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Towards the full Lyapunov spectrum of elementary cellular automata

Jan Baetens (UGent) and Bernard De Baets (UGent)
(2011) AIP Conference Proceedings. 1389. p.981-986
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Abstract
Throughout the decades following the postulation of cellular automata (CA) halfway the previous century, numerous studies have been conducted to gain insight into the dynamical properties of these uttermost discrete dynamical systems. Mostly, these studies were motivated by the fact that CA turned out capable of evolving intriguing spatio-temporal dynamics notwithstanding their intrinsically simple nature. Though many measures have been proposed for gaining insight into CA dynamics, the use of Lyapunov exponents that measure how their phase space trajectories evolve with respect to each other has proved particularly fruitful. Yet, to this day, in all studies relying on this measure the conclusions are drawn upon the so-called maximum Lyapunov exponent (MLE), whereas the determination of the full Lyapunov spectrum has been neglected despite its demonstrated usefulness in elucidating the dynamics of continuous dynamical systems and maps. In this preliminary study, we outline how the Lyapunov spectrum of so-called elementary CA can be obtained, we show the validity of the proposed computational methodology and, finally, we present the full spectra of some renowned elementary CA.
Keywords
Cellular automata, Lyapunov spectrum, Stability, STRANGE ATTRACTORS, DIMENSION, CHAOS, MODEL

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Chicago
Baetens, Jan, and Bernard De Baets. 2011. “Towards the Full Lyapunov Spectrum of Elementary Cellular Automata.” In AIP Conference Proceedings, ed. Theodore E Simos, George Psihoyios, Ch Tsitouras, and Zacharias Anastassi, 1389:981–986. Melvillle, NY, USA: American Institute of Physics (AIP).
APA
Baetens, Jan, & De Baets, B. (2011). Towards the full Lyapunov spectrum of elementary cellular automata. In T. E. Simos, G. Psihoyios, C. Tsitouras, & Z. Anastassi (Eds.), AIP Conference Proceedings (Vol. 1389, pp. 981–986). Presented at the International conference on Numerical Analysis and Applied Mathematics (ICNAAM), Melvillle, NY, USA: American Institute of Physics (AIP).
Vancouver
1.
Baetens J, De Baets B. Towards the full Lyapunov spectrum of elementary cellular automata. In: Simos TE, Psihoyios G, Tsitouras C, Anastassi Z, editors. AIP Conference Proceedings. Melvillle, NY, USA: American Institute of Physics (AIP); 2011. p. 981–6.
MLA
Baetens, Jan, and Bernard De Baets. “Towards the Full Lyapunov Spectrum of Elementary Cellular Automata.” AIP Conference Proceedings. Ed. Theodore E Simos et al. Vol. 1389. Melvillle, NY, USA: American Institute of Physics (AIP), 2011. 981–986. Print.
@inproceedings{8553357,
  abstract     = {Throughout the decades following the postulation of cellular automata (CA) halfway the previous century, numerous studies have been conducted to gain insight into the dynamical properties of these uttermost discrete dynamical systems. Mostly, these studies were motivated by the fact that CA turned out capable of evolving intriguing spatio-temporal dynamics notwithstanding their intrinsically simple nature. Though many measures have been proposed for gaining insight into CA dynamics, the use of Lyapunov exponents that measure how their phase space trajectories evolve with respect to each other has proved particularly fruitful. Yet, to this day, in all studies relying on this measure the conclusions are drawn upon the so-called maximum Lyapunov exponent (MLE), whereas the determination of the full Lyapunov spectrum has been neglected despite its demonstrated usefulness in elucidating the dynamics of continuous dynamical systems and maps. In this preliminary study, we outline how the Lyapunov spectrum of so-called elementary CA can be obtained, we show the validity of the proposed computational methodology and, finally, we present the full spectra of some renowned elementary CA.},
  author       = {Baetens, Jan and De Baets, Bernard},
  booktitle    = {AIP Conference Proceedings},
  editor       = {Simos, Theodore E and Psihoyios, George and Tsitouras, Ch and Anastassi, Zacharias},
  isbn         = {9780735409569},
  issn         = {0094-243X},
  keyword      = {Cellular automata,Lyapunov spectrum,Stability,STRANGE ATTRACTORS,DIMENSION,CHAOS,MODEL},
  language     = {eng},
  location     = {Halkidiki, Greece},
  pages        = {981--986},
  publisher    = {American Institute of Physics (AIP)},
  title        = {Towards the full Lyapunov spectrum of elementary cellular automata},
  url          = {http://dx.doi.org/10.1063/1.3637774},
  volume       = {1389},
  year         = {2011},
}

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