Abstract
Although cellular automata (CAs) have been invented only about three quarter of a century ago, much has been written about the fascinating space-time patterns these utter discrete dynamical systems can bring forth, which can be largely attributed to the fact that it is striking to notice that the dynamics of these overly simple dynamical systems can be so intriguing. Driven by the advances in the theory of dynamical systems of continuous dynamical systems, several quantitative measures have been proposed to grasp the stability of CAs, among which the Lyapunov exponent (LE) has received particular attention. Originally, the latter was understood as the Hamming distance between configurations evolved by the same CA from different initially close configurations, but it suffers from the important drawback that it can grow only linearly over time. In this paper, it will be shown how one should determine the LE of a two-state CA, in such a way that its meaning is in line with the framework of continuous dynamical systems. Besides, the proposed methodology will be exemplified for two CA families.
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References
Baetens, J.M., De Baets, B.: Phenomenological study of irregular cellular automata based on Lyapunov exponents and Jacobians. Chaos 20, 33112 (2010)
Baetens, J.M., De Baets, B.: Towards the full Lyapunov spectrum of elementary cellular automata. In: American Institute of Physics Conference Proceedings, Halkidiki, Greece, vol. 1389, p. 981 (2011)
Baetens, J.M., Van der Weeën, P., De Baets, B.: Effect of asynchronous updating on the stability of cellular automata. Chaos, Solitons and Fractals 45, 383–394 (2012)
Bagnoli, F., Rechtman, R.: Thermodynamic entropy and chaos in a discrete hydrodynamical system. Physical Review EÂ 79, 041115 (2009)
Bagnoli, F., Rechtman, R., Ruffo, S.: Damage spreading and Lyapunov exponents in cellular automata. Physics Letters A 172, 34–38 (1992)
Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Reviews of Modern Physics 57, 617–656 (1985)
Gardner, M.: Mathematical games: The fantastic combinations of John Conway’s new solitaire game ‘Life’. Scientific American 223, 120–123 (1970)
Ilachinski, A. (ed.): Cellular Automata. A Discrete Universe. World Scientific, London (2001)
Langton, C.: Computation at the edge of chaos. Physica D 42, 12–37 (1990)
Lyapunov, A.M.: The general problem of the stability of motion. Taylor & Francis, London (1992)
Moore, E.F.: Machine models of self reproduction. In: Bellman, R.E. (ed.) Proceedings of the 14th Symposium in Applied Mathematics, New York, United States, pp. 17–33 (1992)
Oseledec, V.I.: A multiplicative ergodic theorem. Lyapunov characteristics numbers for dynamical systems. Transactions of the Moscow Mathematical Society 19, 197–231 (1968)
Schrandt, R.G., Ulam, S.M.: On recursively defined geometrical objects and patterns of growth. Technical Report LA-3762, Los Alamos Scientific Laboratory (1968)
Ulam, S.M.: On some mathematical problems connected with patterns of growth of figures. In: Bellman, R.E. (ed.) Proceedings of the 14th Symposium in Applied Mathematics, New York, United States, pp. 215–224 (1968)
Vichniac, G.: Boolean derivatives on cellular automata. Physica D 45, 63–74 (1990)
von Neumann, J.: The general and logical theory of automata. In: Jeffres, L.A. (ed.) The Hixon Symposium on Cerebral Mechanisms in Behaviour, pp. 1–41. John Wiley & Sons, Pasadena (1951)
von Neumann, J.: Theory of Self-Reproducing Automata. University of Illinois Press, Urbana (1966)
Wolfram, S.: Universality and complexity in cellular automata. Physica D 10, 1–35 (1984)
Wolfram, S.: A New Kind of Science. Wolfram Media, Inc., Champaign (2002)
Wuensche, A., Lesser, M.: The Global Dynamics of Cellular Automata, vol. 1. Addison-Wesley, London (1992)
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Baetens, J.M., De Baets, B. (2013). A Lyapunov View on the Stability of Two-State Cellular Automata. In: Zenil, H. (eds) Irreducibility and Computational Equivalence. Emergence, Complexity and Computation, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35482-3_3
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DOI: https://doi.org/10.1007/978-3-642-35482-3_3
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