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Cycle with Transient-like Appearance

Here is an interesting CA using an algebraic expression as rule.
With the given initial conditions it has no transient, and the cycle is 2556 steps long.

But the entire cycle looks like a transient in that it has the dendrites throughout that, with other initial conditions for this rule, are usually seen only in the transient.

In the attached figure the graph is divided into six columns each 426 steps. Notice that each column is a repeat of the previous column, only shifted right 6 cells. This is no artifact. That is the way the CA presents.

The rule is Mod[-(-2 + p)r + q(2 - 2r + s), 3].

The initial conditions are:

{2,0,0,0,0,0,0,0,0, 0,1,0,0,2,0,2,0,2, 0,0,0,2,0,2,0,0,0, 0,0,0,0,0,0,2,0,1}.

With this rule the values for p, q, r, and s are supplied by the two left and right neighbors of a cell. The algebraic expression is then evaluated and the result is stored as the updated version of the cell. The cell’s previous value does not enter into the calculation.

__________________
L. J. Thaden

I found a cycle exactly half that size - that is, it repeats exactly twice in 2556 steps.

I was using the following as my rule, with your p and q the same as mine, my r meaning "center cell" and not used in the rule, here, and my s and t equivalent to your r and s.

newrule = MapThread[
    Rule, {x, Table[Mod[-(-2 + p[i])
      s[i] + q[i](2 - 2s[i] + t[i]), 3], {i, 1, Length[x]}]}];


I am applying that to a partition into blocks 5 wide etc.

My findCycle (the ugly one) returns a cycle length of 1278, transient length of 0.

There do seem to be great similarities on smaller subcycles - the 6 arrays 426 long next to each other look close in many details. As though it has 3 slightly different phases of the same subcycle, that have to "go around" several times to hit exactly in phase, and thus exactly duplicate a previous step, and therefore properly be said to cycle in the strictest sense.

What I like about these large cycles is they illustrate the point that eventually periodic does not exactly correspond to simple. Kovas Bogata put the thought experiment to me once, if you started a simple rule from a relatively simple seed, and it builds something incredibly intricate - say the Taj Mahal - then goes back to its starting point and does it again - is it adequately described as "class 2 simple behavior"?