Boolean Difference Equations, I: Formulation and Dynamic Behavior

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

1984 Society for Industrial and Applied Mathematics0036-1399/84/4401-0009 $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 thresholdsassociated 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 suchsituations: the feedback thresholds result in the discrete, on-off character of the variables, and the interactiontime scales of the feedbacks are expressed as delays.

The initial-value problem for Boolean delay equations (BAEs) is formulated and shown to have uniquesolutions for all times. Examples of periodic and aperiodic solutions are given. Aperiodic solutions haveincreasing complexity which depends on time roughly as l-l, being the number of delays. Stability ofsolutions 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, feedbackmechanisms between variables are highly nonlinear. For some of these systems,thresholds can be associated with the action of the feedbacks, and one may describethe state of the system using a vector of Boolean variables. It is then possible toformulate a set of equations for the logical variables governing the feedback mecha-nisms. Each feedback has a characteristic time scale, which can be incorporated as adelay in the governing equations. This highly simplified formulation allows a moredetailed study of solutions, and one hopes that these provide qualitative informationon the original system’s dynamic behavior.

For biological systems, the idea of thresholds and of a Boolean description wasfirst formulated by Jacob and Monod (1961). Sugita (1963) and Kauffman (1969)formulated simple models with a single delay. R. Thomas (1973), (1978) introducedmultiple delays associated with the different time scales and further expanded thetheory.

A typical example of a system which suggests a Boolean description can be foundin 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 thepresence or absence of other gene products, can change the state of other genes. Eachpossible feedback has an associated delay’ it takes a certain amount of time before agene product exists in sufficient quantity to have an effect on any other gene.

These ideas were recently applied by Nicolis (1982b) to certain systems modelingclimate dynamics. An instance is an elementary self-oscillatory model of glaciationcycles (Kill6n et al. (1979), Ghil and Tavantzis (1983)): global, annually averagedtemperature T decreases as ice extent increases, due to the reduction in the solarradiation absorbed by the system (the ice-albedo feedback), while decreases whenT decreases, due to the reduction of snow accumulation caused by a less activehydrological cycle (the precipitation-temperature feedback). Again, the action of eachfeedback is associated with certain delays. These delays are due to the ocean’s heatcapacity and to the slow, visco-plastic flow of ice sheets, and are of the order ofthousands of years.

* Received by the editors October 12, 1982, and in final form February 28, 1983. This work wassupported by the National Aeronautics and Space Administration under grants NSG-5034 and NSG-5130and 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

112 D. DEE AND M. GHIL

The purpose of this article is to formulate a simple Boolean feedback model withdelays and illustrate the behavior of its solutions. In 2, the mathematical model ofBoolean difference equations (BAEs) is formulated. Section 3 deals with the initial-value problem for BAEs, stating an existence and uniqueness theorem in the largeand giving a simple example with periodic solutions. Section 4 provides a generalsolution algorithm, which proves the theorem in 3 by induction. Lemma 1, neededfor the induction, is proven in the Appendix.

In 5, a solution with aperiodic behavior of perpetually increasing complexity isconstructed; 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). Somegeneralizations are presented in 8.

2. Formulation of the mathematical model. Consider a system with state variables{vl, v2, , v,}, vi R, 1, , n. We associate with each state variable vi a Booleanvariable xi depending on a set of thresholds tri R"

1 if vi-->tri,x 0 ifvi <

The set of Boolean variables x {x 1, x2, ’, x,} gives a simple qualitative descriptionof the original system, in effect reducing the number of possible states to 2". Referringto the example from paleoclimatology, x might indicate whether the Earth is in anice age or not, i.e., whether T is high and low, or vice versa.

Time dependence can be introduced by prescribing a set of delays {tij}, 1,.. , n,] 1, , n, t/> 0, where ti is the time it takes for x to have an effect on x. In thegenetic example, we saw that one gene must have been on for a certain amount oftime before its product can be present in sufficient quantity to change the state of anyother gene. Hence for each pair of state variables there is an associated time delay.Notice that it is not necessary to have tii ti.

