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Differential delay equations in chemical kinetics. Nonlinear models: The cross-shaped phase diagram and the Oregonator

Irving R. Epstein and Yin Luo@ Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254-9110

(Received 12 March 199 1; accepted 29 March 199 1)

Delayed feedback plays a key role in most, if not all chemical oscillators. We derive general results useful in the linear stability analysis of models that explicitly incorporate delay by using differential delay equations. Two models of nonlinear chemical oscillators, the cross-shaped phase diagram model of Boissonade and De Kepper and the Oregonator, are modified by deleting a feedback species and mimicking its effect by a delay in the kinetics of another variable. With an appropriate choice of the delay time, the reduced models behave very much like the full systems. It should be possible to carry out similar reductions on more complex mechanisms of oscillating reactions, thereby providing insight into the role of delayed feedback in these systems.

I. INTRODUCTION

The mechanisms of chemical oscillators are nearly always complex, involving typically a dozen or more elementary reactions. Nevertheless, in many, perhaps even most cases, it is possible to construct relatively simple models, involving only a handful of key species, that accurately mim- ic the most important features of the dynamics. One aspect that has been remarked upon in a number of mechanisms and models for oscillating reactions is the presence of de- Zayedfeedback. In a recent analysis’ of mechanisms of chemical oscillators, we suggested that a delay between initiation of a feedback and its effect plays an essential role in many oscillating reactions.

In the first part of this series,’ one of us (I.R.E.) exam- ined a series of simple, but somewhat artificial linear model systems in which a time lag was explicitly introduced into the kinetic equations. Both analytical and numerical methods were employed to demonstrate that even very simple systems can display oscillatory behavior, if they include a delay. A sequence of consecutive first-order reactions was shown to be reducible to a much smaller set of reactions by replacing inessential intermediates with time lags in the kinetics of the remaining essential species.

Here we consider the effects of delay on two more realistic, nonlinear, but still relatively simple models. We are able to derive some results analytically, but are forced to rely more heavily on numerical methods to extract the detailed behavior ofthese systems. The two systems, the Boissonade- De Kepper (BD) model3 that yields the cross-shaped phase diagram and the Oregonator model4 of the Belousov-Zha- botinskii (BZ) reaction, have played key roles in the devel- opment of nonlinear chemical dynamics. The BD model provides the theoretical underpinning for the systematic search procedure that has led to the discovery of many new oscillatory reactions.5 The Oregonator has served as the pro-

‘) Present address: Department of Physical and Inorganic Chemistry, Uni- versity of Adelaide, Adelaide, SA 5001, Australia.

totype in the study of a wide range of spatial and temporal phenomena.”

We first present some general results applicable to the linear stability analysis of the differential delay equations used to model systems with time delays. We show in each of our two models how the species that generates a delayed feedback can be eliminated and replaced by an explicit time delay in the kinetics of the remaining species. We then ana- lyze the results both analytically and numerically. We consider how the magnitude of the time delay is related to other parameters in the model, and we speculate on how the con- clusions reached here may be generalized to more complex systems.

II. LINEAR STABILITY ANALYSIS OF DIFFERENTIAL DELAY EQUATIONS

By a differential delay, or differential difference equation,’ we mean one in which the values of one or more variables at a given time influence the evolution of the system at some subsequent time, i.e., the rate of change at time t depends not only on the state of the system at t, but also on its state at some earlier time t - r (and possibly at other earlier times t - r’, . . . as well). Thus, in the case where there is a single time delay r, a set of rate equations may be written as

dx(t)/dt = f[x(t),x(t - T,], (1) where x is a vector of concentrations, and f is a vector of rate terms.

To find the steady state(s) x, of a system like that described by Eq. ( l), we simply take x(t) and x(t - T) to be equal and require the time derivatives to vanish:

f(x,,xs) = 0. (2)

A. The secular equation

As in the case of ordinary differential equations (ODES), many of the properties of the system ( 1) are deter- mined by the stability of the steady states (2) to small per- turbations. To assess this stability, one proceeds exactly as for ODES, making the ansatz that

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I. R. Epstein and Y. Luo: Delay equations in chemical kinetics 245

x(t) = x, + a exp(ot) (3) substituting Eq. (3) into Eq. ( 1) and linearizing the resulting system by dropping terms higher than first order in the perturbation a. The familiar secular equation for the eigenvalues w results, but with one crucial difference. The equation is no longer a polynomial in w, but is a transcendental equation, containing terms in exp( - wr).

To see how these exponential terms arise, consider the simple case in which there is only a single dependent variable x, and f in Eq. (1) depends only upon the delayed value x(t - 7). On substituting Eq. (3) in Eq. (l), we have

waexp(wr) =fcx, +aexp[w(t-r)]}

=f(x,) +crJ,(x,)exp[w(t--11 + *.a, (4)

where J, (x,) = (af/ax,) IX,, x, =x(t - r), and the right- hand side of Eq. (4) has been obtained by expanding the functionfin a Taylor Series about the steady state and ne- glecting terms of second and higher order. If we substitute Eq. (2) in Eq. (4) and divide by a exp(ot), we obtain an equation for 0:

J,(x,)exp( -or) -w=O. (5) The meaning of w is exactly the same as for an ODE: if

Re(w) < 0, the solution is stable, if w has positive real part, the solution is unstable. The time delay, however, has introduced the additional term exp( - or) into Eq. (5). In the more general multidimensional Eq. ( l), the secular equation becomes

det[J(x,,xS) + J,(x,,x,)exp( -UT) -WI] =O, (6) where I is the identity matrix and

J,,(XsJ*) =Cd~[x(t),X(t--)l/dx,(t)},(,,=,(,-,,=.~, Ua)

J,,, (x, 9~s 1 =CaJ;x(t),X(t--)l/ax,ct-7)}X(I)=X(f--r)=X,.

