{"docId":2975,"paperId":2975,"url":"https:\/\/dmtcs.episciences.org\/2975","doi":"10.46298\/dmtcs.2975","journalName":"Discrete Mathematics & Theoretical Computer Science","issn":"","eissn":"1365-8050","volume":[{"vid":261,"name":"DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems"}],"section":[{"sid":66,"title":"Proceedings","description":[]}],"repositoryName":"Hal","repositoryIdentifier":"hal-01196142","repositoryVersion":1,"repositoryLink":"https:\/\/hal.science\/hal-01196142v1","dateSubmitted":"2017-01-31 10:21:32","dateAccepted":null,"datePublished":"2011-01-01 00:00:00","titles":{"en":"Bifurcations in Boolean Networks"},"authors":["Kuhlman, Chris,","Mortveit, Henning,","Murrugarra, David","Kumar, Anil,"],"abstracts":{"en":"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."},"keywords":[["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]"]}