Local phase and renormalized frequency in inhomogeneous chemioscillations

P.Ortoleva

Department of Chemistry, Indiana University, Bloomington, Indiana 47401 (Received 3 October 1975)

Inhomogeneous evolution of a chemically reacting system capable of a homogeneous oscillation is analyzed in terms of a nonsecular perturbation method. This "phase diffusion" theory takes account of the local phase shifts and frequency renormalization that occurs due to the interaction of reaction and diffusion. The theory is used to show that oscillator like plane waves are stable to small perturbations. Boundary value problems including those for a finite volume impermeable vessel, a ring shaped vessel (periodic boundary condition) and catalytic walls and membranes are shown to lead to stable oscillatory states some of which are inhomogeneous for all times. The structure of the theory allows for clear qualitative interpretations. Several experiments are suggested for the purpose of verification of the theory and the phenomena predicted.

I. INTRODUCTION

A. Qualitative properties and brief review

Many interesting phenomena in reacting media main­tained far from equilibrium have been found both experi­mentally1 and theoretically. Much of the analysis thus far presented has been based on the nonlinear partial differential equations of phenomenological nonequilib­rium·thermodynamics. 2 The solution of these equations has taken three different tacks based on rather differ­ent physical observations. Let us review them briefly in their application to spatially and temporally varying phenomena such as waves, pulses, and transient phe­nomena.

Bifurcation theory3 is closely tied to the existence of a reference state which under given conditions has be­come weakly unstable to some small perturbation. At a point of marginal stability the given perturbation does not (according to the equations linearized about the ref­erence state) decay. As the system conditions (such as boundary conditions, ambient illumination4•s or temper­ature)6 change, this marginally stable mode may be predicted to grow indefinitely by the linearized equa­tions. Physically we know that the concentrations and other variables are bounded. Thus, bifurcation theory attempts to find solutions to the problem when the sys­tem is weakly unstable by balancing this tendency to­ward linear growth with the nonlinear terms. The smallness parameter used to expand the solutions is the amplitude of the disturbance from the weakly unstable reference state. This approach has been used exten­sively in hydrodynamics. 3 For nonequilibrium chem­ical systems it has been applied to the bifurcation of chemical waves7•8 and static spatial structures9 from a time independent reference state and also from a homo­gene'ous oscillation. 8

When chemical or transport rates vary widely, mul­tiple length and time scales enter the problem. The classical manifestation of this is the relaxation oscilla­tion10 where the oscillation cycle consists of temporal intervals of slow variation separated by much shorter time scale (quasidiscontinuous) jumps in at least one of the descriptive variables. Such phenomena have been seen experimentally in the Belousov-Zhabotinsky-

Zaikin (BZZ) reaction1d and theoretically by classical relaxation oscillation methods and computer simulation. 11

A multiple length and time scale approach has been ap­plied to the analysis of propagating phenomena where it was shown that when the chemical kinetics has multiple time scales the system may sustain propagating discon­tinuities demarking transitions from one regime of the chemical kinetics to another. 12 Qualitative and experi­mental observations on this point have been given with regard to "trigger waves" in the BZZ reaction. 13

A third approach is based on the existence of a stable oscillatory solution-chemical oscillation-for the homogeneous evolution of the system (i. e. , a well stirred tank). Whensuchacycleexists, one might expect that many phenomena such as waves or transient evolu­tion towards overall synchrony of the oscillation may be described in terms of the homogeneous oscillation oc­curring locally at each point in the system. In general we' might expect that not only the phase of oscillation at each pOint will be different but that also the interaction of transport and reaction should lead to shifts in the local frequency of oscillation. This picture of frequen­cy renormalization and spatial distribution in phase has been shown to lead to a diffusion equation for the phase of oscillation from which the name "phase diffusion theory" has been introduced. 8.14 This picture has been applied to problem of inhomogeneous kinetics14 and to the analysis of chemical waves. 7.8 The phase diffusion theory avoids the pitfalls of a straightforward expansion in the effects of transport1s which has the same "sec­ular behavior" as an expansion of sin(wo+ E)f in powers of E,

sin(wo + E)t = sinwot + Et coswo + • •. •

To any finite order of E, the terms in the series diverge as t- 0() no matter how small E is. Clearly this behav­ior may be avoided if we note that the frequency w(= Wo + E) is a function of E.

In this paper we extend the phase diffusion theory to solve a variety of problems including the stability of waves and phenomena in ring shaped, catalytic, and impermeable vessels. For those problems involving catalytiC or membrane boundaries, it is found that the phase diffusion equation derived earlier is supplement-

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1396 P. Ortoleva: Inhomogeneous chemioscillations

ed by a boundary condition that reflects the ability of these active surfaces to speed up or slow down the prog­ress of oscillation in their vicinity. In a companion paper we show that the phase diffusion theory may even be applied to systems such as a dense dispersion of cat­alyst particles as long as one renormalizes the kinetics of reaction in the suspension medium to include the surface reaction. 16

The reader wishing to ignore some of the mathemati­cal details may skip the next subsection (I. B) and pro­ceed to Sec. II.

B. Linear properties of limit cycle systems

1. Dynamics of perturbations

The evolution of the reacting diffusing system is as­sumed to be described by the following set of coupled nonlinear partial differential equations of phenomenolog­ical nonequilibrium thermodynamics:

(I. 1)

Here IJt is a column vector of descriptive variables such as concentrations and temperature. The matrix D is taken to be constant and, by Onsager's reciprocal rela­tions, is symmetriC. The rate contributions from chemical reactions are denoted by the column vector F which is in general a nonlinear function of 1Jt. When the conservation equations (I. 1) are supplemented with ini­tial conditiOns, lJt(r, t=O), and boundary conditions our problem is completely determined.

The chemical kinetics F[ 1Jt] is assumed to yield a homogeneous limit cycle oscillation. Thus the ordi­nary differential equation

(I. 2)

is assumed to have a limit cycle solution lJte with period T,

(I. 3)

In this section we will discuss the evolution of small spatially inhomogeneous perturbations from this limit cycle solution >}Ie. We assume that the homogeneous cycle 'lie satisfies the full system of Eqs. (1.1). Typical boundary conditions consistent with the homogeneous cycle are impenetrable walls, boundaries at infinity, and periodic boundary conditions (as in a ring shaped reaction vessel).

Consider a small initial perturbation 5>}1(r, 0) from an otherwise homogeneously oscillating system IJtc • there­after we have >}I(r,t)=lJtc(t)+51Jt(r,t). If 5>}1(r,0) is a small perturbation it will evolve according to the con­tinuity equation (I. 1) linearized about IJt c(t) (unless IJt c is unstable-see below). We find

(1.4)

where

(1.5)

We now study the behavior of the solutions of (I. 4).

