Long Memory Properties and Complex Systems
Thus , in order to study the conditions where the system is stationary , it is useful to rewrite the previous expression as a difference equation in terms of i , as it follows :
Which has the following system as solution :
As there are no possible negative values , it follows that :
Solving the following simplified system , one obtains :
Hence :
So , for example , if x i , 0
= 0 , without any noise and , with only one x k , 0
= 1 | k ∊ { 0 , N }, the system will remain at its stationary state . The same happens if all the cells are initialized with 0 . Moreover , it is possible to verify that all stationary solutions are those where the cells are
initialized as triplets of the form “ 1 _ 0 ”.
From the stationary state with S ( t ) = 1 ( in other words , with only one black cell ), when a very low level of noise is added , the initial state of the system is preserved with a slow convergence towards the limit density , as verified in Figure 13 . If there is no noise , nothing happens to the system .
However , it is also interesting to notice that if all cells are initialized as one , except for only one cell , being equal to 0 , the long memory behavior becomes more evident , when no noise is present in the system .
On the other hand , when analyzing rule 182 , it is possible to derive the following transition rule :
Repeating the same previous steps , as done for rule 106 , it is possible to notice that :
Consequently , it is important to verify that the first term of the previous equation given by :
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