On the preservation of limit cycles in Boolean networks under different updating schemes

Boolean networks under different deterministic updating schemes are analyzed. It is direct to show that fixed points are invariant against changes in the updating scheme, nevertheless, it is still an open problem to fully understand what happens to the limit cycles. In this paper, a theorem is presented which gives a sufficient condition for a Boolean network not to share the same limit cycle under different updating modes. We show that the hypotheses of the theorem are sharp, in the sense that if any of these hypotheses do not hold, then shared limit cycles may appear. We find that the connectivity of the network is an important factor as well as the Boolean functions in each node, in particular the XOR functions.

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