The recent posts on cellular automata have inspired me to dig up some work I had done on continuous variable cellular automata. The goal was to investigate phase changes. It turned out the math is a bit intractable, but the pictures are interesting. I used the standard 1D CA with 3 inputs per daughter state. The initial step is much like that proposed by kram1032
http://www.fractalforums.com/general-discussion-b77/probability-based-fuzzy-cellular-automata/. Instead of using the polynomials for stochastic updates, I performed a sum of powers followed by a tanh to produce the final value.
a = state[i+1];
b = state[i];
c = state[i-1];
if(rule&1) E = E+pow((1-a)*(1-b)*(1-c),power);
if(rule&2) E = E+pow((1-a)*(1-b)*(c ),power);
if(rule&4) E = E+pow((1-a)*(b )*(1-c),power);
if(rule&8) E = E+pow((1-a)*(b )*(c ),power);
if(rule&16) E = E+pow((a )*(1-b)*(1-c),power);
if(rule&32) E = E+pow((a )*(1-b)*(c ),power);
if(rule&64) E = E+pow((a )*(b )*(1-c),power);
if(rule&128)E = E+pow((a )*(b )*(c ),power);
next_state[i] = tanh(E*beta);
This gives two free variables to play with. In the limiting case where both power and beta are large the original discrete CA is recovered. Interesting things happen when power and beta are in the 2-4 range. The CA tends to melt into another phase that is not present in the discrete CA.
The first simulation result uses rule 105. 105 has a stable chaotic phase as well as a few stationary but unstable phases. For example an alternating all 1s and 0s phase and a polka doted phase. With the proper parameters a third melted phase is introduced. The melted phase tends to yield to the alternating phase. The alternating phase yields to the chaotic phase. And finally the melted phase tends to invade the chaotic phase from the sides. (Time flows from the bottom up.)
The next two images use rule 94 and 179. These two rules tend to produce boring low entropy phases, but the addition of a melt creates a chaotic battle between the boring phase and the melted phase.
March 18, 2010, 07:29:03 PM
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