If we introduce the Green's function 2(r, r';t) such that

a2 = [DY'2 + n(t )]2 at 2(r, r'; 0) = 5(r - r')l,

then we may write

(I. 6)

5>}1(r,t)=jd3Y'2(r,r';t)=0, rES, r'EV. (1.7)

Our system is taken to be in a volume V with boundary surface S. With this formulation 2(r, r'; t) must obey the same boundary conditions as 5>}1(r, t) (for all points r' within the system). Since it is fundamental for solv­ing a broad class of more general boundary value prob­lems, we limit our discussion here to the case of im­penetrable boundaries. The flux is given by - DVIj! and hence we have

n· DVZ(r, r'; t) = 0, rES, r' E V. (I. 8)

In (I. 8), n is a unit normal vector to the boundary sur­face S at point r directed out from the system volume V. We see from 0.7) that 2 contains all the response properties of the limit cycle to small perturbations and we will thus concentrate on determining some of its general properties.

Much progress can be made at understanding the properties of the system by expanding the propagator in terms of the eigenfunctions of the Laplacian. Consider the set of functions W .. (r) and eigenvalues K~ satisfy­ing

with boundary condition

n· VW .. =O, rES.

(1. 9)

(I. 10)

Since the eigenvalues of v2 are all negative or zero, we have K~:;' O. Assuming {Wn } is complete in V, we may write

2(r,r';t)=L C.,(r';f)W .. (r). ., Using completeness and orthonormality,

L W!(r')W .. (r) = 5(r - r'), ..

we find from (I. 5) and (I. 6) that

ac",/ at = (n(t) - K~D)Cm ,

C.,(r', 0) = W!(r')l.

(I. 11)

(1. 12)

(I. 13)

(I. 14)

(I. 15)

Before proceeding further, we examine the properties of the solutions of (1.14).

2. Floquet decomposition. Criteria for stability

Equation (1.14) is an ordinary differential equation with periodic coefficients and hence, according to Flo­quet's theorem, its solutions may be written in the form 8,14

(I. 16)

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P. Ortoleva: Inhomogeneous chemioscillations 1397

where Q(K 2 , t) is a matrix with the periodicity T of the limit cycle and Il(K2) is the "characteristic matrix" for the system. The matrices Q and Il depend parametri­cally on K2D. It is important to note that Q and Il do not depend on the particularities of the shape or size of the boundaries but are only properties of the linearized bulk kinetics n and the constant matrix K 2D.

To understand how the Floquet decomposition may in fact be used in an actual calculation, let us introduce X= Qel"t, where X satisfies

dX/dt=(n-K 2D)X,

X(K 2, 0) =1.

(1.16')

(1. 17)

Let us assume that the matrix X(K 2 ,t) has been calcu­lated numerically or by some analytical metholjl for one period of the homogeneous cycle l/!c. Then since Q(K 2

, t) is periodic and Q(K 2 ,0)=I, we must have

2 el"(K )T=X(K2,T). (1.18)

From this equation it is clear that Il(K2) is defined only modulo an integer multiple of iwl, where w is the fre­quency 21[/ T of the homogeneous cycle.

To proceed further we introduce an (assumed) com­plete set of biorthogonal eigenvectors {IPiK2)"} of Ilj

Il(K2) Ip,K2) = II p(K2) Ip ,K2), (1. 19)

(p,K 21 Il(K 2)= II p(K 2)(p,K 21,

By differentiating (1. 2) with respect to time it is easy to see thatf(t),=d>¥jdt obeys the variational equation (1. 4) for a homogeneous perturbation. Thus we must have f(t) = Q(O,t)el"(Oltf(O). Butf(t) and Q(O,t) are periodic functions. Thus Il (only defined modulo iMwl, M=O, ± 1, ... ) may be chosen to have a zero eigenvalue Ill(O) corresponding to the eigenvector 11,0) = f(O) =diJIc(O)/dt.

The dependence of II p on K2 is in general rather com­plicated. However, for the case of equal diagonal dif­fusion, Dj' = [)()jj' we see from (1.16) that X(K 2 ,t) =X(O, t)e-i2Dt and hence II p(K2) = IIp(O) - K2jj. Also, since III (0) = 0 it may be shown from an expansion in K2 that

(1. 20)

where the so-called phase diffusion coefficient Dp is given by

1 IT Dp=T dt (1,01 ql(0,t)DQ(0,t)11,0), o

(1.21)

Now we introduce a criterion for stability that guaran­tees that all small perturbations will not grow:

Definition. A limit cycle system with transport will be said to have structural orbital stability or simply stability if for all po# 1, Rellp(K2) <0 and Relll(K2) <0 forK 2 >0, Ill(O)=O.

Cases of limit cycle systems which are not stable in­clude (1) the homogeneous instability for which the cycle is unstable to homogeneous perturbations (Rellp(O) >0 for at least one P and 2), the symmetry breaking insta­bility for which the cycle becomes unstable to some in­homogeneous perturbation while remaining stable to

homogeneous perturbations [Rellp(K2) > ° for at least one p, K2 0# 0]. Symmetry breaking instabilities may be of two types, according to whether the length scale of the symmetry breaking perturbations is determined es­sentially only by the reaction vessel or by the reaction transport kinetics (see Ref. 5 for details). Note that for equal diagonal diffusion [for which IIp(K2) = IIp (0) - K2 jj] no symmetry breaking instability is possible.

Consider a system near the bifurcation point of the limit cycle. Hopf's theorem states that if the nonlinear system d>¥/ dt = F~[ >¥] has a complex conjugate pair of stability eigenvalues (for perturbations from a steady state >¥*) whose real part vanishes as (X - Xc) near Xc, then the limit cycle has an amplitude proportional to (X - Xc)1/2 for I X - Xc I > 0, Thus if the equations linear­ized about w* take the form a1i>¥/at=[Dv2 +n*]1iw, where n* = (aF/aw)",*, we obtain

Il(K 2) "" n* _K2D. ~~~c

Thus, symmetry breaking in the cycle near its bifurca­tion point occurs near a point of symmetry breaking predicted by the analysis of the homogeneous steady state >¥* from which the homogeneous cycle >¥ c bifur­cates. This suggests an expansion in (X - Xc)1/2 for studying Il and Q. A second possible expansion is in deviations from the equal diagonal diffusion case. We do not go into these here.

3. Properties of the Green's function

Consider now our final result for the propagator 2:

E(r, r' jt) = LQ(K~,t) e"(K~tw .. (r)w:(r'). (1. 22) .. At this point we may give mathematical expression to the physical statement that the stable limit cycle is only weakly stable to long length scale variations in phase. There is always one eigenfunction of the LaplaCian, de­noted m = 0, corresponding to a homogeneous perturba­tion

( 1 2 Wo r)= VD2' Ko=O' (1. 23)

Furthermore, since 1l(0) has an eigenvector 11,0) = f(O) with corresponding eigenvalue III (0) = 0, we have a con­tribution Eo(r, r' ,t) to 2(r, r' j t) of the form

';:' ( , . t) _ If(t) (1, ° I -or,r, - V (1. 24)

We have written

If(t) '=Q(O, t)11, 0) = d>¥ cldt. (1. 25)

The remaining contributions to E, ~E = E - Eo, may be written

~E(r, r' j t) = L L(1-1i".01iP.l)Q(K~,t) .. p

x Ip K2) el"p(K;lt(p K21 W (r)W*(r'). , m , .. .. .. (1.26)

For a stable limit cycle in a finite volume V, we see that

E(r, r'jf) rv 20(r, r' jf). (1.27) t~ ..

