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Chaos / Volume 21 / Issue 2 / REGULAR ARTICLES

Chaos 21, 023108 (2011); http://dx.doi.org/10.1063/1.3535581 (12 pages)

On the topological sensitivity of cellular automata

Jan M. Baetens and Bernard De Baets

KERMIT, Department of Applied Mathematics, Biometrics and Process Control, Ghent University, Coupure links 653, Gent, Belgium

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(Received 20 October 2010; accepted 15 December 2010; published online 19 April 2011)

Ever since the conceptualization of cellular automata (CA), much attention has been paid to the dynamical properties of these discrete dynamical systems, and, more in particular, to their sensitivity to the initial condition from which they are evolved. Yet, the sensitivity of CA to the topology upon which they are based has received only minor attention, such that a clear insight in this dependence is still lacking and, furthermore, a quantification of this so-called topological sensitivity has not yet been proposed. The lack of attention for this issue is rather surprising since CA are spatially explicit, which means that their dynamics is directly affected by their topology. To overcome these shortcomings, we propose topological Lyapunov exponents that measure the divergence of two close trajectories in phase space originating from a topological perturbation, and we relate them to a measure grasping the sensitivity of CA to their topology that relies on the concept of topological derivatives, which is introduced in this paper. The validity of the proposed methodology is illustrated for the 256 elementary CA and for a family of two-state irregular totalistic CA.

© 2011 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. CELLULAR AUTOMATA ON IRREGULAR TESSELLATIONS
    1. Definition
    2. Subdividing the Euclidean plane
    3. The neighborhood function N
    4. Dynamics governing function ϕ i
  3. GRASPING THE TOPOLOGICAL SENSITIVITY OF CELLULAR AUTOMATA
    1. On topological perturbations of cellular automata
    2. Lyapunov exponents
      1. Classical formulation
      2. Topological Lyapunov exponents
    3. Jacobians
      1. Classical Jacobians
      2. Topological Jacobians
        1. Topological derivatives and Jacobians of totalistic CA.
        2. Topological Jacobians of order-variant CA.
    4. Relating topological topological Lyapunov exponents and Jacobians
  4. ASSESSING THE TOPOLOGICAL SENSITIVITY OF CA
    1. Conventions
    2. Topological Jacobians of elementary CA
    3. Unraveling the topological sensitivity of 2D (2, 7) irregular totalistic CA
  5. CONCLUSIONS

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KEYWORDS and PACS

Keywords

cellular automata, nonlinear dynamical systems, topology

PACS

  • 05.45.-a

    Nonlinear dynamics and chaos

  • 05.50.+q

    Lattice theory and statistics (Ising, Potts, etc.)

  • 02.40.Re

    Algebraic topology

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3535581

PUBLICATION DATA

ISSN

1054-1500 (print)  
1089-7682 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    Bagnoli, F. and Rechtman, R., “Synchronization and maximum Lyapunov exponents of cellular automata,” Phys. Rev. E 59, R1307–R1310 (1999).

    Bagnoli, F. and Rechtman, R., “Thermodynamic entropy and chaos in a discrete hydrodynamical system,” Phys. Rev. E 79, 041115 (2009).

    Urías, J., Rechtman, R., and Enciso, A., “Sensitive dependence on initial conditions for cellular automata,” Chaos 7, 688–693 (1997)CHAOEH000007000004000688000001.

    Wolfram, S., “Statistical mechanics of cellular automata,” Rev. Mod. Phys. 55, 601–644 (1983).


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