Boolean delay equations. II. Periodic and aperiodic solutions

Abstract

Boolean delay equations (BDEs) areevolution equations for a vector of discrete variables x(t). The value of each componentX i (t), 0 or 1. depends on previous values of all componentsx j (t− t ij ), x i (t)=f i (x1(t−t i1),...,x n (t −t in )). BDEs model the evolution of biological and physical systems with threshold behavior and nonlinear feedbacks. The delays model distinct interaction times between pairs of variables. In this paper, BDEs are studied by algebraic, analytic, and numerical methods. It is shown that solutions depend continuously on the initial data and on the delays. BDEs are classified intoconservative anddissipative. All BDEs with rational delays only haveperiodic solutions only. But conservative BDEs with rationally unrelated delays haveaperiodic solutions of increasing complexity. These solutions can be approximated arbitrarily well by periodic solutions of increasing period.Self-similarity andintermittency of aperiodic solutions is studied as a function of delay values, and certain number-theoretic questions related toresonances and diophantine approximation are raised. Period length is shown to be a lower semicontinuous function of the delays for a given BDE, and can be evaluated explicitly for linear equations. We prove that a BDE isstructurable stable if and only if it has eventually periodic solutions of bounded period, and if the length of initial transients is bounded. It is shown that, for dissipative BDEs, asymptotic solution behavior is typically governed by areduced BDE. Applications toclimate dynamics and other problems are outlined.