The feedbacks among the Boolean variables may be described by a set of Booleanfunctions f: " B, 1,. ., n, where {0, 1}, as follows:

x (t) f1(x (t t11), xz(t txz), x. (t tx. )),

x2(t) =f2(x l(t t21), x2(t t22), x, (t t2,)),(1)

x,(t) f, (x x(t- t,x), x2(t t,2),""", x,(t

The system of Boolean difference equations, or BAEs, (1) is perhaps simpler todescribe in.terms of "memorization variables" xj"

(2) xq(t) xj(t tii), 1,. ., n, f 1,. ., n.

In terms of the memorization variables, (1) becomes(3) xi(t) fi(Xl(t), xi2(t), xi,(t)), i= 1, n.

Equation (3) states that at time t, the state x is determined by the memorizationvariables x,1, x,2, , x, or, equivalently, by the states xi at times t- ti, ] 1, , n.

We only consider here for simplicity two-valued variables, x [. The generali-zation to multiple, discrete thresholds, cf. Van Ham (1979), proceeds by the introduc-tion of additional variables for each threshold. Also the systems treated in the presentpaper are autonomous, i.e., the f do not depend explicitly on time. Nonautonomoussystems can also be analyzed within the present framework.

BOOLEAN DIFFERENCE EQUATIONS 113

3. The initial-value problem for BAEs. There is a certain similarity here withthe theory of real-valued delay-differential equations (DDEs) (Bhattacharya and Ghil(1982), Driver (1977), Hale (1971), McDonald (1978)) as well as with that of ordinarydifference equations (OAEs) (Bellman and Cooke (1963), Isaacson and Keller (1966,Ch. 8)).

In order to solve the system of equations (I), we need to prescribe the Booleanvariables on an interval"

(4) xi(t)=x i(t) for to-z=<t<t0, i=l,...,n,

where -=max {t0.}. This z may be called the length of the memory of the system,since the state of the system at time depends only on the states {x(t- t’)}, 0 < ’-< r.After these preliminaries, we are ready to state the following existence and uniquenesstheorem.

THEOREM 1. Let {x(t)} be initial data with jumps from 0 to 1 or vice versa at a

finite number of points on the interval to-" <= < to. Then the system (1) has a uniquesolution for all >= to, for arbitrary delays tij > O.

This theorem is proved by induction, constructing an algorithm which advances thesolution 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) x l(t) =fl(x l(t 1), x2(t-0)),

(5b) x2(t) =f2(x l(t-- 1), X2(t--O)),

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

(6a)

and initial data

(6b)

fl(O, O)= O, f2(O, O)= 1,

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

f,(;,O)=O, f:(t,O)=O,

f(1, 1)= 1, f2(1, 1)=0,

(7a)

(7b)

x(t) =x2(t) 1 for -1 =<t <0.

In terms of the Boolean operation (not), (5), (6a) are equivalent to

Xl(t) =X2(t--O),

x2(t) -qxl(t- 1).

This means that (6a) expresses one positive and one negative feedback, the action ofthe positive feedback, cf. (5), being faster than that of the negative one. Notice that,in general, n x n truth tables like (6a) are in one-to-one correspondence with n Booleanexpressions of n variables connected by the logical operations v (or), ^ (and), and- (not).

The solution of (5), (6) is simply

Xl(t) { 01x2(t) { 01

if t e [0 +(2k-2)(1 +0), 0 +(2k- 1)(1 +0)),if t e [0 + (2k 3)(1 + 0), 0 + (2k 2)(1 + 0)),if t e[(2k -2)(1 +0), (2k 1)(1 +0)),if e[(2k -3)(1 +0), (2k-2)(1 +0)),

k=l,2,...


The solution is a periodic step function of period 1 + O. The behavior of the systemin this example is simple, due to the fact that each fi is a function of one variableonly. As we shall see below, the general case is considerably more complicated.

