Computing Correlation Dimension of Attractor | Hi, Explain?

[penguin007]

Hi,

I read that the dimension of an attractor is not necessarily an integer. I then tried to compute the dimension of the attractor defined by the system I’m studying but I found two definitions for the dimension: -The capacity dimension;
-The correlation dimension;
The second one is said to be easier to compute. However, I couldn’t do it because:
*the definitions I found lean on the existence of an explicit function, but I just have points given by the approached resolution of a differential system;
*they introduce a dimension m and then a limit when m tends to infinity (?).

Could anyone explain me what is the correlation dimension and how I can compute it without an explicit function??

Thanks in advance.

 

[Valerie Park]

The correlation dimension is a measure of the fractal structure of an attractor. It is determined by calculating the correlation integral of the attractor, which measures the number of pairs of points whose distance is less than r. This correlation integral can be used to calculate the correlation dimension, which is the slope of the log-log plot of the correlation integral versus r. The correlation dimension is related to the fractal dimension, which is the degree of self-similarity of the attractor. To compute the correlation dimension without an explicit function, you will need to approximate the correlation integral directly from the data points given by the approached resolution of the differential system. For example, you can use the method developed by Grassberger and Procaccia (1983) to calculate the correlation integral. This method involves counting the number of pairs of points that are within a certain distance (r) of each other and then summing these up over a range of distances. The result is then used to calculate the correlation dimension.