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Stability of cellular automata trajectories revisited : branching walks and Lyapunov profiles

(2016) JOURNAL OF NONLINEAR SCIENCE. 26(5). p.1329-1367
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Abstract
We study non-equilibrium defect accumulation dynamics on a cellular automaton trajectory: a branching walk process in which a defect creates a successor on any neighborhood site whose update it affects. On an infinite lattice, defects accumulate at different exponential rates in different directions, giving rise to the Lyapunov profile. This profile quantifies instability of a cellular automaton evolution and is connected to the theory of large deviations. We rigorously and empirically study Lyapunov profiles generated from random initial states. We also introduce explicit and computationally feasible variational methods to compute the Lyapunov profiles for periodic configurations, thus developing an analog of Floquet theory for cellular automata.
Keywords
GROWTH, EXPONENTS, UNIVERSALITY, PERIODICITY, BEHAVIOR, CHAOS, SEEDS, Asymptotic shape, Branching walk, Cellular automaton, Doubly periodic, configuration, Large deviations, Lyapunov exponent, Percolation, Stability

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Citation

Please use this url to cite or link to this publication:

Chicago
Baetens, Jan, and Janko Gravner. 2016. “Stability of Cellular Automata Trajectories Revisited : Branching Walks and Lyapunov Profiles.” Journal of Nonlinear Science 26 (5): 1329–1367.
APA
Baetens, Jan, & Gravner, J. (2016). Stability of cellular automata trajectories revisited : branching walks and Lyapunov profiles. JOURNAL OF NONLINEAR SCIENCE, 26(5), 1329–1367.
Vancouver
1.
Baetens J, Gravner J. Stability of cellular automata trajectories revisited : branching walks and Lyapunov profiles. JOURNAL OF NONLINEAR SCIENCE. 2016;26(5):1329–67.
MLA
Baetens, Jan, and Janko Gravner. “Stability of Cellular Automata Trajectories Revisited : Branching Walks and Lyapunov Profiles.” JOURNAL OF NONLINEAR SCIENCE 26.5 (2016): 1329–1367. Print.
@article{8510176,
  abstract     = {We study non-equilibrium defect accumulation dynamics on a cellular automaton trajectory: a branching walk process in which a defect creates a successor on any neighborhood site whose update it affects. On an infinite lattice, defects accumulate at different exponential rates in different directions, giving rise to the Lyapunov profile. This profile quantifies instability of a cellular automaton evolution and is connected to the theory of large deviations. We rigorously and empirically study Lyapunov profiles generated from random initial states. We also introduce explicit and computationally feasible variational methods to compute the Lyapunov profiles for periodic configurations, thus developing an analog of Floquet theory for cellular automata.},
  author       = {Baetens, Jan and Gravner, Janko},
  issn         = {0938-8974},
  journal      = {JOURNAL OF NONLINEAR SCIENCE},
  language     = {eng},
  number       = {5},
  pages        = {1329--1367},
  title        = {Stability of cellular automata trajectories revisited : branching walks and Lyapunov profiles},
  url          = {http://dx.doi.org/10.1007/s00332-016-9307-8},
  volume       = {26},
  year         = {2016},
}

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