4. Solution algorithm. The initial data x (t) is a vector step function with jumpsat a finite subset of points in the interval to-" _-< t < to. Let this set of jump points beA {sl to-r, s2, s3, ’, s,,}, sm< to, and let SO c [to, to +’) be the corresponding setof jump points of the memorization variables xj (see (2)) on the interval [to, to +’).It is clear how to construct the "jump set" S: if s is a time at which xj jumps, then

x0 will jump at time s + ti, since

xii(t + tii) xi(t).

S has at most rn x n 2 elements, since for each element s in A there are at most n 2

elements s + t, in So (some of the ti may be equal).The set of jump points of the solution on [to, to +z) must be a subset of S, since

the solution cannot jump unless a memorization variable jumps (see (3)). The algorithmworks by updating the jump set So along with the solution at successive jump pointss as follows’

Given Sk, let tk =infSk. Compute the solution for s =tk using the delayequations (1).

This is possible since the solution is known for all < s; e.g., for S, s to, whilefor S 1, s is the first jump point tl E 81, /’1 > to, etc.

Then let

Sk +1 (S {tk}) (-JMk,where Mk is the set of jump points of the memorization variables due to the jumpof the solution at s t.

Explicitly, if the variable x does jump at s t, but x, l, does not jump there,then Mk {s + t, 1, , n }. In particular, if none of the variables jump at s tk,then Mk is empty.

By repeating this process, one obtains the solution x(t).It is clear that sk is finite for finite k, although the set may grow rapidly, especially

if all the tq are different. In order to prove that the solution can be extended indefinitelyin time, it is sufficient to show that sk will never have an accumulation point. Thisfollows from the fact that sk is a subset of the set

P={s" s =st+ pjt,sEA, p,i,f l, ,n},i,i=1

where is the set of nonnegative integers; the following statement is true about P:LEMMA 1. The set

P [mr, (m + 1)r)contains a finite number ofpoints ]’or any integer m.

This means that there is a finite upper bound for the number of jumps of thesolution in any interval [mr, (m + 1)r). The upper bound depends only on rn and onthe set of delays {ti}. This lemma is proven in the Appendix.

The implementation of the algorithm on the computer is simple, although onehas to take into account the following difficulty. It may happen that Mk (-] Sk ,which means that jump points of the memorization variables, due to the jump of thesolution at s tk, are already present in S k. In this case one has to make sure thatthe computer recognizes these duplicates, or else spurious jumps will be generated.


It is possible to avoid this problem by storing for each element of sk the sequenceof integers which uniquely identifies it: for each s Sk, s st +Y.s.--1 ps.tsi for somest cA and some sequence {Psi}, Psi . It has been our experience that without such adevice, spurious jumps contaminate the solution more and more as time increases,because the system "remembers" all these jumps and each jump may generate futurejumps.

The opposite situation may also occur, in which jumps get arbitrarily close toeach other in finite time; in particular, they may get closer than machine accuracy.This is due to a fact known as Kronecker’s lemma, which states that a number of theform pOl--p202, where pl, p2] and 01/02 is irrational, can get arbitrarily close tozero for sufficiently large, but finite, p and p2. Notice that this does not contradictthe lack of accumulation points of the jump set" only two or some other finite numberof jump points will get arbitrarily close to each other at any finite time. An approachto eliminating this numerical stability problem by "smoothing" will be mentioned inthe last section.

5. Chaotic behavior of solutions. When computing the solution of a system ofBAEs, the size of the set S k and its growth in time provide a measure of the complexityof the solution. In the simple example given by (5)-(7) above, Sk will never containmore than two elements:

A ={-1}, $3={1+0,2},S {0}, S4 {2},

S ={0, 1}, Ss ={2 +0, 3},

S2 ={1}, etc.

However, the following example, with two coupled variables and two delays, is moretypical.