(7b) In other words, whenever a partial derivative is taken with respect to a delayed variable, a factor of exp( - or) will multiply the resulting term in the Jacobian matrix.

B. Three theorems about transcendental equations

The equations that result on expanding Eq. (6) cannot in general be solved analytically, even in the one-dimensional case. Furthermore, they commonly have an infinite number of solutions. Nevertheless, it is possible in some cases to decide, without explicitly solving a transcendental equation oftheformg(w) +h(~)exp( -wr) =O,wheregandhare polynomials, whether or not the roots all have negative real parts, i.e., whether or not the steady state is stable.

To obtain such results, we shall make use of three theorems about the roots of transcendental equations. The first, which will be useful in considering systems with a single concentration variable, is due to Hayes.8 Theorem 1: All roots of the equation

pe’ $ q - ze’ = 0, (8) where p and q are real, have negative real part if and only if

(a)p<L @a) and

(b)p< -q<(a: +p*)“*, (9b) where a i is the root of a = p tan a such that 0 < a < QT. If p = 0, we take a, = IT/~.

A second theorem, given by Bellman and Cooke,9 is useful for analyzing the stability of some two-variable differential delay equations. Theorem 2: Let

H(z) = (2 + pz + q)e’ + rz, (10) wherep is real and positive, q is real and nonnegative, and r is real. Denote by ak (k>O) the sole root of the equation

tana= (q-a*)/pa

which lies on the interval (kn- - n-/2, kn + rr/2). Define the number w as follows: (a) if r>O, w is the odd integer k for which ak lies closest to q”’ ; (b) if r < 0, w is the even integer k for which ak lies closest to q”*.

Then all the roots of H( z) = 0 have negative real parts if and only if

1 + (r/p)cos a, > 0. (11) The restrictions on the coefficients in Eq. (10) make

Theorem 2 less generally applicable to two-variable systems than Theorem 1 is to single variable models. Bellman and Cooke’ give other stability theorems in which the term rz in Eq. (10) is replaced by either a constant or a term propor- tional to z*, while Schiirer” gives a theorem similar to Theorem 2 but with (rz + s) replacing rz and different conditions on p and q. We present Theorem 2 because it is the most relevant to the BZ model discussed below.

Macdonald” presents another, more general approach to evaluating the stability of solutions to differential delay equations based on a geometric method for finding bifurcation points of the transcendental secular equation. We present here one of several useful results he obtains. Theorem 3: The roots of the equation

w*+p’w+q’+ (r’m+S’)e-“’ (12)

have negative real parts for all values of r if the following conditions are satisfied:

q’+s’>O, (13a)

p’ + r’ > 0, (1%) P(f’) = Op*-q’)*+ (p’2-r’2)f2-d2>0 (13c)

for real positivef*. The function Pin Eq. ( 13~) is a quadratic in f * that approaches CO as f * -) k 00. Therefore, inequality ( 13~) is satisfied if and only if at least one of the following two conditions holds:

(p’* - 2q’ - r’*)* <4(q’* - s’*), (14a) (p’* - 2q’ - r’*) > 0 and (q’* - s’*) > 0. (14b)

When inequality (14a) holds, Pv’) has no real roots and thus is always positive. Conditions (14b) ensure that if there are real roots they occur for negativef’ and that P( 0) is positive.

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246 I. Pi. Epstein and Y. Luo: Delay equations in chemical kinetics

Ill. THE CROSS-SHAPED PHASE DIAGRAM

The BD model contains a primary variable x, whose dynamics are governed by a cubic rate law, and a feedback variabley, which, with appropriate values of the parameters, provides a delayed feedback that causes the primary bistable system to become oscillatory. The model is given by

dx/dt= -(x3+x+/2) -ky, (154 dy/dt = (x - y)/T. (15b)

Linear stability analysis3 of Eqs. (15) with T> l/p yields a characteristic cross-shaped phase diagram in the k - A plane with triangular regions possessing a single stable steady state being separated by a region of bistability between those states and a region of stable periodic behavior (see Fig. 1). If the feedback is too slow, i.e., if T-c l/p, one finds either bistability or a single steady state, but no oscillations are possible. Further nonlinear stability analysis”” re- veals more complex behavior in the immediate vicinity of the cross point where the different regions join, but these details are not germane to the present study.

Since the role of the variable y in Eqs. ( 15 ) is to generate a delayed feedback, and since at steady state we have x = y, we constructed a modified version of the BD model by dropping Eq. ( 15b) and replacing y( t) in Eq. ( 15a) by x( t - 7). The new model is then

dx(t)/dt = - [x(t)‘-px(t) +A ] - kx(t - 7). (16)

A. Linear stability analysis The secular equation obtained from Eq. ( 16) is

- 3x: + p - k exp( - wr) - w = 0. (17) Equation ( 17) is easily transformed to the form of Eq. (8) withz=tir,P= (-3x:+p)randq= -kr.Theorem 1

can then be used to determine the stability of the steady state X 5’

Substitution ofp and q into Eq. (9a) yields as a condition for stability

3xf -p + l/r>O. (18) Equation (9b) yields two further conditions for stabil-

ity:

3x:-p+k>O (19) and

[(3x,2 --/A)~ + (a,/r)2]1’2 - k>O,

whereO<a,<nand (20)

a, = ( - 3x3 +p)r tan a,. (21) Condition (19) is identical to one of the two stability

conditions resulting from linear stability analysis3 of the full BD model, while condition (18) is just the other condition with the delay time T replacing the feedback time T. The third condition (20) is new and makes possible additional regions of instability of the steady states in this model.

