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May  2017, 11(2): 283-287. doi: 10.3934/amc.2017019

Determining steady state behaviour of discrete monomial dynamical systems

Dorothy Bollman , and  Omar Colón-Reyes

Department of Mathematical Sciences, University of Puerto Rico at Mayagüez, Mayagüez, Puerto Rico 00681-9018, USA

* Corresponding author

Received  February 2016 Revised  March 2016 Published  May 2017

In previous work [3] Colón-Reyes et al developed criteria for determining when a discrete monomial dynamical system reaches steady state behaviour. These criteria depend on determining when a certain matrix over a finite ring, that is not a field, defines a fixed point system. It was not until recently that criteria to determine linear steady state behaviour over rings have been found. Using these new results we present a new algorithm to determine steady state behaviour of monomial dynamical systems over finite fields. Delgado-Eckert [5] has also obtained an algorithm for the finite field case, but his algorithm does not take into account the result in [3] and requires $O(n^4\; q^2 \log\; q)$ integer operations. Our algorithm requires only $O(n^3 \log(n\; \log \; q))$ integer operations.

Keywords: Discrete dynamical systems, fixed point system, finite fields, loop number, algorithm.
Mathematics Subject Classification: Primary: 37B99; Secondary: 05C20.
Citation: Dorothy Bollman, Omar Colón-Reyes. Determining steady state behaviour of discrete monomial dynamical systems. Advances in Mathematics of Communications, 2017, 11 (2) : 283-287. doi: 10.3934/amc.2017019
References:
[1]

D. Bollman, O. Colón-Reyes, V. Ocasio and E. Orozco, A control theory for Boolean monomial dynamical systems, Discrete Event Dyn. Syst., 20 (2010), 19-35. doi: 10.1007/s10626-009-0086-3.

[2]

D. Bollman, O. Colón-Reyes and E. Orozco, Fixed points in discrete models for regulatory genetic networks Eurasip J. Bioinform. Syst. Biol. 2007 (2007), Article ID 97356.
[3]

O. Colón-Reyes, A. Jarrah, R. Laubenbacher and B. Sturmfels, Monomial dynamical systems over finite fields, J. Complex Syst., 16 (2006), 333-342.

[4]

O. Colón-Reyes, R. Laubenbacher and B. Pareigis, Boolean monomial dynamical systems, Ann. Combin., 8 (2004), 425-429. doi: 10.1007/s00026-004-0230-6.

[5]

E. Delgado-Eckert, An algebraic and graph theoretical framework to study monomial dynamical systems over a finite field, Complex Syst., 18 (2009), 308-328.

[6]

E. V. Denardo, Periods of connected networks and powers of nonnegative matrices, Math. Oper. Res., 2 (1977), 20-24. doi: 10.1287/moor.2.1.20.

[7]

R. Hernández-Toledo, Linear finite dynamical systems, Commun. Algebra, 33 (2005), 2977-2989. doi: 10.1081/AGB-200066211.

[8]

G. Xu and Y. M. Zou, Linear dynamical systems over finite rings, J. Algebra, 321 (2009), 2149-2155. doi: 10.1016/j.jalgebra.2008.09.029.

show all references

References:
[1]

D. Bollman, O. Colón-Reyes, V. Ocasio and E. Orozco, A control theory for Boolean monomial dynamical systems, Discrete Event Dyn. Syst., 20 (2010), 19-35. doi: 10.1007/s10626-009-0086-3.

[2]

D. Bollman, O. Colón-Reyes and E. Orozco, Fixed points in discrete models for regulatory genetic networks Eurasip J. Bioinform. Syst. Biol. 2007 (2007), Article ID 97356.
[3]

O. Colón-Reyes, A. Jarrah, R. Laubenbacher and B. Sturmfels, Monomial dynamical systems over finite fields, J. Complex Syst., 16 (2006), 333-342.

[4]

O. Colón-Reyes, R. Laubenbacher and B. Pareigis, Boolean monomial dynamical systems, Ann. Combin., 8 (2004), 425-429. doi: 10.1007/s00026-004-0230-6.

[5]

E. Delgado-Eckert, An algebraic and graph theoretical framework to study monomial dynamical systems over a finite field, Complex Syst., 18 (2009), 308-328.

[6]

E. V. Denardo, Periods of connected networks and powers of nonnegative matrices, Math. Oper. Res., 2 (1977), 20-24. doi: 10.1287/moor.2.1.20.

[7]

R. Hernández-Toledo, Linear finite dynamical systems, Commun. Algebra, 33 (2005), 2977-2989. doi: 10.1081/AGB-200066211.

[8]

G. Xu and Y. M. Zou, Linear dynamical systems over finite rings, J. Algebra, 321 (2009), 2149-2155. doi: 10.1016/j.jalgebra.2008.09.029.

Figure 1.  State and Dependency Graphs of $(\mathbb{F}_2^3,f=(x_1x_2,x_1x_2x_3,x_3))$
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