\vec{w}}_2\}:{\vec{w}}_1,{\vec{w}}_2\in S_N\}}$$

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The correlation dimension of the orbit $S$ is $d$ where $C(r)\approx r^d$ for small $r$. i.i.

Qualitatively, in higher dimensions, there are more ways for points to be close to each toher. So the number of pairs close to each other will rise more rapidly in higher dimensions.

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Taylor Couette Flow

The attractor is time-delay reconstructed by plotting vectors of the form:

$$\underset{in{\mathbb{R}}^m}{\underbrace{\left(y_t,y_{t+T},y_{t+2T},\dots \right)}}$$

$y_t$ is velocity at the time $t$.

Technique approximates the attractor because quantities not measured are likely to depend (somehow) on velocity(which we are measuring), but lagged at some charecteristic time scale which the delay $T$ is designed to sample.

The embedding dimension, the number of degrees of freedom sufficient to describe properties of the state space via time delay.

This method can also be used to calculate the (model free) Lyapunov Exponents of data by measuring the rate of divergence of nearest neighboors and looking for consistency across embedding dimensions.