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.

Source : oai:HAL:hal-01196142v1

Volume: DMTCS Proceedings vol. AP, Automata 2011 - 17th International Workshop on Cellular Automata and Discrete Complex Systems

Section: Proceedings

Published on: January 1, 2011

Submitted on: January 31, 2017

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]

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