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1398 P. Ortoleva: Inhomogeneous chemioscillations

Thus we have

lJi(r, t) I'-..J lJiJt) + OIdlJijdt ~ lJieU + 01), t~ ..

(I. 28)

Now we see that the inevitable result of any small per­turbation is that the limit cycle is phase shifted by an amount 01, where 01 is the volume average of the initial perturbation projected onto the cycle kinetics {F [lJi e(f)] =dlJie/dt=f(t)}. We shall see that slowly varying local phase shifts may be sustained indefinitely by reacting or membrane boundaries and other heterogeneous ef­fects. They may also persist autonomously in infinite systems or systems with periodic boundary conditions when very special relationships between frequency shifts and phase distributions OI(r) are satisfied. The essential features of these phenomena are, as we shall see, due to the weak stability of the limit cycle system to long length scale variations in phase as seen in Eqs. (I. 20) and (I. 28) and from terms in 6.::; withK~ - 0, P = 1 which yield contributions that persist for times on the order of (K!Dptl.

II. PHASE DIFFUSION THEORY

A. Formulation of the theory

The physical content of the phase diffusion theory, presented earlier, 8.14 is that from among the family of general solutions of the reaction diffusion equations (1.1) there are solutions which locally resemble the homogeneous limit cycle lJie [see Eqs. (1.2) and (1.3) and related text]. Thus we expect that these solutions are of the form

lJi(r, t) = lJi e(¢(r, f)) + E ~lJi(r, f). (n.1)

In this picture ¢(r,t), a local phase variable, fixes the phase of the unperturbed cycle lJie that the concentrations lJi lies "near." The term E6.lJi(r, f) represents the re­mainder which is, in some sense, small. The phase diffusion and related theories give mathematical ex­pression to this concept. In this section we treat infi­nite systems. We assumed the absence of heterogeneous sites of reaction or embedded objects such as impene­trable spheres or catalyst particles. These latter prob­lems are discussed in Sec. In.

The formal development of the phase diffusion theory starts by introducing an ordering parameter E which is a measure of the magnitude of the term lJi -lJie in (n. 1). We next cast the reaction diffusion equations in terms of the local phase ¢ and position r in favor of laboratory time t and position, (r, t) - (r, ¢). The new independent variable ¢ is taken to be related to the old independent variables by a transformation f = t(r , ¢; E) which we write as a development in a sequence in E:

where

li 1· enol 0 m enol = 1m =, E~O E~O en

(11.2)

n >0, (n.3)

(II. 4)

The concentrations, lJi, are also written as a develop­ment in a sequence Pn(E):

00

lJi = L.: Pn(E) x,,(r, ¢), (II. 5) n=O

where the Pn have ordering properties analogous to those of the en. We note that the X" are written explic­itly as functions of the new variables (r, ¢). We will find that the functions tn(r, ¢) are not arbitrary but must be chosen in such a way as to insure that the Xn(r, ¢) do not diverge as t, and hence c/J, becomes large and sim­ilarly for their dependence on r.

This procedure eliminates terms like EtdlJi e/ dt found in straightforward expansions in Eo 15 As discussed in Sec. I. A, straightforward expansions in E with indepen­dent variables (r, t) are not uniformly convergent in E for all time or space. In many cases this secular be­havior may be removed by the physically motivated ex­pansion of the phase diffusion theory which allows for frequency shifts and distribution of the phase of oscilla­tion.

It is central to our zeroth order picture (n. 1) that Xo should obey the same equation as lJie, namely, dxoId¢ = F[Xo]. Thus, in some sense, in zeroth order the dif­fusion term does not enter. Thus we expect the family of solutions we seek has (Vc/J)I of order E or less for all (r; t).

B. Calculations to second order

To proceed further we next rewrite the reaction dif­fusion equations as partial differential equations in the new independent variables (r, ¢). To avoid complicated notation such as (V)<jI to express the gradient with re­spect to r at constant ¢, we will assume that in any equation involving both rand ¢ we may Simply write V for (V)",. For completeness let us give some interme­diate steps in the transformation. The differential op­erators of interest are given by

Although the sequences p" and en may for some prob­lems be a complicated sequence of functions in E, we find that for the present considerations only simple powers, en = Pn = En, are required to keep tn and Xn well behaved. Using (11.2) under the assumption that as E-O, lJi-lJie(t) [so that to(r,¢)=¢], we have, retaining only terms up to second order in E,

( a¢) = 1 _ E at l + E2 r(~)2 -~J + ... af r a ¢ L a¢ a¢ , (1I.8a)

- (Vc/J)t = EVtl + E2 [ Vi2 - -!;- Vii] + ••. (II.8b)

( 2) 2 2[ 2 ~ 2] - v ¢ 1 = EV tl + E V f2 - a¢ v tl + •..• (II.8c)

Equations (11.5)-(11.8) may thus be used to transform

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P. Ortoleva: Inhomogeneous chemioscillations 1399

the reaction-diffusion equations (I. 1) in terms of the new independent variables (r, cf» to each order in E.

To lowest order we obtain (essentially by design)

(n.9)

By the fundamental assumption of this paper we take this equation to have a stable limit cycle solution life

(see definition in Sec.!. B. 2) and hence we have

(11.10)

To first order the structure of the theory becomes more amusing. One finds

where

wl(r,¢)=XI-f(¢)tl

andf(¢) =dllfc/d¢ =F[ilc(¢)].

(11.11)

(11.12)

For an infinite system the first order equation may be most easily solved by Fourier transformation, r - k, lIfl- ~I' With this, (11.11) takes the form

(11.13)

Using the Floquet decomposition [see (1.16)-(1. 18)], we obtain

(II. 14)

For a stable limit cycle ~I (k, ¢) vanishes as ¢ - 00 for k >0. However, for small k, ~I(k,¢) will have a very long lived contribution from the p = 1 branch of eigen­values of J.L(k2

). This branch of eigenvalues vanishes as _k2Dp as k-O. Hence we obtain

XI(k,¢)-f(¢)ll(k,¢) ;::. f(¢)el'l(J,2).t>

'" 1'1 (k 2 )("1

(11.15)

