Taiwanese Journal of Mathematics


Juei-Ling Ho

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Shih and Ho have proved a global convergent theorem for boolean network: if a map from $\{0,1\}^{n}$ to itself defines a boolean network has the conditions: (1) each column of the discrete Jacobian matrix of each element of $\{0,1\}^{n}$ is either a unit vector or a zero vector; (2) all the boolean eigenvalues of the discrete Jacobian matrix of this map evaluated at each element of $\{0,1\}^{n}$ are zero, then it has a unique fixed point and this boolean network is global convergent to the fixed point. The purpose of this paper is to give a global convergent theorem for XOR boolean network, it is a counterpart of the global convergent theorem for boolean network.

Article information

Taiwanese J. Math., Volume 13, Number 4 (2009), 1271-1282.

First available in Project Euclid: 18 July 2017

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Zentralblatt MATH identifier

Primary: 37E15: Combinatorial dynamics (types of periodic orbits) 37E25: Maps of trees and graphs 68R99: None of the above, but in this section

global convergent theorem boolean network discrete Jacobian matrix boolean eigenvalue fixed point XOR boolean network


Ho, Juei-Ling. GLOBAL CONVERGENCE FOR THE XOR BOOLEAN NETWORKS. Taiwanese J. Math. 13 (2009), no. 4, 1271--1282. doi:10.11650/twjm/1500405507. https://projecteuclid.org/euclid.twjm/1500405507

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  • [1.] F. Robert, Théorèmes de Perron-Frobenius et Stein-Rosenberg booléens, Linear Algebra Appl., 19 (1978), 237-250.
  • [2.] F. Robert, Itérations sur des ensembles finis et automates cellulaires contractants, Linear Algebra Appl., 29 (1980), 393-412.
  • [3.] F. Robert, Dérivée discriye et convergence local d'une iteration booléenne, Linear Algebra Appl., 52 (1983), 547-589.
  • [4.] F. Robert, Les systèmes dynamiques discrets, volume 19 of Mathématiques et Applications, Springer, 1995.
  • [5.] F. Robert, Basic results for the behaviour of discrete iterations, NATO ASI Series in Systems and Computer Science, F20, Springer Verlag, Berlin, Heidelberg, New York, 1986, pp. 33-47.
  • [6.] F. Robert, Discrete Iterations, Springer Verlag, Berlin, Heidelberg, New York, 1986.
  • [7.] M.-H. Shih and J.-L. Dong, A Combinatorial Analogue of the Jacobian Problem in Automata Networks, Advances in Applied Mathematics, 34 (2005), 30-46.
  • [8.] M.-H. Shih and Juei-Ling Ho, Solution of the Boolean Markus-Yamabe problem, Advances in Applied Mathematics, 22 (1999), 60-102.
  • [9.] F. F. Soulié, Boolean networks, in: Automata networks in computer science, Princeton University Press, New Jersey, 1987, pp. 20-34.