On the origin of chaos in autonomous Boolean networks

• Published in: Philosophical transactions of the Royal Society of London. Series A: Mathematical, physical, and engineering sciences Vol. 368; no. 1911; pp. 495 - 513
• Main Authors: Cavalcante, Hugo L. D. de S., Gauthier, Daniel J., Socolar, Joshua E. S., Zhang, Rui
• Format: Journal Article
• Language: English
• Published: England The Royal Society Publishing 28.01.2010; The Royal Society
• Subjects: Boolean data; Boolean Networks; Chaos; Chaos theory; Control loops; Delay circuits; Disasters; Feedback; Logic gates; Mathematical Concepts; Mathematical lattices; Neural Networks (Computer); Nonlinear Dynamics; Propagation delay; Pulse duration; Systems Theory; Time-Delay Dynamical Systems; Trajectories
• ISSN: 1364-503X; 1471-2962
• DOI: 10.1098/rsta.2009.0235

We undertake a systematic study of the dynamics of Boolean networks to determine the origin of chaos observed in recent experiments. Networks with nodes consisting of ideal logic gates are known to display either steady states, periodic behaviour or an ultraviolet catastrophe where the number of logic-transition events circulating in the network per unit time grows as a power law. In an experiment, the non-ideal behaviour of the logic gates prevents the ultraviolet catastrophe and may lead to deterministic chaos. We identify certain non-ideal features of real logic gates that enable chaos in experimental networks. We find that short-pulse rejection and asymmetry between the logic states tend to engender periodic behaviour, at least for the simplest networks. On the other hand, we find that a memory effect termed 'degradation' can generate chaos. Our results strongly suggest that deterministic chaos can be expected in a large class of experimental Boolean-like networks. Such devices may find application in a variety of technologies requiring fast complex waveforms or flat power spectra, and can be used as a test-bed for fundamental studies of real-world Boolean-like networks.