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1 Network Dynamics and Simulation Science Laboratory 2 Department of Computer Sciences Virginia Tech 3 Department of Mathematics Virginia Tech

Abstract : 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 ightarrow $1 up-threshold and $1 ightarrow 0$ 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

Author: Chris Kuhlman - Henning Mortveit - David Murrugarra - Anil Kumar -



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