Since we seek approximate solutions of the full equa­tions of reaction and transport which locally lie close to some phase of the homogeneous cycle W c' we limit our initial distributions to the form w(r, 0) = lIf c(€a(r)) +€t.lIf(r,O), where €t.w(r,O) is of order € as €-O. This would include plane wave type distributions of phase for which a=-q' r/w, where €q is the wave vector and w(= 21T/T) is the frequency of life. In Appendix A we show that the present initial value problem may be cast in the phase diffusion theory and leads to the condition tl (r, 0) = - a(r). If a(r) is on a very large length scale, the diffusion operator e-k2Dp'" in (11.15) will lead to large very slowly decaying contributions to X I unless we choose tl(k, ¢) = e-~Dp"'tl(k, 0). {In fact, for plane type distributions [a(r) = - q. r/ w] the contribution to XI from the tl(k,O) term in (11.15) would (for infinite sys­tems) be unbounded in r.} From this we see that tl(r, ¢) obeys the diffusion equation

(II. 16)

and hence X I has the behavior

XI(k,¢) rv f(cb)el'l(k2

)", (1,k 2 1 XI(k,O» "'-~

+ f(cf>)[ e-k2Dprl> - el'l (k2)rI>] (1, k2 11, 0 > ] tl(k, 0). (11.17)

With this we see that XI is well behaved for all rand cf> and hence t. Indeed, if w(r,O)=w.[€ql· r/w+€t.O!I(r)], where t.al(r) is localized to some spatial regions, then although tl(k,O) has a contribution (21T)3i(ql/w), (ao(k)/ ak) it makes no contribution to XI (k, cf». Furthermore, the perSistent part of the synchronization process, that due to the small k contributions from the t.al(r) part of tl(k,O), does not enter XI either. Hence, as our initial picture suggested, solutions starting out with gentle variations in phase are well described as lying close to some phase of the limit cycle for all times since all terms in (II. 17) are bounded.

There does remain one part of XI which persists for all time. If XI(r,O) (=M.'(r,O), see Appendix A) has a nonzero spatial average value (X. = V-I f d3r (1,0 I X.L (r, 0» as V-oo), then XI(k,¢) has a contribution (21T)3 xO(k)Xsf(¢) and hence as t-oo ~(r,t)=lIfc(¢)+€xsf(¢). Recalling thatf(¢)=dwe/d¢, we see that this contribu­tion corresponds to a homogeneous phase shift €X. of the cycle. This can be accounted for by redefining the phase of the function we by shifting it an amount €X •• Thus we may assume X. = O. A similar redefinition of we can be used to remove these terms in all orders, Xn , n:;' 1.

The diffusion equation (II. 16) yields the dynamics of the tendency of the oscillatory system to synchronize by "phase diffusion." Caution must be taken in that not all solutions of the phase diffusion equation corre­spond to solutions of the full equation (1.1). For ex­ample, consider a one-dimensional system. Solutions of the form tl = f3 1¢ + al(x) satisfy (11.16) when

d2a l / dx2 = f31•

As Ix I - 00, a l - t I3 1x2 and hence tl yields infinite gra­

dients in lIf no matter how small € is. Thus, acceptable solutions to (II. 16) have I V tIl bounded for all r. The local phase takes the form, in this order, ¢ = [t - € a l (x)]/ [1+f31€]' and hence the frequency, w*=w/[l+f3I€]' is unaltered to order € since f3 1 = 0 for physical solutions in an infinite medium.

Let us now investigate the theory to order €2 where we find frequency shifts for the infinite system. Col­lecting second order terms and putting lIf2 = X2 - f(¢ )t2 ,

we obtain

~ = [DV2 + n] Wg -DVi j ' V (~) + V2tl ~ a'f' . a¢ a¢

+~ ~ +D [IViI 12n+ v2tl ~] f(¢)+B(¢): XIXh

(II. 18) where B(¢) is the bilinear coefficient

B(¢) = t[ a2F/ a+2]~ (rI»' e

(11.19)

Since tl obeys a diffusion equation and I vttl must be bounded for all r, we must have, as ¢ - 00, i l =ql' r/ w + a, where €ql is the wave vector [entering as a - ql . ao(k)/ak contribution to i1(k;0)] and a is a constant. Thus atl / a¢ and V2il vanish as ¢ - 00, and since X I

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1400 P. Ortoleva: Inhomogeneous chemioscillations

does also, we obtain

Now as ¢ - 00, Vt1 - q1 / wand hence taking Fourier transforms we obtain as <b - 00

88!Z = [0 - kZD]-£z +( 5-)DO(¢ )(27f)30(k) (II. 21)

and hence

~z = Q(kZ, ¢) e,,(kZ)1/> wz(k,O) +.s= t Q(kZ, ¢) Ip, kZ)

[ eIMWI/> _ e"p(kZ)II>] (Q )Z .J (2)3 A (p.M)

X [iMw _ II p(k2)] w 7f '"" , (11.22)

where we have written

Q-1(kZ, ¢)DO(¢)f(¢) = L L Ip,kZ) elMwlI> LI. (P.M) , p M~~

The new feature that arises in second order is due to the term p = 1, M = 0. This yields a contribution

(~lr(27f)30(k)LI.<bf(d», LI.=LI.(1,O). (11.24)

which diverges as ¢ - 00. Hence Xz also diverges unless tz has a contribution - (Q1/w)ZLI.¢. This new feature, in conjunction with analogous arguments to those used for the construction of t1, leads to the condition

~=D VZt _(91)Z LI. (11.25) 8¢ p Z w

in order to insure that Xz is well behaved [bounded for all (r, ¢)].

C. Summary of results: Stability of plane waves

Collecting the results thus far obtained, we find that as t (and hence ¢) - 00,

t f"V ¢-Eq1·r/W+E:Z[(Q1/W)zLI.¢-qz·r/w]+", II>-~

(we have dropped constant terms in t1 or t z which just add a small shift of order E to the space or time origin). Hence to order EZ

t - (Eq1 HZq2) . r/ w

¢(r,t) tr:-:. [1 + LI.(EQ1/W)2] , (11.26)

and W - we(¢) + 0(E2) as t - 00. From this we see that if the system was started close to a limit cycle we at all points with a linear distribution of phase with wave vector q = Eq1 + EZq Z plus arbitrary deviations in phase ELl.a [with I Val bounded for all (r)] and small deviations in amplitude ELl.l/!(r,O)], that in time the system would evolve to a plane wave with frequency w(1 + (Eq/ w)z LI.(l. Plane wave solutions in an infinite system are thus stable to small perturbations (modulo, of course, a small shift in origin, r- r + Ear)o

III. BOUNDARY VALUE PROBLEMS

In this section we consider the effects of boundaries on a system with oscillatory kinetics. Examples of boundary surfaces include the finite impenetrable ves-

sel, the ring shaped vessel (leading to periodic bound­ary conditions), the catalytic surface, and the catalyst particle. We find that these boundary conditions may strongly influence the system behavior. Our treatment will essentially be that of the phase diffusion theory of Sec. II, except that here we find that the phase diffusion equation derived earlier must be supplemented by an appropriate boundary condition.

Our treatment is not valid for all classes of boundary conditions. Consider for example a system where w(r, t) itself is speCified on certain boundaries. Then only for the particular case where the specified bound­ary value is itself close to some phase of the homoge­neous cycle >lie can we consider the exact solution to be locally near some phase of we' For cases where the boundary value is far from We, we might only expe ct the phase diffusion theory to be applicable far from the boundaries; and in this case, therefore, the theory is only expected to be useful for large systems.

