Auchmuty, J.F.G.; Nicolis, G. - Bifurcation Analysis of Reaction-diffusion Eqs. (1976)

BULLETIN OF

IV,-~THEI~IATI C AL BIOLOGY

VOLLn~E 38, 1976

BIFURCATION ANALYSIS OF I~EACTION-DIFFUSION EQUATIONS~I I I . CHEMICAL OSCILLATIONS

[] J. F. G. Avcm~cTY~ Department of Mathematics, Indiana University, Bloomington, Indiana 47401, U.S.A.

G. NICOLIS

Facultd des Sciences, Universit4 Libre de Bruxelles, Brussels, Belgium

1. Introduction. In this paper we shall describe certain oscillatory solutions of a model chemical system. I t is a continuation of two previous papers; Auchmuty and Nicolis (1975a) and Herschkowitz-Kaufman (1975) which will be called papers I and I I respectively.

There are a number of ways of obtaining oscillations in systems which are described by reaction-diffusion equations. When one imposes oscillatory boundary conditions, or prescribes oscillatory concentrations for a particular "buffer" reagent, one obtains forced oscillations of the system. Alternatively, one sometimes obtains transient oscillations in a chemical system during its evolution to a steady state. Here we will not be looking at either of these situations. Instead we shall be interested in oscillations which arise as a natural consequence of the coupling of diffusion and chemical kinetics and which persist over long time intervals.

The systems in which we are interested undergo mass-action type of kinetics. Thus the nonlinearities in the equations are of polynomial form. In systems with exponential nonlinearities, such as chemical reactors with temperature control, certain space-dependent chemical oscillations have been described by several authors. For recent reviews see Amundsen (1974) and Keller (1974).

325

326 J.F.G. AUCI4_MUTY AND G. NICOLIS

Turing (1952) seems to have been the first to point out that diffusion together with nonlinear kinetics can produce organization and that one can obtain stable solutions which are not homogeneous in space or time. Similar ideas have been extensively developed in the theory of dissipative structures (see Glansdorff and Prigogine (1971)). Here we shall s tudy time-periodiv solutions of a model nonlinear system which involves an autocataly~ie step.

The time-periodic solutions to be described may be considered, from a mathematical point of view, as limit cycles in an appropriate function space. Qualitatively, they are quite similar to oscillations described by limit cycles in systems of ordinary differential equations. For example, the stable time- periodic solutions arise as the asymptotic behaviour in time of the system for a set of initial conditions. Moreover, at each point in the medium, the rate of change of concentration of a constitutent may vary considerably during a period. Quite often these time-periodic solutions feature sharp bursts of chemical activity followed by intervals of relative quiescence.

The time-periodic solutions, in general, axe also spatially non-uniform. This depends, however, on the boundary conditions for the system. When one has no flux boundary conditions, one often obtains spatially uniform bulk oscillations. Spatially non-uniform oscillations often give the appearance of a propagating wave over a small time interval in numerical simulations (see Herschkowitz- Kaufman and l~icolis (1972) and Erneux and Herschkowitz-Kaufman (1975)) although in some cases one can show that there cannot be any propagating wave solutions (see Auchmuty and Nicolis (1975b)). Such oscillatory behaviour has been observed in the Belousov-Zhabotinskii reaction amongst others (Zhaikin and Zhabotinskii (1970)).

These chemical oscillations arise in a quite different manner from the chemical waves analyzed by Kopell and Howard (1973), by Ortoleva and Ross (1973, 1974) and, more recently, by Kuramoto and Tzusnki (1975). A compara- tive analysis of these various methods and a more detailed comparison with the observed chemical oscillations is given in Auehmuty and Nicolis (1975b).

