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29 March 2024
 
  » arxiv » nlin/0612047

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Boolean Delay Equations: A simple way of looking at complex systems
Michael Ghil ; Ilya Zaliapin ; Barbara Coluzzi ;
Date 20 Dec 2006
Subject Cellular Automata and Lattice Gases
AbstractBoolean Delay Equations (BDEs) are a novel type of semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can 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. They represent therewith 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 in earthquake modeling and prediction, as well as in genetics. Some of the climatic and solid-earth applications are illustrated; the former have used small systems of BDEs, while the latter have used large networks of BDEs. We introduce BDEs with an infinite number of variables distributed in space (``partial BDEs``) and discuss connections with other types of dynamical systems, including cellular automata and Boolean networks. This research-and-review paper concludes with a set of open questions.
Source arXiv, nlin/0612047
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