Conditions (18) and (19) imply that, for a given set of parameters (,u,R,k) if a steady state of Eqs. ( 15) is unstable with feedback time T, the corresponding steady state of Eq. ( 16) with delay time r will also be unstable. Steady states of the delay equation ( 16) may also be unstable, even if the corresponding states of Eqs. ( 15) are stable, if condition (20) is violated.

We follow Boissonade and De Kepper3 in considering the phase diagram in the (k - A) plane withy and r fixed. In systems described by Eqs. ( 15 ) , two cases need to be distin- guished: ( 1) T> l/p, in which a cross-shaped diagram with regions of monostability, bistability and oscillations is obtained; and (2) T< I/,u in which only steady state behavior (monostability or bistability ) is found. Since conditions ( 18) and ( 19) when applied to systems governed by Eq.

FIG. 1. Phase diagram in the (k-/2) plane obtained from linear stability analysis ofthe delayed BD model, Eq. ( 16) with p = 3, T = 1. Dotted line separates regions in which there are one or three steady states. Dashed lines are stability boundar- iesgiven by Eqs. ( 18) and (19), equivalent to the ordinary BD model with T = 1. Sol- id lines are stability boundary for full models Eqs. (18)-(20).

k

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( 16) yield a phase diagram identical to that found by Bois- sonade and De Kepper, we focus on the consequences of condition (20). In particular, we shall look for the boundary of stability specified by making condition (20) an equality and consider its location relative to the stability boundaries already generated by the other two conditions.

Substitution of Eq. (21) into Eq. (20) and some alge- braic manipulation yields a simpler expression for the stability condition:

a,/sin a,u(a,) > kr. (22) We observe that in the allowed range 0 <a, < P, u(a, )

increases monotonically from 1 to CO. Thus if r < l/k condition (20) or (22) can never lead to instability, regardless of the other parameter values. To trace out the stability boundary in the (k - R ) plane we proceed as follows. Allow a, to increase incrementally from 0 to 7~. For each value of a,, with y and r fixed, Eqs. (20) and (22) treated as equalities and the steady state form of Eq. ( 16) yield successively

k = a,/(rsin a,), (23a)

x, = f C[p - a,/(7-tana,)]/3]“2, (2%) A= -x:+(/i-k)x,. (23~)

For each a,, Eqs. (23a) and (23~) give a pair of points on the curves separating the stable and unstable regions for the steady state(s) given by x, in Eq. (23b). In case 1, shown in Fig. 1 for the same parameters discussed by Boissonade and De Kepper, these curves start at k = l/r and are initially indistinguishable from the stability boundary given by conditions (18) and (19). As a, increases, the stability boundaries separate, and that provided by Eq. (23) yields new regions of instability. In particular, as shown in Fig. 1, the cross point P [k = (2,~ + l/7)/3,/2 = 0] at which the monostability, bistability and oscillatory regions meet is shifted to the left (P’), thereby creating a larger region of oscillation’” as well as regions that have just one stable steady state instead of the two predicted by conditions ( 18) and (19) or found in the Boissonade-De Kepper model (15).

The location of the new cross point P’ is of particular interest. It may be found by solving Eqs. (22) and (24) si- multaneously for k and a, at the cross point P’. Equation (24) is obtained by setting il = 0 in Eq. (23~) and substituting the resulting nonzero value of x, in Eq. (21).

(3k - 2p)r tan a, = a,. (24) In Fig. 2 we show how P’ moves toward smaller k as r is

increased; i.e., longer delays increase the region of parameter space in which oscillatory behavior may occur. How far can the cross point move as the delay is made indefinitely large? To answer this question, we divide Eq. (22) by Eq. (24) and rearrange to obtain an expression for k,. as a function of p and a,:

k,. = 2,~~/(3 - cos a,). (25) The limit of small r corresponds to a, -+O and yields, as ex- pected, k,+ = ,u. The large r limit corresponds to a, --+ rr and shows that the cross point can never move to k values less than ,u/2, no matter how long we make the time delay.


tau

FIG. 2. Locationofcrosspoint P’ (k,,O) inphasediagramofthe typeshown in Fig. 1 as a function of delay time T when Q-> l/p (case 1). Also shown is parameter a, in Eq. ( 2 1) . Here, /J = 3.

In case 2, r < l/p, the BD phase diagram obtained from a linear stability analysis is quite simple. There is a wedge of bistability at low values of k and ]il 1. The rest of the parameter space has only a single steady state. No oscillatory behavior is found. Our delayed version Eq. ( 16) gives rise to a new region of oscillation as a result of condition (20). As shown in Fig. 3, this region begins on the k axis at k = l/r and broadens as k increases. Thus the phase diagram for the delayed BD model becomes a double wedge, with a region of bistability separated from a region of oscillation by an inter- vening region of monostability. As r-+ l/p, the width on the k axis of the monostable region shrinks to zero, and the phase diagram approaches the cross-shaped diagram in ap- pearance. So long as kr > 1, the system can oscillate for at least some value of /2. A result given by MacDonald14 guar- antees that if the fixed point is unstable for r,, it will be unstable for r2 > rr at the same k, R and ,u.

