Boolean delay equations: A simple way of looking at complex systems

PHYSICA D-NONLINEAR PHENOMENA

卷:237

Review

Ghil, Michael^1,2,3,4,5,6  Zaliapin, Ilya^6,7  Coluzzi, Barbara⁴ 

[1] Ecole Normale Super, Dept Geosci, F-75231 Paris 05, France

[2] Ecole Normale Super, Meteorol Dynam Lab, CNRS, F-75231 Paris 05, France

[3] Ecole Normale Super, IPSL, F-75231 Paris 05, France

[4] Ecole Normale Super, Environm Res & Teaching Inst, F-75231 Paris 05, France

[5] Univ Calif Los Angeles, Dept Atmospher & Ocean Sci, Los Angeles, CA 90095 USA

[6] Univ Calif Los Angeles, Inst Geophys & Planetary Phys, Los Angeles, CA 90095 USA

[7] Univ Nevada, Dept Math & Stat, Reno, NV 89557 USA

关键词: Discrete dynamical systems;    Earthquakes;    El-Nino/Southern-Oscillation;    Increasing complexity;    Phase diagram;    Prediction;   

DOI  :  10.1016/j.physd.2008.07.006

来源: Elsevier

【 摘 要 】

Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs call be classified into conservative or dissipative, in a manner that parallels the classification of ordinary or partial differential equations. Solutions to certain conservative BDEs exhibit growth of complexity in time; such BDEs call be seen therefore as metaphors for biological evolution or human history. Dissipative BDEs are structurally stable and exhibit multiple equilibria and limit cycles, as well as more complex, fractal Solution sets, Such as Devil's staircases and fractal sunbursts. All known Solutions of dissipative BDEs have stationary variance. BDE systems of this type, both free and forced, have been used as highly idealized models of climate change on interannual, interdecadal and paleoclimatic time scales. BDEs are also being used as flexible, highly efficient models of colliding cascades of loading and failure in earthquake modeling and prediction, as well as in genetics. In this paper we review the theory of systems of BDEs and illustrate their applications to climatic and solid-earth problems. The former have used small systems of BDEs, while the latter have used large hierarchical networks of BDEs. We moreover introduce BDEs with all infinite number of variables distributed in space (partial BDEs) and discuss Connections with other types discrete dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions. (C) 2008 Elsevier B.V. All rights reserved.

【 授权许可】

Free