The model system to be studied is due to Lefever and Prigogine (1968) and is described in paper I. I t is given b y the pair of nonlinear reaction-diffusion equations

~X = D 1 A X - (B+ 1 ) X + X g ' Y + A ,

at ~ y (1.1)

= D 2 A Y + B X - X 2 y . ~t

Here A, B are the concentrations of certain "initial" substances which are assumed constant throughout the system, X, Y are the concentrations of the two intermediates and D1, D2 are their diffusion coefficients. A represents the Laplacian. The nonlinear term X 2 Y describes the autocataiytic step in the

BIFURCATION ANALYSIS OF REACTION-DIFFUSION EQUATIONS 327

reaction scheme. The equations (1.1) are to be solved in a bounded region g~ of n-dimensional space with 1 < n < 3 and subject to given initial and boundary conditions.

I t will be assumed that on the boundary either one has given concentrations,

X = A, Y = B/A, (1.2)

or one has no flux conditions

a x aY - - - - - - 0 . ( 1 . 3 ) ~v ~v

Hero v is the direction of the outward normal to the boundary, l~or either of those sots of boundary conditions the system admits a uniform steady-state solution

Xo -- A, Yo - B /A . (1.4)

This holds for all positive values of B and this solution is the continuation of the equilibrium-like bohaviour as the distance from thermodynamic equilibrium increases and the reaction stops become completely irreversible. These solutions (1.4) will be referred to as the thermodynamic branch of solutions (Glansdorff and Prigogino (1971)).

In Section 2 we shall summarize some results about the stability of the thermodynamic branch (see also Auchmuty and Nieolis 1975a, b). From this stability analysis one can deduce the onset of certain time-periodic solutions and these will be constructed in Sections 3 and 4 using the Hopf bifurcation. Such solutions are temporal dissipative structures. We shall use the concentration B as the bifurcation parameter and show that, under certain conditions, there is a critical value BH of B such that, when B crosses BH, a new branch of time- periodic solutions of (l.1) arises. Their amplitude depends continuously on B and goes to zero as B goes to BH. The period of these solutions also depends continuously on B near B g . The solutions will, in general, be spatially inhomogeneous as well as time-periodic, although this depends on the boundary conditions.

The bifurcation of time-periodic solutions from steady-state solutions was developed for ordinary differential equations by E. Hopf (1942). Recently it has been used for partial differential equations by Sattinger and Joseph (1972) and Iooss (1972). See also Fife (1975) and Marsden and McCracken (1975). The Hopf bifurcation has already been used, though in a very different manner, for reaction-diffusion equations by Kopo]l and Howard (1973).

Section 5 describes some of the qualitative properties of the solutions. We show that, with fixed boundary conditions, the oscillations obtained are not propagating waves, although for a portion of each period they appear to have propagating fronts. When periodic boundary conditions are imposed, (i.e.

328 J .F.G. AUCHMUTY AND G. NICOLIS

one solves the system in a ring or on the surface of a sphere), one can get propagating waves of constant velocity as solutions. The part icular case of a ring is analyzed in Section 6, while Section 7 summarizes our conclusions.

2. Onset of Time-Periodic Solutions. Equat ions (1.1) subject to the boundary conditions (1.2) or (1.3) have the uniform steady-state solution (1.4) for every value of B. As was shown in paper I, there is a critical va lue /~ of B such that for B < / ~ , this solution is stable, while for B > / ~ it is unstable. Under certain conditions, when B > •, there is a stable time-periodic solution of the equations. Such a phenomenon is a l ready known for this system in the absence of diffusion. Here we shall discuss the si tuat ion where diffusion is present.

First we shall derive criteria for a time-periodic solution to bifurcate from the thermodynamic branch of solutions given by (1.4) via a Hopf bifurcation.

A Hopf bifurcation occurs at a value BH when the eigenvalue problem for the linear stability equations about the solution (1.4) admits a simple pair of complex conjugate eigenvalues with zero real parts at B = BH and provided the sign of the real part of this pair of eigenvalues changes as B crosses BH.