B. Numerical analysis

The linear analysis offered above does not yield the complete bifurcation behavior of which this system is capable. As in the case of the BD mode1,37’2 a full nonlinear stability analysis may reveal a more detailed structure than that shown in Figs. 1 and 3. This analysis is beyond the scope of the present work, but we have investigated the behavior of Eq. ( 16) numerically for several sets of parameter values, and we present the results in this section. The algorithm used for the numerical integration of the differential delay equations in this section was the modified Gear method described in Appendix A.

Our preliminary examination of several parameter sets for case 2 suggests only minor differences from the predic- tions of the linear stability analysis. For k only slightly greater than l/r, the bifurcation from steady to oscillatory behavior as/z is varied appears to be of the supercritical Hopf type, with relatively constant period and oscillatory amplitude increasing roughly as the square root of the distance from the bifurcation point as shown in Fig. 4. This behavior

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248 I. I?. Epstein and Y. Luo: Delay equations in chemical kinetics

x

5.0

-2.5

-5.0

monostable

I I / I I I I I , I I 0 1 2 3 4 5 6

k

is in agreement with the linear stability analysis. At somewhat larger values of k, the bifurcation apparently becomes subcritical; our numerical calculations show a narrow region of hysteresis between the steady and oscillatory states. For example, with the other parameters as in Fig. 4, at k = 6, the region of hysteresis extends about 0.2 units on either side of the value /z = f 5.15 given by the linear analysis.

Boissonade and De Kepper3 found that a linear stability analysis was adequate for case 2, but a full bifurcation analysis revealed a far richer structure for case 1. We obtain similar results for our delayed model. A detailed phase diagram for one set of parameters is shown in Fig. 5. The region around the new cross point P’ is similar to that found in the neighborhood of the cross point Pin the BD model (cf. Ref.

1

-1

0 0.2 0.4 0.6 Od

lambda

FIG. 4. Variation of amplitude of oscillation with bifurcation parameter R withy = 3, k = 4, r = 0.3. Triangles show maximum and circles show min- imum x in oscillatory state; filled circles show steady state values.

FIG. 3. Phase diagram in the (k - A) plane obtained from linear stability analysis of the delayed BD model, Eq. (16) withy = 3, r= 0.3. Oscillatory region results from stability condition (20).

3, Fig. 3). In particular, we find small areas of tristability in which both steady states as well as the oscillatory state are stable.We also see a rather broad region of bistability between one steady state and the oscillatory state; the Hopf bifurcation is subcritical. Although we cannot make a complete comparison of the two models without further analysis of the delay model, our preliminary results suggest that the bifurcation structure of the BD and the delay models in case 1 are the same in most, perhaps all details. The major difference is that the regions of oscillation in the delay model at a given rare larger than the corresponding regions of the BD model for the same value of T.

IV. THE OREGONATOR

The Oregonator is undoubtedly the most thoroughly studied model in nonlinear chemical dynamics. It is not only simple enough to allow for a considerable amount of analysis, but it faithfully models much of the chemistry as well as the dynamical behavior of the real Belousov-Zhabotinskii reaction. The Oregonator in the form first proposed by Field and Noyes consists of five “elementary reactions:”

A+Y+X, (Ml) X+Y+P, CM21 B+X-+2X+Z, (M3)

2X-Q (M4)

z+fy, (MS)

where A = B = BrO; are taken to be present at constant concentrations, P = HOBr and Q = BrO, + HOBr are treated as inert products, X = HBrO,, Y = Br- and Z = Ce(IV) are the variable species, andfis a stoichiometric coefficient that specifies how many bromide ions are generated for each Ce(IV) consumed during the organic portion of the reaction. Typical values for the concentrations and rate constants for conditions under which oscillations


\ J’\

-14 I , I , I , 1 I I I 1 I 1 2 3 4 5 6

k 6

0.4

0.2

4 0.0

-0.2

-0.4 1

b

/

.85 \ I 1

1.95 2.05 2.15 2 k


occur are A= B=0.06 M, k,, = 1.34 M-’ s-‘, k,, = 1.6~10~ M-‘s -l, kM3 =8x103 M-Is-‘, k,, =4x107 M-‘s-l, k,, = 1 s - ’ , f = 1. These reactions

yield the following set of rate equations for X, Y and Z:

dX/dt = k,,AY - k,,XY + k,,BX - 2kM4X2, (26)

dY/dt = - k,,AY - k,,XY j-j&Z, (27) dZ /dt = k,, BX - k,, Z. (28) Examination of Eqs. (Ml)-(M5) suggests that if Eq.

(M5) is faster than Eq. (M3), the two equations can be combined into the single step

B +X+2X +fY (M3’) thereby making it possible to eliminate Z and reduce the system to two variables. Field and Noyes attempted such a reduction, but found that the resulting two-variable system was incapable of oscillatory behavior. We proposed’ that species Z is essential for oscillation, because it provides a mechanism for introducing a delay in the regeneration of Y from X that occurs in reactions (M3) + (M5). The single step (M3’) forces this feedback to occur instantaneously, thereby removing the possibility of oscillation.

