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21
Monomial dynamical systems over finite fields
 COMPLEX SYSTEMS
, 2006
"... 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, one can obtain information about t ..."
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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, one can obtain information about the limit cycle structure 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.
The dynamics of conjunctive and disjunctive Boolean networks
, 2008
"... The relationship between the properties of a dynamical system and the structure of its defining equations has long been studied in many contexts. Here we study this problem for the class of conjunctive (resp. disjunctive) Boolean networks, that is, Boolean networks in which all Boolean functions are ..."
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Cited by 10 (2 self)
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The relationship between the properties of a dynamical system and the structure of its defining equations has long been studied in many contexts. Here we study this problem for the class of conjunctive (resp. disjunctive) Boolean networks, that is, Boolean networks in which all Boolean functions are constructed with the AND (resp. OR) operator only. The main results of this paper describe network dynamics in terms of the structure of the network dependency graph (topology). For a given such network, all possible limit cycle lengths are computed and lower and upper bounds for the number of cycles of each length are given. In particular, the exact number of fixed points is obtained. The bounds are in terms of structural features of the dependency graph and its partially ordered set of strongly connected components. For networks with strongly connected dependency graph, the exact cycle structure is computed.
The steady state system problem is NPhard even for monotone quadratic Boolean dynamical systems
, 2006
"... In [2], the authors give a polynomialtime algorithm for deciding for a Boolean dynamical system in which each regulatory function is a monomial whether every limit cycle is a steady state. We show that the corresponding problem is NPhard if the class of permissible regulatory functions contains th ..."
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Cited by 5 (0 self)
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In [2], the authors give a polynomialtime algorithm for deciding for a Boolean dynamical system in which each regulatory function is a monomial whether every limit cycle is a steady state. We show that the corresponding problem is NPhard if the class of permissible regulatory functions contains the quadratic monotone functions xi ∨ xj and xi∧xj. We also show that the problem is NPhard if the set of permissible regulatory functions includes all functions of the type xixj and xj +1. 1
Discrete dynamical systems on graphs and boolean functions
 Math. Comput. Simul
, 2004
"... Abstract. Discrete dynamical systems based on dependency graphs have played an important role in the mathematical theory of computer simulations. In this paper, we are concerned with parallel dynamical systems (PDS) and sequential dynamical systems (SDS) with the OR and NOR functions as local func ..."
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Abstract. Discrete dynamical systems based on dependency graphs have played an important role in the mathematical theory of computer simulations. In this paper, we are concerned with parallel dynamical systems (PDS) and sequential dynamical systems (SDS) with the OR and NOR functions as local functions. It has been recognized by Barrett, Mortveit and Reidys that SDS with the NOR function are closely related to combinatorial properties of the dependency graphs. We present an evaluation scheme for systems with the OR and NOR functions which can be used to clarify some basic properties of the dynamical systems. We show that for forests that does not contain a single edge the number of orientations equals the number of dierent ORSDS.
Large attractors in cooperative biquadratic Boolean networks
, 2007
"... Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960’s as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular impor ..."
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Cited by 3 (1 self)
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Boolean networks have been the object of much attention, especially since S. Kauffman proposed them in the 1960’s as models for gene regulatory networks. These systems are characterized by being defined on a Boolean state space and by simultaneous updating at discrete time steps. Of particular importance for biological applications are networks in which the indegree for each variable is bounded by a fixed constant, as was stressed by Kauffman in his original papers. An important question is which conditions on the network topology can rule out exponentially long periodic orbits in the system. In this paper we consider cooperative systems, i.e. systems with positive feedback interconnections among all variables, which in a continuous setting guarantees a very stable dynamics. In Part I of this paper we presented a construction that shows that for an arbitrary constant 0 < c < 2 and sufficiently large n there exist ndimensional Boolean cooperative networks in which both the indegree and outdegree of each for each variable is bounded by two (biquadratic networks) and which nevertheless contain periodic orbits of length at least c n. In this part, we prove an inverse result showing that for sufficiently large n and for 0 < c < 2 sufficiently close to 2, any ndimensional cooperative, biquadratic Boolean network with a cycle of length at least c n must have a large proportion of variables with indegree 1. Such systems therefore share a structural similarity to the systems constructed in Part I.
Fixed Points in Discrete Models for Regulatory Genetic Networks
, 2007
"... It is desirable to have efficient mathematical methods to extract information about regulatory iterations between genes from repeated measurements of gene transcript concentrations. One piece of information is of interest when the dynamics reaches a steady state. In this paper we develop tools that ..."
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It is desirable to have efficient mathematical methods to extract information about regulatory iterations between genes from repeated measurements of gene transcript concentrations. One piece of information is of interest when the dynamics reaches a steady state. In this paper we develop tools that enable the detection of steady states that are modeled by fixed points in discrete finite dynamical systems. We discuss two algebraic models, a univariate model and a multivariate model. We show that these two models are equivalent and that one can be converted to the other by means of a discrete Fourier transform. We give a new, more general definition of a linear finite dynamical system and we give a necessary and sufficient condition for such a system to be a fixed point system, that is, all cycles are of length one. We show how this result for generalized linear systems can be used to determine when certain nonlinear systems (monomial dynamical systems over finite fields) are fixed point systems. We also show how it is possible to determine in polynomial time when an ordinary linear system (defined over a finite field) is a fixed point system. We conclude with a necessary condition for a univariate finite dynamical system to be a fixed point system.
Monomial dynamical systems of dimension one over finite fields, Acta Arith.148
, 2011
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A mathematical formalism for agentbased modeling
 in Encyclopedia of Complexity and Systems Science
, 2009
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Parallel Dynamical Systems over Graphs and Related Topics: A Survey
"... In discrete processes, as computational or genetic ones, there are many entities and each entity has a state at a given time. The update of states of the entities constitutes an evolution in time of the system, that is, a discrete dynamical system. The relations among entities are usually represent ..."
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In discrete processes, as computational or genetic ones, there are many entities and each entity has a state at a given time. The update of states of the entities constitutes an evolution in time of the system, that is, a discrete dynamical system. The relations among entities are usually represented by a graph. The update of the states is determined by the relations of the entities and some local functions which together constitute (global) evolution operator of the dynamical system. If the states of the entities are updated in a synchronous manner, the system is called a parallel dynamical system. This paper is devoted to review the main results on the dynamical behavior of parallel dynamical systems over graphs which constitute a generic tool for modeling discrete processes.
Algebraic Models in Systems Biology
 ALGEBRAIC BIOLOGY
, 2005
"... Algebra and discrete mathematics play an important role in the study of biological networks, through the use of discrete models and the use of computational and theoretical tools for their analysis. This paper contains a survey of algebraic models in systems biology and mathematical tools for their ..."
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Algebra and discrete mathematics play an important role in the study of biological networks, through the use of discrete models and the use of computational and theoretical tools for their analysis. This paper contains a survey of algebraic models in systems biology and mathematical tools for their construction and analysis. A particular focus is on polynomial dynamical systems models as well as the logical models proposed by Thomas.