boolean difference equations, i: formulation and dynamic behavior

SIAM J. APPL. MATH. Vol. 44, No. 1, February 1984

1984 Society for Industrial and Applied Mathematics 0036-1399/84/4401-0009 $01.25/0

BOOLEAN DIFFERENCE EQUATIONS, I: FORMULATION AND DYNAMIC BEHAVIOR* D. DEEr

AND

M. GHILt

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 delaydifference 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 (BAEs) 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 roughly as l-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 BAEs is compared with related work and generalizations are suggested. A classification of BAEs and rigorous periodicity and aperiodicity results will follow in a companion paper.

1. Introduction. In certain physical as well as biological systems, feedback mechanisms between variables are highly nonlinear. For some of these systems, thresholds can be associated with the action of the feedbacks, and one may describe the state of the system using a vector of Boolean variables. It is then possible to formulate a set of equations for the logical variables governing the feedback mechanisms. Each feedback has a characteristic time scale, which can be incorporated as a delay in the governing equations. This highly simplified formulation allows a more detailed study of solutions, and one hopes that these provide qualitative information on the original system’s dynamic behavior. For biological systems, the idea of thresholds and of a Boolean description was first formulated by Jacob and Monod (1961). Sugita (1963) and Kauffman (1969) formulated simple models with a single delay. R. Thomas (1973), (1978) introduced multiple delays associated with the different time scales and further expanded the theory. A typical example of a system which suggests a Boolean description can be found in genetics. A set of interacting genes shows a strong threshold behavior: a gene is "on" or "off"; if a gene is "on" it produces a product which, in combination with the presence or absence of other gene products, can change the state of other genes. Each possible feedback has an associated delay’ it takes a certain amount of time before a gene product exists in sufficient quantity to have an effect on any other gene. These ideas were recently applied by Nicolis (1982b) to certain systems modeling climate dynamics. An instance is an elementary self-oscillatory model of glaciation cycles (Kill6n et al. (1979), Ghil and Tavantzis (1983)): global, annually averaged temperature T decreases as ice extent increases, due to the reduction in the solar radiation absorbed by the system (the ice-albedo feedback), while decreases when T decreases, due to the reduction of snow accumulation caused by a less active hydrological cycle (the precipitation-temperature feedback). Again, the action of each feedback is associated with certain delays. These delays are due to the ocean’s heat capacity and to the slow, visco-plastic flow of ice sheets, and are of the order of thousands of years. * Received by the editors October 12, 1982, and in final form February 28, 1983. This work was supported by the National Aeronautics and Space Administration under grants NSG-5034 and NSG-5130 and by the National Science Foundation under grants ATM-8018671 and ATM-8214754. 5" Courant Institute of Mathematical Sciences, New York University, New York, New York 10012. 111

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D. DEE AND M. GHIL

The purpose of this article is to formulate a simple Boolean feedback model with delays and illustrate the behavior of its solutions. In 2, the mathematical model of Boolean difference equations (BAEs) is formulated. Section 3 deals with the initialvalue problem for BAEs, stating an existence and uniqueness theorem in the large and giving a simple example with periodic solutions. Section 4 provides a general solution algorithm, which proves the theorem in 3 by induction. Lemma 1, needed for the induction, is proven in the Appendix. In 5, a solution with aperiodic behavior of perpetually increasing complexity is constructed; some quantitative aspects of this increase in complexity are discussed. Section 6 outlines the algebraic and topological structure of the solution space. In 7, the present formulation is compared with that of Thomas (1978), (1979a). Some generalizations are presented in 8.

2. Formulation of the mathematical model. Consider a system with state variables n. We associate with each state variable vi a Boolean v,}, vi R, 1, variable xi depending on a set of thresholds tri R"

,

,

{vl, v2,

x

1 if v i-->tri, 0 ifvi
O. This theorem is proved by induction, constructing an algorithm which advances the solution in time. Before describing the algorithm, it is helpful to consider an example. Consider the 2 x 2 system of BAEs with two delays

(5a) (5b)

x l(t) =fl(x l(t

1), x2(t-0)),

x2(t) =f2(x l(t-- 1), X2(t--O)),

with 0 < 0 < 1, fi given by the "truth table"

fl(O, O)= O, fl(O, 1)= 1, f,(;,O)=O, f(1, 1)= 1,

(6a)

f2(O, O)= 1, f(O, 1)= 1, f:(t,O)=O, f2(1, 1)=0,

and initial data

(6b)

x(t) =x2(t)

1 for -1 =