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Correlation Dimension

Define the correlation integral as

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 $H$ is the Heaviside Step Function. When the below limit exists, the correlation dimension is then defined as
D_2\equiv d_{\rm cor} \equiv \lim_{\epsilon, \epsilon'\to 0^...
\end{displaymath} (2)

If $\nu$ is the Correlation Exponent, then
\lim_{\epsilon\to 0}\nu\to D_2.
\end{displaymath} (3)

It satisfies
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 $M$-dimensional system with accuracy $(1-Q)$ requires $N_{\rm min}$ data points, where

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 $D_2$ saturates above some $M$ given by
M\geq 2D+1,
\end{displaymath} (6)

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

See also Correlation Exponent, q-Dimension


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

© 1996-9 Eric W. Weisstein