As a test of the hypothesis that the key role played by Z is to generate a delay in the production of Y and in order to assess where that delay might be most important, we exam- ined two modified two-variable “delayed Oregonators.” In both versions, steps (M3) and (M5) were replaced by (M3’). In version A, we substituted the “delayed concentration” Y( t - 7) for the instantaneous concentration Y(t) in step (M 1) . In version B, the delay was introduced by replacing Y(t) in step (M2) by Y( t - 7). The timer represents the delay produced by species Z. The resulting rate equations can be written as

dX/dt = k,,AY, - k,,XY, + k,,BX - 2k,,X2, (29)

dY/dt = - k,,AY, - k,,XY* +jk,,BX, (30) where YA = Y(t - 7) or Y(t) and Ya = Y(t) or Y(t - 7) in versions A or B, respectively.

A. Linear Stability Analysis

FIG. 5. (a) Phase diagram in the (k - 1) plane obtained from nonlinear (numerical) analysis of the delayed BD model, Eq. (16) with p = 3, r = 0.7. Symbols: f, steady state 1; I, steady state 2; 0, oscillation; 0, bistability between steady states 1 and 2. Combinations ofsymbols signify multi- plicity between the corresponding states, e.g., a diamond with a filled circle in it signifies tristability of the two steady states and the oscillatory state. (b) expanded view of the neighborhood of the cross point at (2.08,0),

Before analyzing Eqs. (29) and (30) we transform from X, Y, and t to dimensionless variables4 a, ~7, and 0 where

X= (k,,A/k,,)a= 5.025~10~“a,

Y= (k,,B/k,2)n=3.000x10-7~,

t = O/(k,,k,,AB)“’ = 0.16100.

In these variables the rate equations become

da/d@ = ~(7~ - vBa + a - qa’), (31)

dr]/dO = s-‘( - vA - plea +fa), (32) where f = 1, s = (k,, B/k,, A) “’ = 77.27 and q = 2k,, k,,A/(k,,k,,B) = 8.375~ 10e6. The delayed variables qa and vn are defined in a fashion analogous to their dimensional counterparts YA and YB, respectively, in Eqs. (29) and (30). The two steady states in either version A or B are easily found to be (a~~ ) = (0,O) and”


250 I. Ft. Epstein and Y. Luo: Delay equations in chemical kinetics

a, = [ - 1 + (1 + 8/q)“*]/2 = 488.18,

1, = a,/( 1 + a,) = 0.997 96.

Using Eqs. (6) and (7)) we obtain the secular equations associated with Eqs. (3 1) and (32). For version A we have

w2 + w[s(?j - 1 + 2qa) + a/s] + (f- 1-t 2qa)a

+oe -“T/s + (277 - 1 -f+ 2qa)e-“‘= 0.

Version B gives (33)

o*+w[s(~- 1+2qa) + l/s] +2qa+2v-- 1 -f

+ aLwecoT/s + a(f- 1 + 2qa)e-“‘= 0. (34) Note that when r goes to zero, Eqs. (33) and (34) both become identical to the secular equation for the instantaneous version of Eqs. (3 1) and (32) in which 7,rA = 778 = q(t):

62 + w[s(v - 1 + 2qa) + (a + l)/sl + If- 1+2q(a+ l)]a+27- 1 -f=O. (35)

On inserting the numerical values of the parameters, we find that at the nonzero steady state, Eq. (35) becomes

a* + 6.805~ + 3.995 = 0

for which both eigenvalues w have a negative real part, i.e., the steady state away from the origin is stable, and the system will not oscillate (the steady state at the origin is always a saddle point). This behavior persists for any reasonable choice of the concentrations, stoichiometric factor and rate constants, which is consistent with Field and Noyes’ obser- vation that the reduced model without delay does not yield oscillations.

Equations (33) and (34) can be transformed to the form of Eq. (10) by the procedure outlined in Appendix 2. The coefficients p, q and r for the two steady states in versions A and B are given in Table I. We see that Theorem 2 can be used to investigate only the stability of the steady states of version A, since at the origin version B yields a polynomial secular equation without exponential terms, and for the other steady state of version B, p < 0, so the conditions of Theorem 2 are not fulfilled.

We consider first version A. For the steady state at the origin, d/c < 0, so according to Appendix B, Theorem 2 offers the possibility of proving this state to be unstable. How- ever, we also observe that r/p < 1 for all positive values of 7, so the stability condition of Eq. ( 11) will always hold, regardless of the value of u,. Thus, all that we can conclude in

*:11 \ ’ I I I,

G 2 :’ ‘\ .I

tz 1 -I -0 I

I s o -J

m 5 z -1 -

> -2 -, ‘L

0 2 4 6 6

tau FIG. 6. Real (v) and imaginary (w) parts of stability eigenvalue o for the nonzero steady state in version B of the delayed Oregonator.

general is that Re( w) < - d/c > 0 for all r. This result says nothing about the stability of the origin. Our numerical results suggest that the origin, which we know to be unstable for r = 0, is in fact unstable for all positive r.

Theorem 3 is more enlightening with respect to the other steady state of version A. Equation (33) is of the same form as Eq. ( 12), with the coefficientsp’, q’, r’, and s’ in that equation taking the values 6.793, 3.992, 0.012 94 and 0.004 08, respectively. Inserting these values into Eqs. ( 14) shows that both conditions ( 14b) are satisfied, so this state is stable for all values of 7.

For version B, the secular equation for the steady state at the origin yields a quadratic equation in w independent of r, and it is easily shown that one of the two roots has positive real part; the origin is a saddle point.

Neither Theorem 2 nor any similar result” is applicable to the transcendental secular equation at the other steady state. One can, however, employ another, somewhat more tedious and less comprehensive approach outlined in Ap- pendix B. If we write w = v -t iw, where v and w are real, we obtain from the secular equation in w two simultaneous transcendental equations for v and w. These can be solved numerically for v and w. The number of roots is in general infinite, but the existence of a single root with v > 0 suffices to demonstrate that the steady state is unstable.

We carried out the above procedure on the nonzero steady state of version B of the delayed Oregonator. In Fig. 6,

TABLE I. Coefficientsp, 4, r in JZq. ( 10) for steady states of the delayed Oregonator.

Version A Version B

Steady state (%?7s) P 4 r Conclusion

(070) (488.18,0.997 96)

231.08~ 6.1601~ 1194072 1.94583

0.0129427 e - ‘w’~ 0.012942re”“‘q’ d/c<O, Stable for all r by

but r/p < 1 Theorem 3 See text

VW)

. . .

. . .

.. No effect of time delay

Saddle point

(488.18,0.997 96)

- 0.7765~ 0.09552i

6.3178~e~~~'*' Numerical

analysis required

J. Chem. Phys.,Vol.95, No. 1,l July 1991 Downloaded 15 Jun 2006 to 129.64.99.137. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

0


I -

-6

-9

d

/ ,, 1 . . . .

10 20

time (s)

0

1

-7 -

-a -

-9 -

-lO-

1 I . . . I . .

10 20

t ime(s)

FIG. 7. Complex oscillations in the delayed Oregonator, Version B, with short delay times. (a) Simple oscillation (period l), 7 = 0.06 s; (b) per iod 3, ~=0.08s;(c)period4,r=0.10s;(d)period2,~=0.15s.

we show how v and w for the root of interest vary with r. Note that v changes sign from negative to positive as 7 increases through a value of about 0.245 or 0.039 s. We were unable to find a smaller value of 7 which gave rise to a root of the secular equation with positive real part.

B. Numerical analysis

While the above linear stability analysis offers some in- formation about the dynamics of the delayed Oregonator, the limited applicability of Theorem 2 necessitates numerical investigation of Eqs. (3 1) and (32) over a range of parameters. We wish to know whether the value Q- = 0.04 s obtained from our “brute force” linear stability analysis ac- tually coincides with the onset of oscillations in Version B. What sort of oscillations are obtained from the delayed models,and is there a choice of T that yields oscillatory behavior similar to that displayed by the full Oregonator?

The linear stability analysis suggested that oscillatory behavior should occur in Version B even at relatively small values of the delay time. We find that, in agreement with those results, the nonzero steady state is stable for T less than about 0.045 s, and that oscillatory behavior appears for all delay times longer than that.

Our numerical studies reveal the existence of two qual- itatively different kinds of oscillation. For very short delay times we observe complex, mult ipeaked oscillation with periods of a few seconds and relatively low amplitudes. Some examples are shown in Fig. 7. We initially anticipated that the period 2 and 4 oscillations were part of a period doubling sequence. However, our subsequent discovery of the period 3 behavior and our failure thus far to find any chaotic behavior suggest that a different kind of bifurcation sequence is in- volved. We have not yet explored this sequence fully. This behavior does not appear to be related to that of the normal Oregonator, since the amplitude and period of the oscillations are quite different. It may be that these oscillations can yield insight into the observations of Gyijrgyi and FieldI that spatially coupled BZ reactions can lead to complex periodic and chaotic behavior.

As the delay time is increased, the complex oscillations give way first to similar high frequency oscillations with a single nearly symmetric peak and then to relaxation oscillations resembling those seen in the ordinary Oregonator. An example with 7 = 2.00 s is compared with a simulation of the Oregonator in Fig. 8. While the oscillations in the two models are certainly not identical, the similarities of the periods,

J. Chem. Phys., Vol. 95, No. 1,i July 1991 Downloaded 15 Jun 2006 to 129.64.99.137. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

252 I. Ft. Epstein and Y. Luo: Delay equations in chemical kinetics

-0 ’ f I I I I 60 80 100 120

time (S)

-4 -

-6 -

-a 1 I I I I 60 60 100 120

time (s)

FIG. 8. Oscillatory behavior in (a) the ordinary Oregonator (T = 0 s), and (b) the delayed Oregonator, version B with T = 2 s. All other rate constants and concentrations as specified in the text and Ref. 5.

amplitudes and waveforms are striking. It appears that Ver- sion B provides a useful model for the behavior of the system.

V. CONCLUSION

We have demonstrated that two of the classic models in the study of nonlinear chemical dynamics can be reduced by deleting one variable and replacing its feedback effects by a delay in the rate term(s) for another variable. With a proper choice of the delay time, the qualitative dynamics of the original model are maintained, though there is a general ten- dency for the delay models to show oscillatory behavior over

a broader range of parameters. Our calculations on the Boissonade-De Kepper model

suggest that a reasonable choice for the delay time T is the characteristic time T for the feedback that is to be mimicked by the delay. Another choice, which is usually of a similar magnitude (0.7 s in the BD model withy = 3, k = 6, /2 = 0, T = 1 ), is the lag in the full model between the maxima in the variable to be deleted and the variable whose kinetics are to be delayed. For the BZ reaction, the characteristic time for Version A is (k,, A) - ’ z 12 s. For Version B, N.,,~,) - ’ ~0.025 s for the steady state or 12 s for the pseu- do-steady state X at which the Oregonator spends most of its


time. The lag time between the maxima in Y and Z is 2.2 s, which suggests that this quantity may be the most appropriate estimate of the time lag introduced by the feedback.