A. Inhomogeneous evolution in a finite impermeable vessel

Let us consider the evolution of a limit cycle system in a finite vessel of volume V and impenetrable surface S. On the surface the flux J (= - DVW) has no normal component

rE S, (IlL 1)

where n is a unit normal vector to S at r taken to point outward from the reaction volume V. Using the phase diffusion expansion (11.1)-(11.5), the boundary condition takes the form

t Emn' [VXm + (8XM/ 8¢ )(V¢ it] = 0. (III. 2) m.O

To order zero this equation is trivially satisfied since Xo=we(ct» and (Vd»t is of order E. To first order we obtain

rES, (III. 3)

where w1 = Xl - f(dJ)t 1• As in Sec. II, w1 obeys the equa­tion

(III. 4)

and hence we may use the propagator introduced in Sec. 1. B. Note that this guarantees that (III. 3) will hold, and we obtain

(III. 5)

Recall that in the phase diffusion theory we assume that at time zero we may take the deviation ELl.w(r, 0) from some phase of the limit cycle to be of order E, i. e., w(r, 0) = We(ct>i) + EA w(r, 0), where

ct>i(r)=<b(r,O). (III. 6)

Applying the inversion process J d3r'E-1(r, r'; ¢) on (III. 5) we obtain [see (B4)-(B6)]

w1(r,0)= jd3r'E(r,r'; -¢)w1(r,¢),

and noting that Xl (r, ¢ i) = LI. w(r, 0) we obtain

w1(r, 0) = jd3r'E(r, r' ,- ¢i(r'»

(III. 7)

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P. Ortoleva: Inhomogeneous chemioscillations 1401

x [t..'lt(r', 0) - f(qi(r')) 11(r', cf>i(r'))]. (Ill. 8)

Since t..'lt(r,O) is bound for all r and in a finite system cf>i(r) is bound for all r we see that 'lt1(r,0) is bound for all rEV.

Using the asymptotic property (1.27) we obtain as c/>-oo

Since li is a constant, we can assume it to be zero by shifting the phase in the definition of the function 'It c by an amount li. There are two types of phenomena em­bedded in (Ill.9), (IlL 10). The first has already been discussed in Sec. I. B, and our result here verifies the conclusion that the eventual effect of a small perturba­tion on a stable homogeneous cycle is at most a homo­geneous phase shift. This corresponds to the trivial solution of the phase diffusion equation, tl = constant. However, in some cases such as the torus, the reac­tion volume bends back upon itself and there may be other nondecaying solutions to the phase diffusion equa­tion which do depend on r. We investigate this point in the next section for the case of the ring shaped vessel.

B. Phenomena in a ring: Periodic boundary conditions

Let us consider our oscillatory medium to be enclosed in a ring shaped vessel. The tube itself is assumed narrow and impenetrable. The radius of the ring R is taken to be large. If we take x to measure distance along the length of the ring and neglect lateral varia­tions, then 'It(x, t) must obey periodic boundary condi­tions

'It(x + 27TR ,f) = 'It(x, f) (III. 11)

for all x. In this section we examine the properties of temporally periodic evolution in such a ring by applica­tion of the phase diffusion theory. The latter theory is based on the smallness of the contributions to the dif­fusion term. Hence the radius of the ring must be large compared to the characteristic length Ac (the cycle diffusion length), A~ = Dp T. Hence we write

(III. 12)

guaranteeing that solutions with even gentle gradients (of order 10) will still be able to obey the periodic bound­ary conditions.

The periodic boundary conditions must first be cast in terms of the r, cf>, X variables of the phase diffusion theory (11.2)-(11.5). To the first two orders we obtain [noting that Xo='ltc(cf»]

'ltc(cf> (x + 27TR ,f)) = 'It c(cf>(x, t)),

Xl (x + 27TR, cf>(x + 27TR, f)) = Xl(X, cf>(x, t)).

(III. 13)

(III. 14)

Since 'lta has period T, (Ill. 13) may be satisfied as long as

cf>(X+27TR,t)=cf>(x,t)+nT; n=O, ±1, .... (III. 15)

Note that this condition must hold to all orders in 10.

The most general solution of the phase diffusion equa­tion (It 16) for fl corresponding to temporally periodic solutions takes the form

(III. 16)

where {31o Vl, and °1 are constants. With this we obtain

(III. 17)

The phase shift - 10°1 we shall henceforth assume to be absorbed in the definition of 'ltc' However, (III. 5) can only be satisfied if

{31 = 0, (III. 18)

vl(n)=(27TAC /T)/n, n=O, ±1, ...

[where we have used the radius scaling (III. 12)] and hence

(III. 19)

We see that these solutions are waves propagating around the ring with velocity v(n, 10) such that

v(n 10) rv v 1 (n) = 27TR , .-0 10 nT' (III. 20)

and wavelength A given by

A(n,€)= Iv(n,€)1 T(n,€)-27TR/n, as 10-0, (III. 21)

where T(n, 10) is the period of the wave which to order 10 is just T since (31 =0 in (III. 17). Carrying these re­sults to order 102, we find the same frequency renor­malization as in Sec. II and hence to second order

T(n, 10)= [1- t..(nT/27TR)2]T. (III. 22)

At this point we should carefully examine the mean­ing of the result (III. 19)-(III. 22). We see that our so­lutions are spatially periodic trains of waves and hence 27TR/A must obviously be an integer and hence is in a sense quantized. What is noteworthy is that because of the presence of the limit cycle dynamics, the velocity of propagation for large rings (R - 00) and small n is essentially determined by geometric considerations. In this limit v(n, 10) is apprOximated by that velocity cal­culated on the assumption that the wave goes around the ring, a distance 27TR, in n periods T of the homogeneous cycle 'ltc' Thus a measure of the importance that trans­port has on the waves is simply how much v( n, €)T/27TR differs from n, the number of wavelengths around the ring.

C. Membranes and catalytic surfaces

Qualitatively, one would expect surface reactions to tend to speed up or slow down limit cycle dynamics in the vicinity of these surfaces. A similar comment holds in the vicinity of a membrane allowing the system to exchange molecules with another system. Thus gradients of phase and frequency shifts are expected in these boundary value problems. We shall term these boundaries active surfaces and study their effects on OSCillating kinetic systems within the framework of the phase diffusion theory.

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1402 P. Ortoleva: Inhomogeneous chemioscillations

At the active surface the amount of material trans­ported away from the surface n· DV'¥ must be just bal­anced by the rate of surface reaction or transmembrane transport yG[r, '¥],

n· DV'¥ = yG[r, '¥]. (III. 23)

Note that G may depend explicitly on r since the surface reaction may not be the same at all points. If the active surface is a linear membrane, then G will have the form G = H['¥O - '¥] where '110 is the column vector of concentrations on the other side of the membrane and H is a matrix of membrane permeability factors.

