Correlation Integral

Consider a set of points ${\bf X}_i$ on an Attractor, then the correlation integral is

$$C(l)\equiv \lim_{N\to\infty} {1\over N^2} f,$$

where $f$ is the number of pairs $(i,j)$ whose distance $\vert{\bf X}_i-{\bf X}_j\vert<l$. For small $l$,

$$C(l)\sim l^\nu,$$

where $\nu$ is the Correlation Exponent.

References

Grassberger, P. and Procaccia, I. ``Measuring the Strangeness of Strange Attractors.'' Physica D 9, 189-208, 1983.