On the Existence of a Spectrum of Equilibria in Adjustment Processes with Persistent Randomness. A Bak-Sneppen Evolution Model Approach

Most distributed systems and systems depending on their sensing of their surroundings operate under a great deal of uncertainty from many sources. The dynamic behavior of such systems can assume very complex forms [1] inducing complex dynamics that are challenging to observe and control. Their stability analysis in general depends upon the system underlying design and implementation because of the evolutionary character of the models upon which they are built. The nonlinearity of circular causations and feedback effects of their internal dynamics moves the center of mass or energy of such systems from one accumulation point to another within a set when the system is supposed to have achieved its thermodynamic equilibrium. In this paper, we prove the existence of such set and provide a cross-fertilization approach using tools from Physics and Biology to show how one can practically approach the equilibria of this class of systems in their full dynamic complexity. To do this, and to show the originality of our technique, we depart from Freidlin and Wentzell technique that is considered to be the standard in game theory to tackle similar problems, and apply our cross-fertilization technique to an implementation of an extension of the Olafsson´s model. We then show the limitation of Freidlin and Wetzell technique and devise our stability analysis method using the same baseline. Stability analysis from which we prove the existence of a spectrum of equilibria for such systems.