Random Boolean network model exhibiting deterministic chaos

This paper considers a simple Boolean network with [Formula presented] nodes, each node’s state at time [Formula presented] being determined by a certain number of parent nodes, which may vary from one node to another. This is an extension of a model studied by Andrecut and Ali [Int. J. Mod. Phys. B 15 17 2001], who consider the same number of parents for all nodes. We make use of the same Boolean rule as Andrecut and Ali, provide a generalization of the formula for the probability of finding a node in state 1 at a time [Formula presented], and use simulation methods to generate consecutive states of the network for both the real system and the model. The results match well. We study the dynamics of the model through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show that the route to chaos is due to a cascade of period-doubling bifurcations which turn into reversed (period-halving) bifurcations for certain combinations of parameter values.