Let

(8a)

(8b)

with 0 < 0 < 1, )es given by

(9a)

x (t) =fx(x (t 1), x2(t -o)),

x(t) =A(x(t-o), x(t- )),

fl(O, O) O, .f2(O, O) O,

fx(O, 1)= 1, .f2(O, 1)= 1,

fa(1, O)= O, f2(1, O)= 1,

fi(1, 1)= 1, f2(1, 1)= 0,

and initial data

(9b) x(t)=x2(t)=l,

The Boolean expressions for (8), (9a) are

(10a) x(t) =x2(t-O),

-l=<t<0.

(10b) x2(t) xx(t-O)Vx2(t- 1).

Here g is the "exclusive or",

(11) X17Xz--(X1VX2)A --](X1AXz)--(X1V X2) A ()1V 32),

116 D. DEE AND M. HIL

with i xi. Clearly the use of the equivalent notations (9a) or (10) is a matter ofrelative convenience.

The solution of this system was computed for different values of O up to t 100.The number of jumps per unit time increases with time until all possible state transitionstake place and there is no preferred state or sequence of states of the system (Fig. 1).

In fact, it is proven in the Appendix that the number of jumps per unit time isbounded from above by a function linear in m for the case of two delays. Figure 2shows the number of jumps per unit time for the solution of (8)-(10), as well as thetheoretical upper bound.

Clearly, the number of jumps per unit time appears to increase linearly, on theaverage. This number is not a monotone function of time, as might have been suspected.On the other hand, the average slope of the solution graph is rather close to that ofthe upper bound.

The surprising fact is that all solutions of BAEs with rational delays are periodic(Ghil and Mullhaupt (1983)). Hence all numerical solutions of BAEs will be periodic,due to finite computer word length. The period of the numerical example in Figs. 1and 2, however, is certainly very long and could be as large as O(103) (Ghil andMullhaupt (1983)).

This is a strong warning against premature inferences drawn even from carefuland extended numerical computations. Such inferences, which are frequent in theliterature of dynamical systems and nonlinear phenomena, can also be very fruitful.The theoretical results of Ghil and Mullhaupt (1983) about the existence of aperiodicsolutions and their approximation by periodic solutions with increasing period weremotivated by Figs. 1 and 2. Solutions were shown to exist whose growth of complexityis sustained indefinitely and behaves asymptotically like a fractional power of time.

X 1

X2

t zl zz, 27 28, 29 30

-r-llll-lllllllNilllll’- ii[ ml,llii llitmnmllilmmmnml timlll hililllmntlnill iiiillllliilliiiiillliiniiiiiiiil iIiiiiiiililiiilimlVllllliiil Vrll-lttlhlvlmnllnmnll I/ill illlllnllllllll

X 1

X2

FIG. 1. The solution of (8), (9) for 0 0.977, on 0 =< < 40. The marks on the t-axis indicate times atwhich state transitions take place.


BAE solutions of increasing complexity seem to have properties which changeindefinitely with time, tending to no asymptotic behavior whatsoever, not even in astatistical sense. We suspect that such solutions are connected to aperiodic, noncentralorbits of systems of autonomous ordinary differential equations which are stable, butnot uniformly stable. Such orbits were mentioned by Lorenz (1963) as the excludedcase in his theorem that central orbits are unstable if they are aperiodic. We considernext the appropriate notions of stability for BAEs.

100

80

60

40

20

o- zb 40 abFI. 2. The number of[umps of the solution of (8), (9) per unit time as a function of time, as compared

to the theoretical upper bound derived in the Appendix. The abscissa of a point on the graph corresponds to

m, its ordinate indicates the number ofjumps of the solution during the interval [m, m + 1).

