Boolean delay equation

A Boolean Delay Equation (BDE) is an evolution rule for the state of dynamical variables whose values may be represented by a finite discrete numbers os states, such as 0 and 1. As a novel type of semi-discrete dynamical systems, Boolean delay equations (BDEs) are models with Boolean-valued variables that evolve in continuous time. Since at the present time, most phenomena are too complex to be modeled by partial differential equations (as continuous infinite-dimensional systems), BDEs are intended as a (heuristic) first step on the challenging road to further understanding and modeling them. For instance, one can mention complex problems in fluid dynamics, climate dynamics, solid-earth geophysics, and many problems elsewhere in natural sciences where much of the discourse is still conceptual.

One example of a BDE is the Ring oscillator equation: X(t-τ) = X(t), which produces periodic oscillations. More complex equations can display richer behavior, such as nonperiodic and chaotic (deterministic) behavior.[1]

References

1. Cavalcante, Hugo L. D. de S.; Gauthier, Daniel J.; Socolar, Joshua E. S.; Zhang, Rui (2010). "On the origin of chaos in autonomous Boolean networks". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 368 (1911): 495–513. arXiv:0909.2269. Bibcode:2010RSPTA.368..495C. doi:10.1098/rsta.2009.0235. ISSN 1364-503X. PMID 20008414. S2CID 426841.

Further reading

• Wright DG, Stocker TF, Mysak LA (1990). "A note on quaternary climate modelling using Boolean delay equations". Climate Dynamics. 4 (4): 263–7. Bibcode:1990ClDy....4..263W. doi:10.1007/BF00211063. S2CID 128603325.
• Oktem H, Pearson R, Egiazarian K (December 2003). "An adjustable aperiodic model class of genomic interactions using continuous time Boolean networks (Boolean delay equations)". Chaos. 13 (4): 1167–74. Bibcode:2003Chaos..13.1167O. doi:10.1063/1.1608671. PMID 14604408. Archived from the original on 2013-02-23.
• Ghil M, Zaliapin I, Coluzzi B (2008). "Boolean Delay Equations: A simple way of looking at complex systems". Physica D. 237 (23): 2967–86. arXiv:nlin.CG/0612047. Bibcode:2008PhyD..237.2967G. doi:10.1016/j.physd.2008.07.006. S2CID 12652082.