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.
A minimum cellular automaton, carrying precise biophysical significance in each rule, is presented to model pigmentation patterns on molluscan shells. We find the following types of modes: self-organisation into stationary (Turing) structures, travelling waves, chaos, and so-called class IV behaviour. The latter consists of a disordered spatio-temporal distribution of periodic and chaotic patches; it differs from chaos in that it has no well-defined error propagation rate. The calculations of the modes agree well with observations in natural shells. In particular, our results suggest evidence in nature for class IV behaviour, a mode that had so far been reported only as the result of simulations. Moreover, we show that patchiness results in a class IV mode from the same algorithm that renders chaos and periodicity; thus, there is no need to invoke two competing pattern generators, as in previous approaches.
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