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Correlation Dimension
Define the correlation integral as
 |
(1)
|
where 
is the Heaviside
step function. When the below limit exists, the correlation dimension is then
defined as
![D_2=d_(cor)=lim_(epsilon,epsilon^'->0^+)(ln[(C(epsilon))/(C(epsilon^'))])/(ln(epsilon/(epsilon^'))).](/web/20140328162843im_/http://mathworld.wolfram.com/images/equations/CorrelationDimension/NumberedEquation2.gif) |
(2)
|
If 
is the correlation
exponent, then
 |
(3)
|
It satisfies
 |
(4)
|
where 
is the capacity
dimension and 
is the information
dimension (correcting the typo in Baker and Gollub 1996), and is conjectured
to be equal to the Lyapunov dimension.
To estimate the correlation dimension of an 
-dimensional system
with accuracy 
requires 
data points,
where
![N_(min)>=[(R(2-Q))/(2(1-Q))]^M,](/web/20140328162843im_/http://mathworld.wolfram.com/images/equations/CorrelationDimension/NumberedEquation5.gif) |
(5)
|
where 
is the length of the "plateau
region." If an attractor exists, then an estimate
of 
saturates above some 
given by
 |
(6)
|
which is sometimes known as the fractal Whitney embedding prevalence theorem.
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are (1,i), (i,-1) linearly independent?