Transient problem for Boolean Monomial Dynamical Systems
Towards a solution of the transient problem for boolean monomial dynamical systems
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A problem of interest in finite dynamical systems is to determine when such a system reaches equilibrium, i.e., under what conditions is it a fixed point system. Moreover, given a fixed point system, how many time steps are required to reach a fixed point, i.e., what is its transient? Dorothy Bollman and Omar Colón have shown that a Boolean Monomial Dynamical System (BMDS) f is a fixed point system if and only if every strongly connected component of the dependency graph G_f of f is primitive and in fact, when G_f is strongly connected, the transient of f is equal to the exponent of G_f. Furt...
A problem of interest in finite dynamical systems is to determine when such a system reaches equilibrium, i.e., under what conditions is it a fixed point system. Moreover, given a fixed point system, how many time steps are required to reach a fixed point, i.e., what is its transient? Dorothy Bollman and Omar Colón have shown that a Boolean Monomial Dynamical System (BMDS) f is a fixed point system if and only if every strongly connected component of the dependency graph G_f of f is primitive and in fact, when G_f is strongly connected, the transient of f is equal to the exponent of G_f. Furthermore, every directed graph gives rise to a unique BMDS and hence every example of a primitive graph with known exponent gives us an example of a fixed point BMDS with known transient. Unfortunately, the general problem of determining the exponent of a primitive graph is unsolved. In this work we give several families of primitive graphs for which we can determine the exponent and hence thetransient of the corresponding BMDS.
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