Skip to main content

Dynamics of linear systems over finite commutative rings


The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous publication, the last two authors developed an efficient algorithm to determine whether a linear dynamical system over a finite commutative ring is a fixed point system or not. The algorithm can also be used to reduce the problem of finding the cycles of such a system to the case where the system is given by an automorphism. Here, we further analyze the cycle structure of such a system and develop a method to determine its cycles.

This is a preview of subscription content, access via your institution.


  1. The assumption on the module considered in Theorem 2.1 [9] is somewhat different, but a quick check of the proof there reveals that it also applies to the case here.


  1. Bollman, D., Colón-Reyes, O., Orozco, E.: Fixed points in discrete models for regulatory genetic networks. EURASIP J. Bioinform. Syst. Biol., On-line ID 97356 (2007)

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

    MathSciNet  MATH  Google Scholar 

  3. Deng, G.: Cycles of linear dynamical systems over finite local rings. J. Algebra 433, 243-261 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Elspas, B.: The theory of autonomous linear sequential networks. IRE Trans. Circuit Theory 6, 45-60 (1959)

    Article  Google Scholar 

  5. von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge Press, Cambridge (2003)

    MATH  Google Scholar 

  6. Hernández-Toledo, A.: Linear finite dynamical systems. Commun. Algebra 33, 2977-2989 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hungerford, T.W.: Algebra, GTM 73. Springer, Berlin (1974)

    Google Scholar 

  8. Neunhöffer, M.M., Praeger, C.E.: Computing minimal polynomials of matrices. LMS J. Comput. Math. 11, 252-279 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Xu, G., Zou, Y.M.: Linear dynamical systems over finite rings. J. Algebra 321, 2149-2155 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Zhang, R., et al.: Network model of survival signaling in large granular lymphocyte leukemia. PNAS 105, 16308-16313 (2008)

    Article  Google Scholar 

Download references


Part of the work in this paper was done during the visit of Y.J.W. to the Department of Mathematical Sciences at the University of Wisconsin-Milwaukee in the summer of 2015. Y.J.W. wishes to thank the University of Wisconsin-Milwaukee and its faculty for the hospitality she received during her visit.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Yi Ming Zou.

Additional information

Y.J.W. acknowledges the support of the National Natural Science Foundation of China (11461010) and the Guangxi Natural Science Foundation (2014GXNSFAA118005).

G.W.X. acknowledges partial support from the National 973 Project of China (2013CB834205).

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wei, Y., Xu, G. & Zou, Y.M. Dynamics of linear systems over finite commutative rings. AAECC 27, 469–479 (2016).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification