Recall a CA rule project a circular-boundary state space back to it self, that is

where , and more strictly , for simplicity we let , thus

In practical, we take n=20^2=400;

Consider where

In practical, we sought a big enough representative subset, from , so that

Denote , we have

,

We then define a profile with Pearson correlation so that

, where , denotes standard deviation of X and Cov[X,Y] denotes covariance between X and Y.

We assert that classify the dynamics of the CA rule by capturing the periodic behavior in the underlying dynamics.

Importantly, we can find a good by taking a random and obtain

where T is a variable parameter.