Wolfram’s cellular automata [1986] can be classified according to their asymptotic behavior: class I (homogeneous), class II (periodic), class III (chaotic) and class IV (“undecidable”, i.e. erratically changing between periodicity and chaos). While these automata are purely number-theoretical and suffer from ill-defined parametrization, we present here examples of automata describing actual physical systems and governed by well-defined control parameters: resting states between earthquakes, pigmentation on the shells of molluscs, and two-dimensional reaction–diffusion systems. We find that the dynamics of these three systems can be classified analogously to Wolfram’s automata. Moreover, we find agreement between class IV simulations and real shell patterns, indicating that these shells indeed present evidence for class IV behavior in nature. In addition, we performed an intuitively appealing quantification by averaging the fluctuations of the borders of error-propagation patterns.
EVOLVING LOCALIZATIONS IN REACTION-DIFFUSION CELLULAR AUTOMATA