6. Stability of solutions. To study the stability properties of BAEs, we need toconsider a suitable function space in which the solutions will be imbedded. Thesolutions x(t) of BAEs are piecewise constant functions, with the value 0 or 1 oversubintervals of the real axis. It will be convenient to let to z, so that the functionsare all defined on the positive real axis

Algebra ofsolutions. Real-valued functions defined on + become a vector spaceover the field of real scalars (e.g., Halmos (1958, pp. 1-4)) when introducing additionof two functions as their usual pointwise sum, and multiplication by scalars in theobvious way. For Boolean-valued functions, the role of addition is played by thelogical "or",(12) (x v y)(t)=x(t) v y(t),

which, like usual addition, is associative and commutative. It also has a neutral element,namely the function x (t) 0, but no unique inverse "-x (t)"; in particular, y (t)(x)(t) is not such an inverse.


Multiplication by Boolean scalars is defined using the logical "and" pointwise:

(13) (a ^ x)(t) =a ^x(t).

This scalar multiplication is distributive with respect to the addition of functions,

(14) ^ (x(t) v y(t)) (a ^ x(t)) v (a ^ y (t)),

and has a neutral element, a 1. Again, no unique inverse ,,a-l,, for the multiplicationof and by scalars exists.

The algebraic structure defined by (12), (13) is called a module over a ring ofscalars (Halmos (1958, p. 5)). All the ingredients necessary for the definition of anorm are present.

Distance between solutions. To define closeness between, and hence stability of,solutions, we introduce a norm by

(lSa) Ilxll-}irn? x(s)ds.

The integral is to be taken in ttie Lebesgue sense and clearly

(15b) 0 _-<][xll <_- 1.

Since x(t)>-0, no absolute value is needed in (15a). Given a scalar a s B, we havethe homogeneity property

(16) Ila ^ x (t)ll- llx (t)ll,as required (e.g., Halmos (1958, p. 122), Naylor and Sell (1982, pp. 215tt.)).

In fact, our functions x (t) are just the characteristic functions h’a (t) of the subsetA e N/ on which they are nonzero"

(17a) A ={t+ :x(t)= 1},

(17b) x(t) gA(t).

This puts them in one-to-one correspondence with the power set of R/, i.e., the set(/) of all subsets of R/. Thus (15) defines properly speaking a seminorm, since

IIx 0

for the characteristic function of a set of Lebesgue measure zero. We shall consideras usual equality to mean almost everywhere (a.e.) equality, ignoring sets of measurezero.

To define the distance between two functions x(t), y(t), one needs to introducea suitable "difference", circumventing the previously mentioned difficulty of the lackof (-y(t)). We let

(18a) (xy)(t) x (t)Vy (t)

using the "exclusive or" of (11). This can be easily computed as

(18b) (x y)(t) Ix (t) y (t)l,where on the right-hand side the subtraction and absolute value refer to x (t) and y(t)as real numbers. In other words, (xy)(t) for some equals 0 if x and y are equalfor that t, and it equals 1 if they are not. Thus

(19) IIxyll Jim ; Ix(s)-y(s)l ds.


With these definitions, the usual triangle inequalities for norms (Naylor and Sell(1982, pp. 215ff.)) are satisfied:

(20a) IIx v

(20b) I)lx II-Ilylll < IIx vyll.Likewise, to study systems of BAEs, we introduce the norm of a vector of Boolean-valued functions x(t) (Xl(t), x2(t), ", Xn(t)) by

(21) Ilxll-- a__ IIx,ll.ni=l

With the algebraic structure of (12)-(14) and the norm given by (15)-(21), theset of n-tuples of characteristic functions {xi(t)=xA(t)’A R/} becomes a normedfunction space X. To consider the stability of solutions x(t) to (1) in X, we still needto define the restriction of the topology to functions supported on a subset A R /.

This is easily achieved by

(22a) IIx vylla

(22b)

[.,lx(s)-y(s)lds(a)

is infinite, one considers instead

(22c) IlxVylla lim XA (S )IX (S y(s)l ds,-. I x,(s cts

The power set (A) with the norm (22c) becomes a subspace of X. Notice that, forany x, y X,A +,(22d) 0 <-Ilx Vylla <= 1.

