Correlation Dimension

Define the correlation integral as

$\begin{displaymath} C(\epsilon) \equiv \lim_{n\to \infty} {1\over N^2} \sum_{\sc... ...tyle i\not=j}^\infty H(\epsilon-\vert\vert x_i-x_j\vert\vert), \end{displaymath}$

(1)

where

is the Heaviside Step Function. When the below limit exists, the correlation dimension is then defined as

$\begin{displaymath} D_2\equiv d_{\rm cor} \equiv \lim_{\epsilon, \epsilon'\to 0^... ...')} \right]}\over{\ln\left({\epsilon\over\epsilon'}\right)}}. \end{displaymath}$

(2)

If $\nu$ is the Correlation Exponent, then

$\begin{displaymath} \lim_{\epsilon\to 0}\nu\to D_2. \end{displaymath}$

(3)

It satisfies

$\begin{displaymath} d_{\rm cor} \leq d_{\rm inf} \leq d_{\rm cap} \hskip3pt {\rl... ...\hbox{\hskip2pt$\scriptscriptstyle ?$}}}\hskip6pt d_{\rm Lya}. \end{displaymath}$

(4)

To estimate the correlation dimension of an -dimensional system with accuracy requires $N_{\rm min}$ data points, where

$\begin{displaymath} N_{\rm min}\geq \left[{R(2-Q)\over 2(1-Q)}\right]^M, \end{displaymath}$

(5)

where $R\geq 1$ is the length of the ``plateau region.'' If an Attractor exists, then an estimate of

saturates above some

given by

$\begin{displaymath} M\geq 2D+1, \end{displaymath}$

(6)

which is sometimes known as the fractal Whitney embedding prevalence theorem.

See also Correlation Exponent, q-Dimension

References

Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. New York: Wiley, pp. 547-548, 1995.