The Origin of Complexity

Nature abounds with complex patterns and structures emerging from homogenous media operating far from thermodynamic equilibrium. Such phenomena, which are widely observed in both inanimate (non-biological) 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, is locally active in a precise mathematical sense. This local activity principle is of universal generality and is responsible for all symmetry breaking phenomena observed in a great variety of non-equilibrium media ranging from the emergence of negative differential conductance in bulk semiconductor materials (e.g., Gallium Arsenide in Gunn Diodes) to the emergence of artificial life itself. The main result of this paper consists of a set of explicit analytical conditions for calculating the parameter ranges necessary for the emergence of a non-homogeneous 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.