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Coexistence of Periods in Parallel and Sequential Boolean Graph Dynamical Systems over Directed Graphs

Author

Listed:
  • Juan A. Aledo

    (Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain)

  • Luis G. Diaz

    (Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain)

  • Silvia Martinez

    (Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain)

  • Jose C. Valverde

    (Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain)

Abstract

In this work, we solve the problem of the coexistence of periodic orbits in homogeneous Boolean graph dynamical systems that are induced by a maxterm or a minterm (Boolean) function, with a direct underlying dependency graph. Specifically, we show that periodic orbits of any period can coexist in both kinds of update schedules, parallel and sequential. This result contrasts with the properties of their counterparts over undirected graphs with the same evolution operators, where fixed points cannot coexist with periodic orbits of other different periods. These results complete the study of the periodic structure of homogeneous Boolean graph dynamical systems on maxterm and minterm functions.

Suggested Citation

  • Juan A. Aledo & Luis G. Diaz & Silvia Martinez & Jose C. Valverde, 2020. "Coexistence of Periods in Parallel and Sequential Boolean Graph Dynamical Systems over Directed Graphs," Mathematics, MDPI, vol. 8(10), pages 1-14, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1812-:d:429564
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    References listed on IDEAS

    as
    1. Barrett, Chris L & Chen, William Y.C & Zheng, Michelle J, 2004. "Discrete dynamical systems on graphs and Boolean functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 66(6), pages 487-497.
    2. Chiaselotti, G. & Gentile, T. & Oliverio, P.A., 2014. "Parallel and sequential dynamics of two discrete models of signed integer partitions," Applied Mathematics and Computation, Elsevier, vol. 232(C), pages 1249-1261.
    3. Juan A. Aledo & Luis G. Diaz & Silvia Martinez & Jose C. Valverde, 2017. "On the Periods of Parallel Dynamical Systems," Complexity, Hindawi, vol. 2017, pages 1-6, January.
    4. Toroczkai, Zoltán & Guclu, Hasan, 2007. "Proximity networks and epidemics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(1), pages 68-75.
    5. Aledo, Juan A. & Diaz, Luis G. & Martinez, Silvia & Valverde, Jose C., 2019. "Solution to the predecessors and Gardens-of-Eden problems for synchronous systems over directed graphs," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 22-28.
    6. Sara D. Cardell & Amparo Fúster-Sabater, 2019. "Binomial Representation of Cryptographic Binary Sequences and Its Relation to Cellular Automata," Complexity, Hindawi, vol. 2019, pages 1-13, March.
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