Chris Kuhlman ; Henning Mortveit ; David Murrugarra ; Anil Kumar
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Bifurcations in Boolean Networks
dmtcs:2975 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2011,
DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems
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https://doi.org/10.46298/dmtcs.2975
Bifurcations in Boolean NetworksArticle
Authors: Chris Kuhlman 1,2; Henning Mortveit 1,3; David Murrugarra 3; Anil Kumar 1,2
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Chris Kuhlman;Henning Mortveit;David Murrugarra;Anil Kumar
1 Network Dynamics and Simulation Science Laboratory
2 Department of Computer Sciences [Blacksburg]
3 Department of Mathematics [Blacksburg]
This paper characterizes the attractor structure of synchronous and asynchronous Boolean networks induced by bi-threshold functions. Bi-threshold functions are generalizations of standard threshold functions and have separate threshold values for the transitions $0 \rightarrow $1 (up-threshold) and $1 \rightarrow 0$ (down-threshold). We show that synchronous bi-threshold systems may, just like standard threshold systems, only have fixed points and 2-cycles as attractors. Asynchronous bi-threshold systems (fixed permutation update sequence), on the other hand, undergo a bifurcation. When the difference $\Delta$ of the down- and up-threshold is less than 2 they only have fixed points as limit sets. However, for $\Delta \geq 2$ they may have long periodic orbits. The limiting case of $\Delta = 2$ is identified using a potential function argument. Finally, we present a series of results on the dynamics of bi-threshold systems for families of graphs.
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