eng
episciences.org
Discrete Mathematics & Theoretical Computer Science
1365-8050
2011-01-01
DMTCS Proceedings vol. AP,...
Proceedings
10.46298/dmtcs.2975
2975
journal article
Bifurcations in Boolean Networks
Chris Kuhlman
Henning Mortveit
David Murrugarra
Anil Kumar
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.
https://dmtcs.episciences.org/2975/pdf
bifurcation
bi-threshold
threshold
Boolean networks
graph dynamical systems
synchronous
asynchronous
sequential dynamical systems
[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]
[MATH.MATH-DS] Mathematics [math]/Dynamical Systems [math.DS]
[NLIN.NLIN-CG] Nonlinear Sciences [physics]/Cellular Automata and Lattice Gases [nlin.CG]
[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]