We expect that when the strength parameter y be­comes small there exists a family of solutions which locally looks like the limit cycle, L e. ,

(III. 24)

The phase function cp(r,t) reflects.the frequency re­normalization and spatial distribution of phase. We as­sume our renormalized development to be in simple powers of y and hence write

'" t = L t.(r, cp )yn, (III. 25)

.=0

(III. 26)

With these expansions the boundary condition (III. 23) to the first two orders in y becomes

n· DVXo=O,

n· DV'¥I = G[r, Xo] =g(r, cp),

(III. 27)

(III. 28)

where '¥t = X 1- tl (axo / acp). The condition (III. 27) is satisfied by construction in accordance with our zeroth order picture (III. 24).

For pOints within the reaction volume V we obtain the same first order equation as in Sec. II,

(III. 29)

Here, however, '¥l obeys the boundary condition (III. 28)

1. Reaction sheath method

For the purpose of analysis we replace the differen­tial equation (III. 28) and boundary condition (III. 29) by a more convenient equivalent mathematical problem. Consider our boundaries at S to be impenetrable and nonreactive, n· DV'¥l =0. At a distance I) >0 into the reaction volume (direction - n) we construct a mathe­matical surface S+ (see Fig. 1) on which a reaction

FIG. 1. Reacting system in a volume V with boundary surface S is shown with out­ward normal n and mathe­matical surface S+ a distance {, into V. For formal devel­opments surface reactions are replaced with Dirac-func­tion source terms on S" and in Sec. III. C we show that as (, - 0 the two problems are equivalent.

source term of strength g(r, cp) is placed. As I) -0 the surface S+ becomes identical with S. Our modified equation becomes

(III. 30)

Here I)s+(r) is a one-dimensional Dirac delta function. Its property is defined operationally; if we integrate in r space along some path r that crosses the surface S+ , then an integral of the form f dsA(r)l)s+(r) has the value A(r+); here r+ is the point where the path r intersects the surface S+ and ds is the element of arc length along r. The boundary condition for our modified problem is n· DV'¥I = 0 for all points ron S. Casting the problem this way allows us to make use of the convenient prop­erties of the propagator E introduced in Sec. 1. B.

Before proceeding further we note that our modified problem is equivalent to our original problem. To show this we make a line integration over both sides of (III. 30) along a path that passes through a point r+ on S' and runs along the direction n(r+) from r+ - B'n to r+ + I)'n, where I)' is some small distance such that 0 < I)' ~ I), [; being the distance between S+ and S. With this we obtain

(III. 31)

Now taking the limit as I) - 0: 0 < I)' ~ I) and noting that the first gradient term in (III. 31) vanishes because of the boundary condition on '¥ 1 at S we obtain

(III. 32)

Thus we obtain the proper boundary condition. Rigor­ously speaking our formulation is such that (n· DV'It)r =0 for r on Swhile for r-O+U (III. 32) holds. At all points r within the reacting volume the two problems yield identical results.

2. Formulation of the integral equation

The modified first order equation may be solved in terms of the propagator E(r, r'; cp) discussed in Sec. I. B. The details of the calculations are given in Ap­pendix B and the final result is

'¥I(r, cp) = f dV E(r, r'; cp )'¥t(r' ,0)

+ I d2r' {' dcp'E(r,r';cp'-ctJ')g(r',cp'). (III. 33) "5 0

The first term in (III. 33) is that contribution from the evolution (with respect to cp) due to the initial distribu­tion calculated as if the system had nonreactive imper­meable walls. The second term accounts for the ef­fects of the surface reaction propagated into all points r within the system volume V.

Let us now investigate the effect of the active sur­faces as manifest in the last term of (III.33), the lat­ter denoted A(r, cp). Using the eigenfunction develop­ment (1.22) for E and expanding the quantity ql(K~, cp') g(r',cp') in terms of the eigenvectors Ip,K~ > of jJ.(K~) and its Fourier series (Q-Ig has the periodicity 2n/w of the undisturbed cycle '¥c), we obtain

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P. Ortoleva: Inhomogeneous chemioscillations 1403

A(r,¢)= L t L w .. (r)g!,p,M) m M=-fIO P

eiMw </) _ e"'p (K!)</>

x . 2 Q(K!,¢)lp,K~), tMw- J.lp(Km)

(1lI.34)

where

xg(r,¢). (III. 35)

The purpose of our formulation in terms of the local phase function was to avoid secular behavior in X n [terms diverging as t (and hence ¢) - 00] and to remove terms in the amplitude functions Xn which are not uni­formly small for all r. The terms with p = 1, M = ° in (1lI. 34) have contributions to both of these effects.

3. Secular terms in I/J: Frequency renormalization

Consider the term p=1, M=O, m =Oin(III. 34). Since Wo = v-lI 2, we obtain a contribution to >111 of the form

(III. 36)

Barring any accidental degeneracies, i. e. ,

iMW-J.lp(K;)*O (M*O, m*O, p*1) (III. 37)

(automatically ruled out in a stable cycle), there are no other secular terms in A as ¢ - 00. Hence we must choose tll ¢ - - gcil,O) Wo as ¢ - 00 to avoid secular be­havior in Xl' Hence the frequency w*(y) to order y be­comes

(III. 38)

where the renormalization factor f3 l = gcil,O) V-1/2 is given by

Note that f3 l is essentially proportional to the surface to volume ratio. Thus the frequency shift caused by some slowly acting catalyst would obviously be greatest if the catalyst was added as a powder than as a single plate. (Note, of course, our results are only strictly valid for the unstirred fluid.) More surprising is the fact that the frequency shift to this order is independent of the diffusion coefficient matrix D since all terms in (III. 39) are independent of D. In higher order, however, there will be additional contributions to the frequency shift because of reaction transport interaction effects as were found in Secs. II and III. B.

4. Spatial distribution of phase

Here we consider large systems wherein we expect that although locally >11 may be near some phase of the limit cycle {w-wcr¢(r,t)]}, points widely separated in space may be very much out of phase. Thus, for ex­ample, consider a planar catalytic wall bounding a semi-infinite medium. We expect that even if the sys­tem was initially synchronized, the points near the catalyst wall would eventually get very far out of phase from the oscillation at large distances from the wall.

For a system of largest dimension L, the eigenval­ues of the LaplaCian, K~, are separated by an amount of the order L-2

• Thus, since

J.ll (K2);:'o _K2Dp

(see Sec. LB.), the terms withp=1, M=O, K!=0(r2) are of order L [since (J.lltl =0(L2

), Wm =0(r3/2 ), and g~l,O)=0(Ll/2)] and hence may be very large.

Before drawing general conclusions let us consider the case of scalar diffusion D=DIfor which f.J.l(K 2

)

=_K2jj, Q(K 2 ,¢)=Q(0,¢) and Ip, K2) = Ip,O) for all values of K (see Sec. I. B). With this the contribution to >111 from A(r,¢) for M=O, p=1 is in the form -f(ljJ) xTl(r, ¢), where

(III. 40)

With this definition it may be verified that Tl(r, ¢) obeys the diffUSion equation

aTl = D'il2T a¢ 1

with boundary and initial condition

n·jj.VT1=g(l,O)(r), rES; Tl(r,O)=O, rEV;

g(l'O)(r)=~ IT d¢ e- iMw</) (1, 01 Q-l(O, I/J)g(r, ¢)

(III. 41)

(III. 42)

(III. 43)

Consider, for example, the half-space, x >0 bounded by a planar catalyst surface at x = O. We see that Tl has the time independent solution, always obtained asymp­totically as ¢ - 00,

(III. 44)

No matter how small y becomes, Xl is not uniformly convergent for all x unless we include Tl (r, ¢) in tl (r, cf». with this (III. 44) corresponds to the emission (absorp­tion) of plane waves by the wall for g(l,O) ~ O. The speed of these waves is Dlyg(l,O) and hence decreases with increasing catalyst strength.

More generally one can see from (I. 20) and (III. 34) that all secular behavior in rand IjJ is removed if tl is chosen to obey the phase diffusion equation,

atl =D V 2t a¢ p 1

with boundary condition

n' DpVtl =g(l,O)(r), rES.

(1lI.45)

(III. 46)

The remainder of >111, given by (III. 33), is allocated to Xl since it is bounded for all ¢ and, even for infinite sys­tems, for all r with this procedure.

5. Asymptotic behavior and stability of inhomogeneous periodic states

We are now in a position to show that as t - 00 a limit cycle kinetic system always approaches a stable period­ic state as long as its initial state is not (locally) too far from some phase of the cycle and that the initial gradient of phase is smooth. First the phase correc­tion tl as solution to (III. 45) and (III. 46) has a contribu­tion like Tl in (III. 40) with D replaced by Dp) plus an

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1404 P. Ortoleva: Inhomogeneous chemioscillations

initial value term. With this we have the following asymptotic behavior:

tl(r,rp),......, -a+Lw",(r)g~I.O)/K~DI>-f3lrp, (III. 47) q,"'OQ mifO

where a = v-If d 3r a(r), and €a(r) is the initial distri­bution of phases (see Appendix A) and f3 1 is given in (III. 39). Thus we see that as t (and hence rp) - 00, the phase function rp(r,t)-t-€tl(r,t) has a time independent

We see that after long times (and hence rp - 00) the first term just corresponds to a constant phase shift v-If d 3r (1,0/.6. lJI(r, 0», whereas the second term is periodic in rp and independent of the initial condition.

6. Application to a one-dimensional problem

Consider a test tube of length L. One end, denoted x = L, is taken to be impermeable, n· DVIJI = o. At the other end, x = 0, we put a catalytic plug or we put the tube in contact with a stirred vat through a permeable membrane or consider surface reactions to take place at an air-liquid interface. In either case we have n· DVIJI = yG at x = O.

Taking the concentrations to only depend on the direc­tion x along the tube, the phase diffusion theory yields

atl _ D aZtl arp - P ax2 ,

D atl(O rp)=g(1.0) aaxtl(L,rp)=O. I> ax' ,

(III. 49)

(III. 50)

One solution of (III. 49) and (III. 50) denoted t~, is given by

(III. 51)

If we write tl = t~ + .6.t l , then fl.tl obeys (III. 49) and sat­isfies the condition of impenetrable boundaries, Mt I (L, dJ )/ ax = aM I (0, rp)/ ax = O. Since all eigenvalues of the Laplacian for these boundary conditions are neg­ative, except for the trivial zero eigenvalue solution fl.tl = constant, we see that for rp - 00 we have

t1(r,rp) ""'-Jt~(r,rp)+a, 4)-00 (III. 52)

where a = L -I n a(x)dx, €a(x) being the initial distribu­tion of phase.

To lowest order lJI(r,t) = IJIc(rp) , and from (111.49), (III. 51), and (III. 52) we may arrive at a clear picture of the concentration pattern that the system inevitably attains. For x« L the concentration pattern looks like a plane wave with velocity Dp/yg(l·O) directed away from/toward the active end for yg(1.o, ~ O. (Note for yg(1.0) = 0 we have infinite phase velocity, 1. e., homo­geneous OSCillation.) Near the impenetrable end at x = L the waves speed up to infinite phase velOCity as they annihilate/are emitted for yg(l.O) ~O. These ob-

distribution of phase. The only effect of the initial con­dition is seen to give a position independent phase shift, a.

Now let us show that the concentration profile settles down to an oscillatory state whose form is independent of the initial condition. Using (III. 33) and the asymp­totic properties of :=: (1. 24) and (I. 27), we obtain (with the aid of Appendix A in determining the appropriate initial values)

servations are consistent with experiments on the bromine-malonic acid system carried out in a tube kept in contact with a stirred tank at one end. 17

This sample system contrasts the effects of active and impenetrable boundaries. As we saw in Secs. III. A and III. B, impenetrable boundaries tend to lead to homogeneous states (except in periodiC boundary valve problems like the ring). Here we see that active boundaries such as membranes and catalytic surfaces lead to gradients of phase and after sufficiently long times to inhomogeneous states of temporally periodic evolution.

IV. REMARKS

The conclusions drawn in the various sections of this paper may be briefly summarized by saying that kinetic mechanisms with a stable homogeneous limit cycle os­cillation may sustain stable periodiC phenomena such as plane waves (in an infinite medium). waves traveling around an impermeable ring shaped vessel, (only) homo­geneous oscillations in a simple impenetrable vessel (i. e., not a torus or Klein bottle), and inhomogeneous patterns in the presence of active surfaces. Our phase diffusion treatment is only valid for smooth cycles not having multiple time scales. This is in contrast to the relaxation oscillation where each period of oscillation may be described as having slow processes separated by at least one short time, quasidiscontinuous jump. Such multiple time scale processes have been treated earlier, 12 and it would be quite interesting to apply the matched asymptotic expansion method to the problems studied in this paper and contrast the results. This is important since many experiments on oscillatory sys­tems have shown the presence of multiple time scales. lc.d

It should be pOinted out that our conclusions are "local," not "global" in the mathematical sense. Thus we have discussed the behavior of solutions of the form lJI(r,t)=lJIc(rp(r,t»+€fl.-t(r,t), where € is small and rp(r,t) does not vary too rapidly in space. In many sys­tems the homogeneous kinetics is such that no matter what the initial condition is (except at isolated, unstable steady states), the system will eventually evolve to the cycle, modulo a phase shift (globally stable cycle). For such systems the results of this paper are expected in many cases to describe the eventual evolution of the

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P. Ortoleva: Inhomogeneous chemioscillations 1405

system for any initial condition. However, in general, the homogeneous kinetics may have multiple stable so;. lutions (cycles or steady states), and for these systems we do not expect our results to hold globally. The strength of our results is that the inhomogeneous tem­porally periodic states studied in this paper are stable. We may not conclude, however, that they are the unique asymptotic (t- 00) states.