In particular, we shall be concerned with sets of the form A {a < < 00} as a + +00.Stability results. Consider first a single BAE(1), given in terms of its right-hand

side i=(fl,f2,""" ,f,) and delays 0=(01, 02,’’ ", Or); initial data x(t) are suppliedon A [0, r). Stability of solutions x(t), y(t) with respect to initial data means thatsolutions are close,

(23a) IIx(t)y(t)ll < e,

provided the initial data are close,

(23b) IIx(t)y(t)l[ < a.The simplest possible example is n 1,

(24a) f 0.

The solution

(24b) x(t)--O

is stable in the sense (23): no matter what the values, and hence the norm, o x onthe initial interval, the solution is (24b) and its norm is zero.


A slightly less trivial example is

(25a) fl=xl(t-1).

Both the solutions

(25b,c) x (t) 0, x x(t) --- 1,

are stable: if the initial data on [0, 1) differ from 0, say, on a subinterval of length e,the norm of the corresponding solution will be e, so (23) holds with 8 e. In fact,any solution of (25a), say, for

(25d) x o 0, 0 _-< < a,

(25e) x=l, a<=t<l,

is stable.Likewise, the periodic solution of (5)-(7) is stable in this sense: small deviations

from the constant initial data (6b) will propagate, but not amplify, as increases.On the other hand, in none of these examples is there an actual decay of initialperturbations.

The concept which is needed is that of asymptotic stability, i.e., given (23), onerequires furthermore that

(26) IIx Vyll -, 0

for B [b, ) and b +o. A simple example is

(27a) x(t) Xl(t- 1) v x2(t-O),

(27b) x2(t) x(t- 1).

The solution tends asymptotically to

(27c) Xl-X2l.The most interesting questions in this context are those about the stability of

aperiodic solutions. We believe that the solutions of increasing complexity are allunstable. It is not clear whether any aperiodic solutions of asymptotically constant,or asymptotically periodic complexity do exist, and if so, whether they are stable.These questions are related to introducing an appropriate concept of forced-dissipative,as opposed to conservative, BAEs (Ghil and Mullhaupt (1983)).

Structural stability. We have considered so far the stability of solutions of a singleBAE with respect to initial data. The next question concerns families of BAEs, or thestability of solutions with respect to parameters. The obvious parameters in thisframework are the delays 0 .

Clearly, for the trivial example (24a), the unique solution (24b) will be stablewith respect to an arbitrary delay 0 introduced in f. The same is true if in (25a) theunit delay is replaced by an arbitrary delay 0, since there is no other characteristictime in the problem.

Things become more interesting for _-> 2. Still, the periodic solution of (5)-(7)is structurally stable with respect to O, even when 0 _-> 1; this is due to the fact thatthe solution depends only on the period (1 + 0). The same structural stability holdsfor the unique, asymptotically stable solution (27c) of (27a,b).

The questions about the solution set of BAEs with aperiodic behavior, like(8)-(10), are the most intriguing ones. We believe that aperiodic solutions are allindividually unstable, but whether invariant manifolds of such solutions exist which


are stable, or asymptotically stable, is not clear. Likewise, the structural stability ofsuch manifolds, if they exist, is an open question. In any case, we suspect that aperiodicbehavior is pervasive for BAEs, i.e., that the equations giving rise to such behavioroccupy a "large set" in the space of all possible BAEs.

A possible approach to the study of the set of aperiodic solutions is through theirtopological entropy (e.g., Takens (1980)). This in turn is connected with predictabilityvia algorithmic information theory (e.g., Chaitin (1977)).

7. The Thomas formulation. The mathematical model presented here was formu-lated by R. Thomas (1978) in a slightly different form. The difference can best beexplained in terms of the memorization variables xo(t)= x(t- t). In our formulationthis is a purely delayed state variable; that is, xi(t) is determined only by the state ofthe system at time t-t. Thomas allows the memorization variable x(t) to dependon the state of the system up to time t: when a change takes place in the time interval(t- t0, t), then the memorization variable may change also.

