Design rules for application specific dynamical systems
This paper explores different forms of custom dynamical systems for decision making. Curiously, covering and association constraints can be rather easily met, and it can sometimes be guaranteed that all feasible solutions (hence including the optimal solution) exist among the attractors. As yet, no general theory exists for designing such systems, but some titillating special cases can be described. Theorems for the pure differential equation layer can lead to simple and direct electronic implementation of new digital devices. The key strength of the application specific dynamical systems (ASDS) approach lies in the fact that all variables are updated in parallel. Thus ASDS method offers the potential of high speed solutions using modern high density integrated circuits hardware. Custom dynamical systems can be devised to solve certain operations research (OR) problems. Such systems might be called ASDS after the application specific integrated circuits (ASIC) of modern digital electronics. ASDS can be regarded in three ways, from the pure to the practical: differential equations as a mathematical abstraction, difference equations running on general purpose computers, and integrated digital circuits with finite bit accuracy. In OR, solutions are represented by constant attractors (stable equilibrium states) of ASDS in all three layers. In general, the structure of the problem determines a dynamical system format and problem parameters such as instances of costs are used to define an initial state in state space. A trajectory then evolves and asympotically approaches one of several attractors, that is, a decision. Different initial states derived from different benefits and costs characteristic of the problem lead to different decisions. Applications include nearest neighbor finders (as in error-correction of binary codes), resource to task assignments, and optimization of functions of constrained binary domains.
dynamical systems equilibrium attractor trajectory neural network