Edge of Chaos Explains Prigogine’s Instability of the Homogeneous

Abstract

One of the complex phenomena, which most attracted the attention of the scientific community over the past few decades, is the emergence of diffusion-driven instabilities in homogeneous cellular media. Explaining this symmetry-breaking process, which the Russian luminary Ilya Prigogine referred to as the Instability of the Homogeneous, is only possible upon invoking the Physics Law of the Edge of Chaos. While the emergence of inhomogeneous static or dynamic periodic solutions in biological reaction-diffusion systems was recently reproduced in two-cell networks, leveraging the Negative Differential Resistance of certain volatile memristors, so far, to the best of our knowledge, no cellular array was ever found to undergo dissipation-induced dynamical phenomena of higher complexity, including a route to chaos, or an uncontrollable upsurge in the respective physical variables. This manuscript determines all the possible sets of conditions on the parameters of a one-port, extracted from the classical Chua circuit, enabling to poise it on some stable and locally-active operating point, employing a deep mathematical analysis, focusing on the properties of its local impedance. The polarisation of the one-port on some Edge of Chaos operating point is the Conditio Sine Qua Non for the Destabilisation of the Homogeneous Solution, resulting in the steady-state appearance of high-order dynamical phenomena, as outlined above, in a simple Reaction-Diffusion Cellular Nonlinear Network, where a linear and passive resistor, interposed between two identical copies of the one-port, lets them interact by means of a diffusion process. Notably, as demonstrated here, whether or not the dissipative coupling path may truly succeed in moving each of the two one-ports of the two-cell array, under study, away from the respective common Edge of Chaos operating point, depends critically upon the satisfaction of the logical disjunction between three inequalities, which enforces the instability of the denominator polynomial of the rational function associated to the local input impedance of the cellular array, as seen across either of the two one-ports.