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Passivity and complexity. (English) Zbl 0948.92002
Summary: Nature abounds with complex patterns and structures emerging from homogeneous media operating far from thermodynamic equilibrium. Such phenomena, which are widely observed in both inanimate (nonbiological) and biological media, can be modeled and studied via the CNN (cellular neural/nonlinear network) paradigm in an in-depth and unified way. Whether a homogeneous medium is capable of exhibiting complexity depends on whether the CNN cells, or its couplings, are locally active in a precise circuit-theoretic sense. This local activity principle is of universal generality and is responsible for all symmetry breaking phenomena observed in a great variety of nonequilibrium media ranging from the nucleation of domain oscillations in bulk semiconductor materials (e.g., gallium arsenide in Gunn diodes) to the emergence of artificial life itself. The long forgotten yet classic P. R. (positive real) criterion is resurrected and given new prominence in this paper by invoking its “negative” version and deriving a set of analytical inequalities for calculating the parameter range necessary for the emergence of a nonhomogeneous static or dynamic pattern in a homogeneous medium operating under an influx of energy and/or matter. The resulting “complexity related” inequalities are applicable to all media, continuous or discrete, which have been mapped into a CNN paradigm.

MSC:
92B05General biology and biomathematics
92B20General theory of neural networks (mathematical biology)
92C15Developmental biology, pattern formation
35Q80Applications of PDE in areas other than physics (MSC2000)
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