The results obtained with delay times less than l/p in the BD model suggest that it should be possible to generate oscillations from a bistable system by introducing a simple feedback with a sufficiently long delay and/or a sufficiently strong coupling. Experiments in progress in this laboratory on the arsenous acid-iodate system,” which displays cubic kinetics” like the BD model, confirm this prediction.

Perhaps surprisingly, our study of delay models has shed light on the chemistry of the Oregonator and, by impli- cation, of the BZ reaction. The fact that Version A cannot oscillate for any value of the delay time, while Version B gives oscillations of roughly the same period and amplitude as the full Oregonator with a delay time roughly equal to the lag between the peaks in Y and Z strongly implies that the key dynamic role of species Z (ceric ion) is to introduce a delay into step (M2), the removal of the autocatalytic species HBrOz by the bromide regenerated in step (M5). Delay effects on step (M 1 ), the generation of the autocatalyst from bromate and bromide, appear to be of no consequence for the oscillatory behavior.

There remain many interesting and difficult questions to explore about the role of delay in complex kinetic systems. The models we have considered here are caricatures of actual chemical oscillators and contain, even before the simplifi- cations we introduce, no more than three variables. Realistic models require a dozen or so variables. How many of these can be replaced by delays and how to choose those delays in any particular system are issues that have yet to be ad- dressed. The complex oscillations found with short time delays in the Oregonator are intriguing, whether or not they have any relation to the behavior of the actual BZ reaction. Dynamically, short delay times correspond to a high value of k,. Perhaps organic substrates that yield such k, values might also give rise to the sort of rapid, complex oscillations seen in our calculations with small r. What kinds of complex dynamics are possible if one adds delay to a reaction mechanism? What relationships between the delay time and the characteristic times of the mechanism result in complex behavior? Can chaos result, and under what conditions? The existence of a period three solution [see Fig. 7 (b) ] is certainly suggestive.” Since even single variable differential delay equations are in a sense infinite dimensional (one must speci- fy the initial condition on an interval, i.e., at an infinite number of points), there is no inherent limitation on the com- plexity of the behavior that can arise. We are only just beginning to understand the dynamic complexities offered by systems containing delays.

ACKNOWLEDGMENTS

This work was supported by the National Science Foun- dation. We thank Joachim Weiner, Craig Hacker, and Ist- v&n Lengyel for helpful discussions, Frank Buchholtz for suggesting the Taylor series approach to numerical integration, and Bard Ermentrout for calling our attention to Mac- Donald’s text on differential delay models.

APPENDIX A: ALGORITHMS FOR NUMERICAL INTEGRATION OF DlFFERENTlAL DELAY EQUATIONS

A critical evaluation of methods for the numerical solution of stiff ordinary differential equations2’ suggests that the availability of more than one method is an important element in the numerical analyst’s arsenal. Since we antici- pate the need to solve equations that not only contain delay terms but are also stiff, we have developed two quite different approaches.

The first is a modification of a method designed for numerically integrating stiff ordinary differential equations. We start from the GEAR program, which employs a back- ward difference formula with adaptive step size, as imple- mented by Hindmarsh. 2’ To deal with differential delay equations we require, in addition to the form of the equations, several further pieces of input: the length(s) of the time delay ( s ) and the values of the delayed variable ( s ) over the initial interval(s).

Instead of allowing the program to integrate over the largest time increment consistent with the error parameters, we force it to compute and store the values of the variables each ET,, where E is a small number (e.g., 10 - ’ or 10e3) and 7, is the smallest time delay in the problem. Each delayed variable is treated as a parameter in the subroutine that contains the “differential” equations, e.g., the differential delay equation

dx/dt = - kc( t)y( t - T)

would become

DX( 1) = - PAR( 1)*X( l)PAR(2),

where X and DX are the arrays containing the variables and rates, respectively, and PAR is the array that contains the parameters (rate constants, temperature, time delays).

At each integration step, a test is made to find the appropriate value of the delayed variable to insert into the parameter array. Both the parameter array and the array of stored variable values are updated at each time step. Higher accuracy can be achieved at the cost of increased computation time by interpolating the delayed variable from the previously computed stored values.

Our second approach utilizes a fourth order Taylor series. Instead of treating delayed values as parameters, we treat them as variables, but save values for one complete delay interval. The interval 7 is divided into a large number N of steps (we used N = 32 000 for most of the BZ calculations). The non-delayed variables are stored as arrays of length N, while delayed variable arrays have length 2N, where the first N locations hold the values for the previous interval. The integration proceeds N steps at a time, moving the solution along by 7 units of time. Variables are output at specified intervals and updated as the calculation proceeds.