Specifically, if for some t’ (t- t0., t) the state of the system is such that

(28a) xi(t’) f.(x x(t’), xz(t’), ", x,(t’)) xi(t tii),

then the memorization variable xii(t), previously equal to xj(t-tii), is changed to

(28b) xii(t) xi(t’).

This adjustment of the variables xi(t) in effect selectively erases some of the memoryof the system. The resulting solutions are usually simpler and hence easier to studythan the typical solution of our BAEs. Referring to the example (8)-(10) above, theincreasing complexity of the solution reflects the fact that the memory of the systemcontains more and more information as time goes on.

In Thomas’s "kinetic logic" formulation quasiperiodic solutions, i.e., solutionswith two or more incommensurate periodicities, are possible. But no truly aperiodicsolutions, for which lagged correlations are exponentially decaying, rather than beingthemselves quasiperiodic functions of the lag (Ghil and Tavantzis (1983), Joseph(1976, App. A), Lorenz (1973)) appear to exist.

8. Concluding remarks. The BAEs presented here, as well as Thomas’s relatedkinetic logic formulation, have certain shortcomings. There seems to be no generalbiological or physical justification for adjusting the memorization variables to erasememory selectively, as in "kinetic logic". On the other hand, the complexity andstability of solutions to BzEs, as introduced here, appear to be somewhat difficult tostudy in many cases of practical importance.

It would be interesting to introduce a more general formulation of the mathemati-cal model and then to choose a particular specification based on the physical orbiological system which is being studied. The present formulation of BAEs, with"discrete" delays, and the formulation of Thomas, with selective memory erasure,would appear as special cases of such a general formulation.

One direction of generalization is to allow the memorization variable x to be afunction of the state variables x over a time interval (ti t, t + At) centered aroundthe delay:

(29a) xii(t) f


where the symbol stands for a properly defined weighted averaging operator

(29b) f" X X

depending on t.Another approach would be to let the time delays 0 be random variables with

prescribed probability distribution. One may expect that such an approach, like (29),has a smoothing effect on the solutions.

The most interesting generalized framework seems to be a randomization in aslightly different sense. Define x(t) as random variables with values 0 or 1. Discretizetime so that a jump occurs within a fixed, elementary interval At h; time t, as wellas the delays , are only defined within a multiple of h. Assign a certain continuousprobability distribution to jumps, rather than the previous yes/no discrete distribution.The result is a generalization of Markov chains, in which distinct delays, rather thana single, unit delay for all variables, are possible. This provides also a connection withother simplified approaches to climate dynamics, via nonlinear stochastic differentialequations with multiple equilibria for the deterministic problem (cf. Benzi et al. (1982),Nicolis (1982a)).

Generalized Markov chains, as suggested above, could provide a formulationwhich is sufficiently flexible to allow the study of stable periodic and quasiperiodicmodel behavior, as well as of individually unstable aperiodic solutions associated withstatistically stable strange attractors. At the same time much of the simplicity of theprevious formulations would be conserved.

We conclude by noticing that increasing complexity is counterintuitive in manyphenomena on the human time scale: these are perceived as having stationary statistics,even if they are individually periodic or aperiodic. Evolution on the longest timescales, on the other hand, is perceived as an increase in complexity, whether in cosmic,biological or social terms.

Modeling of increase in spatial complexity and diversification is an active fieldof research in various sciences. BAEs seem to provide a relatively simple example ofincrease in temporal complexity, which reflects very crudely some aspects of evolution-ary phenomena. Periodic or quasiperiodic complexity, on the other hand, could be ametaphor for intermittence in fluid dynamic turbulence, and for related phenomena.

Appendix. The number of jumps o| solutions to BAEs.Proof o]’Lemrna 1 in 4. Suppose there are l distinct time delays. Without loss

of generality, we take - 1. To simplify the notation, we relabel the set of time delaysas follows:

{tx, t2, , tt}, 0 < tt < tt-x < < t2 < tx 1.

