Complex Systems

Monomial Dynamical Systems over Finite Fields Download PDF

Omar Colón-Reyes
Electronic mail address: ocolon@math.uprm.edu
Mathematics Department,
University of Puerto Rico at Mayagüez
Mayagüez, PR 00681

Abdul Salam Jarrah
Electronic mail address: ajarrah@vbi.vt.edu

Reinhard Laubenbacher
Electronic mail address: reinhard@vbi.vt.edu
Virginia Bioinformatics Institute,
Virginia Tech,
Blacksburg, VA 24061-0477, USA

Bernd Sturmfels
Electronic mail address: bernd@math.berkeley.edu
Department of Mathematics,
University of California at Berkeley,
Berkeley, CA 94720, USA

Abstract

An important problem in the theory of finite dynamical systems is to link the structure of a system with its dynamics. This paper contains such a link for a family of nonlinear systems over an arbitrary finite field. For systems that can be described by monomials, information about the limit cycle structure can be obtained from the structure of the monomials. In particular, the paper contains a sufficient condition for a monomial system to have only fixed points as limit cycles. The condition is derived by reducing the problem to the study of a boolean monomial system and a linear system over a finite ring.