Boolean Difference Equations, I: Formulation and Dynamic Behavior

Abstract

In many biological and physical systems, feedback mechanisms depend on a set of thresholds associated with the state variables. Each feedback has a characteristic time scale. We suggest that delay-difference equations for Boolean-valued variables are an appropriate mathematical framework for such situations: the feedback thresholds result in the discrete, on–off character of the variables, and the interaction time scales of the feedbacks are expressed as delays.The initial-value problem for Boolean delay equations (BΔEs) is formulated and shown to have unique solutions for all times. Examples of periodic and aperiodic solutions are given. Aperiodic solutions have increasing complexity which depends on time t roughly as $t^{l - 1} ,l$ being the number of delays. Stability of solutions is defined and some examples of stability analysis are given; additional stability questions are raised. The present formulation of BΔEs is compared with related work and generalizations are suggested. A classification of BΔE...