Define the set of linear combinations with integer coefficients of k delays

Pr : t= Y. nt, <m, {ntei=1

It is clear that it suffices to prove that the set P is finite, since the set P[m-1, m]in the lemma is iust a subset of the direct sum of the finite set A and the set P. Inparticular, the number of elements in P [0, m) is at most the product of the numberof elements in the two sets, A and P.

LetN be the number of elements in P". We have

P ={0, 1, 2,... m-l}

and so N" m. We prove that N7 < oo by induction.


If Nkm_l < x3, then we may write

Pk-1 ={P,P2,""", Pr},

where N N_. Then the set P’ is generated from P’-I as follows:

P’ {p" P pi + nktk, pi e PkQl, nk N, p < m}.

For each element p ofP we must have

so that

nk >= O, P Pi + nktk < m

where [q denotes the greatest integer less than or equal to q.For each pi P’-I there are [(m --p)/tk]+ 1 elements in P, so that

N -P=Nk-, + ,--iX ""Thus N < when N_, <. In particular, N? <. This completes the proof ofthe lemma.

An upper bound ]’or N?. We derive an upper bound for N’ as follows"

N’= +Ii= tk

and since N’ m we have

+1 =N_ +1i=1

)N’ < +1i--1 ?

The number oflumps per unit time. It is interesting to consider the upper boundfor the number of jumps of the solution per unit time, as a function of time and ofthe delays. Recall that the actual jump points of the solution are a subset of the setP in the lemma. Thus, an upper bound for the number of jumps in the intervalIra, m + 1) is given, in the case of constant initial data (cf. remark about P fq[0, m)being the direct sum of the "initial" jump set A and of P?), by

K? =N?+’

Consider the case 2 (two distinct time delays). We have

P7 ={0, 1,..., m-l},

so

Now

i= t2

re-(i-l) m-(i-1)]-I< <-t2 t2 t2

m-(i-1)


so that

m-(i-1i=x t2 i=x t2 +1),

or

<N < +2.t2 t2

Therefore

K2__Nr+l_N2[.lYt-[l(m-[-g "[-22 tz - t2

+m+l,t2

which is linear in m.We would like to find an upper bound forK for arbitrary I. We conjecture that

the maximum number of jumps per unit time in the case of distinct delays dependson time as l-x. Our conjecture is based on a geometrical argument for 2.

We can represent the set

by a grid c c R2"

P ={p ’p nxt, +n2t2; nl, n2EN, p <m}

( {(x, y)’x nit1, y n2t2; n,n2EN, x +y <m}

(see Fig. 3). Clearly N’ is equal to the number of gridpoints in , which may beestimated as follows. Consider the two rectangular boxes B1, B2 c R2.

BI= (x,y)’x-nltx, y=n2t2;n,n2l,O<=x<= tx, 0=<y < t2,

B2 {(x,y) x nxt,y n2t2, n,n2_[,O<=x<[ ] [22+ ] }+1 ta, O--<y <- 1 t2

From Fig. 3 it is apparent that the points in lie within a triangle whose area isbounded below and above by half the area of B1 and of BE, respectively. Thus, takinginto account the points which lie on the axes,

1[1+1][2 ] 21[ Jim+1 <NT<= +2 --+22 t2

or

1 m m., 1(<N2 < +2 +22 tl t2

Therefore,

l(m+lK’ N2 + -N2 <tl

+ 2)(m+lt2

1 m mO(m)+2

2 tl t2


B2

FIG. 3. The correspondence of the setP to a two-dimensional grid of points.

The argument generalizes to dimensions from the rectangles B1 and B2 toappropriate l-parallelepipeds, and from the triangles to the corresponding higher-dimensional tetrahedra. A technical difficulty lies in the fact that, as increases, "most"points will lie on the faces of the tetrahedra. The desired result is

K’ O(ml-1).Acknowledgments. It is a pleasure to acknowledge stimulating discussions and

correspondence with A. Mullhaupt, C. Nicolis and R. Thomas.

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Boolean Difference Equations, I: Formulation and Dynamic Behavior

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