The basic element in the algorithm is the Taylor series approximation, which we may write

X(J + 1) =X(J) + TAU*(DX(J) + TAU/2.*{D2X(J)

+ TAU/3.*[D3X(J) + TAU/4.*D4X(J)]}) The arrays X, DX, D2X, . . . contain the variable and its

derivatives. For example, a rate equation



254 I. R. Epstein and Y. Luo: Delay equations in chemical kinetics

dx(t)/dt = - k,x(t)y(t - 7) - kg(t)

leads to the equations

DX(J) = - PAR( l)*X(J)*Y(J) - PAR(2)*Y(N + J),

DXZ(J) = - PAR(l)*[DX(J)*Y(J) + X(J)*DY(J)]

- PAR(2)*DY(N + J), etc.,

where the array PAR contains the rate constants, the X and DX, etc. arrays are of length Nand the Yand DY, etc. arrays are dimension 2N with the first N elements holding the values from the previous interval. To treat stiffer equations or to increase the accuracy, one can either go to more terms in the Taylor series or decrease the step size by choosing a larger N.

APPENDIX B: TRANSFORMATIONS OF TRANSCENDENTAL EQUATIONS RESULTING FROM STABILITY ANALYSIS OF DIFFERENTIAL DELAY EQUATIONS IN TWO VARIABLES

In analyzing the stability of a pair of coupled differential difference equations, one often encounters a transcendental equation of the form

W*+uW+b+cwe-“‘+de-“‘=O. 031) In order to apply Theorem 2 to such an equation, it is neces- sary to transform the variables.

We first multiply Eq. (B 1) by ?eor and set y = or. The equation becomes

(Y*+uyr+b~)eY+c~+d~=o. (B2) To convert Eq. (B2 ) to the form of Bq. ( lo), we make

the further substitution z = y + dr/c. After collecting terms and multiplying through by edr’c, we obtain Eq. (10) with

p+-yr, q=(b+$ -$)I? r = credr’c.

Note that the variable z is offset from the true stability variable w, i.e., z = (w + d/c) r. Theorem 2, if applicable, can tell us only if the roots of Eq. ( 10) for z have negative real parts. Therefore, if d /c > 0, and Theorem 2 implies that all the roots z have negative real parts, we can conclude that Re(w) < - d/c < 0, and the steady state is stable. However, if the Theorem shows that Eq. ( 10) has roots z with positive real parts, we cannot deduce that Re( w) > 0, and the question of stability remains open. Similarly, if d/c ~0, the Theorem will only allow us to prove instability, never stability.

In cases where Theorem 2 is not applicable, e.g., because the requirementsp > 0 or q)O are violated, one may investigate the stability of the steady state by solving directly for the roots of Eq. (Bl ) . This is best done by writing w as a complex variable w = u + iw. On substituting this form into Eq. (B 1)

and separately equating the real and imaginary terms we obtain two coupled equations for u and w:

v2- w*fav+b+cve-“cos(wr) +cwe-“‘sin(wr)

+ de-“‘cos(w7) = 0,

2vw + aw + ewe-“‘cos(w7) - cue-“sin(w7) (B3)

- de-“sin(wr) = 0. 034) In general, Bqs. (B3) and (B4) can have an infinite

number of solutions for any value of r. We have found it most efficient to use a standard numerical technique like Newton’s method starting from small r with an initial guess based on the solution of the quadratic equation at r = 0. Once a solution [v(r), w (r) ] has been found, it can be used as the initial guess for the solution at the next value of r. In this manner, one can generate a branch of solutions for v and w as functions of r. To generate further branches requires choosing a r already investigated and searching for a new root to serve as a starting point for the branch. It is this step which is usually the most timeconsuming.

’ Y. Luo and I. R. Epstein, Adv. Chem. Phys. 79,269 ( 1990). *I. R. Epstein, J. Chem. Phys. 92, 1702 (1990). ‘J. Boissonade and P. De Kepper, J. Phys. Chem. 84,501 (1980). 4R. J. Field and R. M. Noyes, J. Chem. Phys. 60, 1877 ( 1974). ‘I. R. Epstein, K. Kustin, P. De Kepper, and M. Orban, Sci. Amer. 248,

112 (1983). ‘R. J. Field and M. Burger, Oscillations and Traveling Waves in Chemical

Systems (Wiley, New York, 1985). ‘R. Bellman and K. L. Cooke, Differential-Difference Equations (Aca- demic, New York, 1963).

*N. D. Hayes, J. London Math. Sot. 25,226 ( 1950). 9 Reference 7, p. 449. ‘“A similar theorem, applicable under somewhat different conditions on

the coefficients in Eq. (lo), is given by F. Schiirer, Math. Nachr. 1, 295 (1948).

’ ’ N MacDonald, Biological Delay Systems: Linear Stability Theory (Cam- bridge University, Cambridge, 1989).

“5. Guckenheimer, Physica D 20, 1 (1986). “Strictly speaking, we can conclude only that all the steady states are un-

stable and that therefore the asymptotic dynamics are time-dependent. We are not aware of a result like the PoincarbBendixson theorem for ordinary differential equations in two variables that would permit one to conclude that the behavior must be periodic.

I4 Reference 11, p. 86. “The value given in Ref. 4 for a, is too large by 0.500 00. lbL. Gyorgyi and R. J. Field, J. Phys. Chem. 92, 7081 (1988); 93, 2865

(1989). ” I. R. Epstein and J. Weiner (to be published). ‘“N Ganapathisubramanian and K. Showalter, J. Phys. Chem. 87, 1098

(i983). “T. Li and J. Yorke, Am. Math. Monthly 82,985 (1975). “‘G. D. Byrne and A. C. Hindmarsh, J. Comp. Phys. 70, 1 (1987). ” A C Hindmarsh, Gear-Ordinaty Dtrerential Equation System Solver, .

UCID-30001, Rev. 3, (Lawrence Livermore Laboratory, Livermore, CA, 1974).


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