Geodesic
Bifurcations in Boolean Networks
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems, DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems (2011).

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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. 
@article{DMTCS_2011_special_261_a7,
     author = {Kuhlman, Chris and Mortveit, Henning and Murrugarra, David and Kumar, Anil},
     title = {Bifurcations in {Boolean} {Networks}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems},
     year = {2011},
     doi = {10.46298/dmtcs.2975},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2975/}
}
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Kuhlman, Chris; Mortveit, Henning; Murrugarra, David; Kumar, Anil. Bifurcations in Boolean Networks. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems, DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems (2011). doi : 10.46298/dmtcs.2975. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2975/

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