An Algebraic and Graph Theoretic Framework to Study Monomial Dynamical Systems over a Finite Field

Edgar Delgado-Eckert, ETH Zürich, Department of Biosystems Science and Engineering
2009 Complex Systems  
A monomial dynamical system f : K n Ø K n over a finite field K is a nonlinear deterministic time discrete dynamical system with the property that each component function f i : K n Ø K is a monic nonzero monomial function. In this paper we provide an algebraic and graph theoretic framework to study the dynamic properties of monomial dynamical systems over a finite field. Within this framework, characterization theorems for fixed point systems, that is, systems in which all trajectories end in
more » ... ajectories end in steady states, are proved. These characterizations are stated in terms of connectedness properties of the dependency graph. Our formalism allowed us to develop an algorithm of polynomial complexity for testing whether or not a given monomial dynamical system over an arbitrary finite field is a fixed point system. In addition, we were able to identify a class of monomial dynamical systems, namely, the Iq -1Mfold redundant monomial systems. Within this class of systems a characterization of fixed point systems is proved that represents a generalization of previous work on Boolean monomial dynamical systems. Complex Systems, 18 The study of dynamical systems generally addresses the question of the system's long-term behavior, in particular, the existence of fixed points and (limit) cyclic trajectories. In this paper we provide an algebraic and graph theoretic framework for studying a specific class of nonlinear time discrete dynamical systems over a finite field, namely, monomial dynamical systems over a finite field. In such systems, every component function f i : K n Ø K is a monic nonzero monomial function. Some types of monomial systems and their dynamic behavior have been studied before: monomial cellular automata [6, 7] , Boolean monomial systems [8], monomial systems over the p-adic numbers [9, 10], and monomial systems over a finite field [11|13]. A necessary and sufficient condition for Boolean monomial systems to be fixed point systems, that is, systems in which all trajectories end in steady states, is proved in [8] . This condition could be algorithmically exploited. Indeed, the authors make some suggestive comments in that direction [8, Section 4.3]. Moreover, they describe the structure of the limit cycles of a special type of Boolean monomial systems. In [13] the authors present a necessary and sufficient condition for monomial systems over a finite field to be fixed point systems. However, this condition is not easily verifiable and therefore the theorem does not yield a tractable algorithm in a straightforward way. In [13] it is explicitly stated that a tractable algorithm to determine whether or not a monomial dynamical system over an arbitrary finite field is a fixed point system has still to be developed [13, Section 3]. Nevertheless, while the present paper was being peer-reviewed, [14] was published, providing the missing computational-algebraic ingredient toward an algorithmic application of the approach in [13] . However, the concrete details of an algorithm exploiting the results in [13] and [14] still need to be developed. Our work was influenced by [8, 13, 12] . However, we took a slightly different approach. The mathematical formalism we developed allows for a deeper understanding of monomial dynamical systems over a finite field. Indeed, we were able to circumvent the complicated Glueing-procedure developed in [8, Section 5]. Our formalism allows formulating a very simple algorithm of polynomial complexity for testing whether or not a given monomial dynamical system over an arbitrary finite field is a fixed point system. Additional theorems that complement the work in [8, 13] are obtained. Furthermore, we were able to identify a class of monomial dynamical systems, namely, the Hq -1L-fold redundant monomial systems (to be defined later). Within this class of systems, a very satisfying characterization of fixed point systems can be reached. Boolean systems are trivial examples of Hq -1L-fold redundant systems. As a consequence, many of our results about Hq -1L-fold redundant monomial systems provide a generalization of theorems proved in [8] for Boolean systems. Last but not least, our formalism also constitutes a basis for the study of monomial control systems [15] . It is pertinent to mention the work of [16] regarding linear time discrete dynamical systems over a finite field, in which the number of limit cycles and their lengths is linked to the factorization (in so-called elementary divisor polynomials) of the characteristic polynomial of the matrix representing the system. (See also [17] for a more mathematical exposition and [2, 18] for applications of the Boolean case in 308 E. Delgado-Eckert Complex Systems, 18
doi:10.25088/complexsystems.18.3.307 fatcat:lf3v3rskxrgldgqbvbycncernu