Finally, we note that the calculations of this paper suggest a variety of experiments on a system such as the bromine-malonic acid system or the glycolytic oscillator. In particular, the ring reactor and the one­dimensional tube with catalytic plug would be interesting tests of the theory. We stress that some care should be taken in interpreting experiments that propose to test the predictions regarding the asymptotic inhomo­geneous periodic solutions. Two relaxation times must be exceeded. One is a consequence of the homogeneous relaxation properties of the cycle, i.e., the time the homogeneous evolution takes to settle down to the limit cycle >lie. In many experiments this is less than a few periods, and for a system like the bromine-malonic acid system this can only be a few minutes. A typically much longer time enters from the phase diffusion equa­tiono This time TD is of order (L2/Dp ). From its def­inition (1.21), Dp may be much larger than any given diffusion coefficient if D is not equal diagonal. Thi~ can arise when the kinetics involves relatively different time scales, and hence the matrix Q-1DQ in (1. 21) may have large terms from Q divided by small terms from Q-1. Taking Dp'" 10-3 cm2

/ sec, this is of order 17 min for a 1 cm tube and on the order of 28 h for L = 10 cm. The bromine-malonic acid system typically lasts 90 min or so without the aid of external supplies of reac­tants. These experiments must either be limited to small tubes or the tubes must be constructed of a per­meable material and placed in baths which will keep the system sufficiently far from equilibriumo

ACKNOWLEDGMENT

I would like to thank Betty Jo Robertson for help in preparation of the manuscript.

APPENDIX A: INITIAL VALUE PROBLEMS

In order to solve our initial value problem we must cast it in terms of the new set of independent variables (r, 1» and corresponding functions X n(r, 1» and tn(r, 1». We present here the results to order € (assuming Pn and en may be chosen as simple powers €n). First we in­vert (11.2) in the form

'" 1>(r, t) = L 1>n(r, l)€'. (Al)

• .0

Setting t = 0 in (II. 2) yields for the first two terms

0= to(r, 1>0), (A2)

0= (ato(r, 1>o)/a1>O)cf>l +t1(r, 1>0). (A3)

For all problems considered in this paper, we have to = <b, and hence we have

<bo(r,t=O)=O, (A4)

(A5)

Since we seek solutions which locally lie close to some phase of the limit cycle for all times, we may write the initial profile as

(A6)

We may include a homogeneous phase shift in the def­inition of >lie and hence may assume that V-1fd3r(1,01 xt.>II(r,O» =0.

Equating the amplitude functions in the development (II. 5) to the terms of similar order in (A6) we obtain [using (A4) and (A5)]

Xo(r,O)=>IIc(O), (A7)

Xl (r, 0) - t1 (r, O)f(O) = t.>II(r, 0) Hl'(r)f(O). (AS)

We want Xl to be well behaved for all r, and since a(r) may be very large (for infinite systems it can go as q. r/w for plane waves), we must make the following separation of (AS) to yield

t 1(r,0)= - a(r),

X1(r, )=t.>II(r,O).

(A9)

(A10)

APPENDIX B: DERIVATION OF (111.33)

In this Appendix we solve the equation

by introducing the propagator E(r, r' ; t) of Sec. I. B, hence deriving (III. 33). First we define a function V(r, t) such that

U(r,t)= jd3rE(r, r';t)V(r't).

(Bl)

(B2)

Inserting this into (B1) and applying the inverse of E, we obtain

aV(r,t) = J d2r,;;,-1(r r' ·t)g(r' t) at s. - " "

(B3)

where the inverse E-1(r, r'; t) is a kernel with the rep­resentation

2 E-1(r, r'; t) = L e-,,(Km )tQ-1(K~, t) W",(r)W!(r / )

'" and has the reCiprocal property

jd 3r I E- 1(r, r' ;t)E(r', rl'l ;t) = 6(r - r")!.

(B4)

(B5)

Noting that the quantity Q(k2 , t) e" (k2 )(t-t' )Q-1(k2

, t') obeys the same equation as Q(k2 , t - t') e,,(h

2)(t-t') and both have

the same value at t = t', we may show that

jd3r'E(r, r' ;t)E-1(r', r" ;t") = E(r, r/;t - til). (B6)

In particular, on setting t = 0 and then t" - t we see that the inverse E-1 actually propagates the system back­ward in time,

E(r,r', -t) =E-1(r,r' ;t). (B7)

Finally, we may obtain the desired result by integrating (B3) from ° to t, and using (B2) and (B6) we obtain

J. Chern. Phys., Vol. 64, No.4, 15 February 1976

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1406 P. Ortoleva: Inhomogeneous chemioscillations

U(r, t) = jd3r''E.(r, r'; t)U(r', 0)

t

+ 1 d2r 1 dt''E.(r,r';t-t')g(r',t'), (B8)

where we have taken the limit (; - 0, in which case S· -5. Q.E.D.

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2D• D. Fitts, Nonequilibrium Thermodynamics (McGraw-Hill, New York, 1962); S. R. deGroot and P. Mazur, Nonequilib­rium Thermodynamics (North-Holland, Amsterdam, 1962).

3A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973); D. H. Sattinger, Topics in Stability and Bifurcation Theory (Springer, New York, 1973).

4A. Nitzan and J. Ross, J. Chem. Phys. 59, 291 (1973). 5A. Nitzan, P. Ortoleva, and J. Ross, J. Chem. Phys. 60,

3134 (1974). 6D• Perlmutter, The Stability of Chemical Reactors. (Pren-

tice Hall, Englewood Cliffs, NJ, 1972). 7N. Koppel and L. Howard, Studies Appl. Math. 52, 291 (1973). 8p • Ortoleva and J. Ross, J. Chem. Phys. 60, 5090 (1974). 9J . F. C. Auchmuty and G. Nicolis, Bull. Math. Bio!. (to be

published); M. Herschkowitz-Kaufman, Bull. Math. Bio!. (to be published).

ION. Minorsky, Nonlinear Oscillations (Van Nostrand, Prince­ton, NJ, 1962).

11M. Herschkowitz-Kaufman and G. Nicolis, J. Chem. Phys. 56, 1890 (1972).

12p. Ortoleva and J. Ross, J. Chem. Phys. 63, 3398 (1975). 13A. T. Winfree, "Physical Chemistry of Oscillatory Phenom-

ena," Faraday Symp. 9 (1974). 14p. OrtolevaandJ. Ross, J. Chem. Phys. 58,5673 (1973). 15G• Rosen, J. Chem. Phys. 63, 417 (1975). 16p. Ortoleva, "Dressed Kinetics for Chemioscillations in

Catalytic Dispersions," J. Chem. Phys. (submitted for pub­lication).

17D• F. Tatterson and J. L. Hudson, Chem. Eng. Commun. 1, 3 (1973).

J. Chern. Phys., Vol. 64, No.4, 15 February 1976

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