The eigenvalue problem is to find the eigenvalues ~ and eigenvectors of the linearized rate equations (c.f. paper I)

D1Au+ ( B - 1)u+A2v = Au, D ~ A v - B u - A 2 v = Av, (2.1)

in ~ subject to the boundary conditions (1.2) or (1.3). When ~ = [0, 1], and one imposes no flux boundary conditions, one sees tha t

(2.1) has _+ i# as eigenvalues when

B = B m = l+A2+m2~2(Dl+D2) , (2.2)

provided

4#2 = 4#2m = 2(A2+ 6m)Bm-(Ag"-5m)2-B2m (2.3)

is positive. Here, 5m = l+m2u2(D1-D2) must be positive and m is any integer including 0.

One can observe t ha t p2 m is a decreasing function of m, so t ha t i f (2.1) has a pair of purely complex eigenvalues then i t has them when m = 0.

l~ow p02 = A 9.

so this system always has a bifurcating time-period solution as B crosses B0 = 1 + A 2.

When f~ = [0, 1] and one imposes fixed boundary conditions, one gets the same criteria (2.2) and (2.3) bu t now m cannot be zero. This implies tha t if (2.1) has a pair of purely complex eigenvalnes t han it has them when B = B1. For this to be true one must have (see paper I)

D 2 - D 1 < 1/~ 2 (2.5)

and


A D~ < ~-~ (1 + u2(D1-D~)l/2). (2.6)

These are only necessary conditions for the existence of bifurcating time- periodic solutions. Their existence also depends on the relative magnitudes of A, D~ and (D1-D2), as will be shown later.

For other choices of the region ~, similar analyses hold. One replaces m2u 2 throughout by Am where - ~ m is the ruth eigenvalue of the Laplacian in the region subject to the given boundary conditions.

The basic solution (2.4) is unstable when any eigenvalue of (2.1) has positive real part.

When no flux boundary conditions are imposed on ~ = [0, 1], instability occurs when B > / ~ = rain (Be, Bo).

Here

Bo = I + A 2 and Be = minm__>l 2 + - ~ - A 2 + D2m2~2--+D1m2~ 2 .

Similarly, when the constant boundary conditions are prescribed on [0, 2] the basic solution (2.4) is unstable when

B > / ~ = min (Be, B1),

where Be and B1 are defined as previously. These facts follow from results in papers I and II.

I t is informative to find when Bo is less than Be. Using the approximation (paper I, equation 4.22) for Bo, one sees that Bo < Be when

2 + A 9. < ( I + 0 A ) 2,

with 0 = (D1/D2)l/% This holds provided

O >

Similarly, B1 < Be provided

1 0 > (2.s)

with ~ = (DI+D2)~ 2. In particular if D1 = D~ = D then (2.7) holds for any value of A and D,

but (2.8) only holds if A and D satisfy

A > D~ 2 (2.9)

When the thermodynamic solution (1.4) first becomes unstable through a simple real eigenvMue changing sign, then 1~ = Be. In this case the nonlinear

330 J.F.G. AUC]~IVIUTY AND G. ~ICOLIS

equations have a branch of steady-state solutions which bifurcate from B = Be. This situation was described in detail in papers I and II.

When, however, the thermodynamic solution (1.4) first becomes unstable because a simple pair of complex eigenvalues crosses the imaginary axis, then one has a Hopf bifurcation and the stability is transferred as B crosses the critical value/~ to a new family of bifurcating time-periodic solutions.

As ~ consequence (2.7) and (2.8), in general, describe the conditions under which time-periodic solutions bifurcate from the thermodynamic branch before any steady-state solutions. In particular, from (2.9), one sees tha t even with equal diffusion coefficients the time-periodic steady-state solutions may appear first. This differs from the results of Balslev and Degn (1974) amongst others. This difference occurs because we are imposing constant boundary conditions and we are working in a bounded region. Also from (2.2) and (2.3) one observes that when D1 ¢ D2, there can only be a finite number of complex eigenvalues of these equations (see paper I equation (4.9)), so there can only, in general, be a finite number of branches of time-periodic solutions which bifurcate from the thermodynamic branch.

Finally, one might compare these results with those obtained when diffusion is neglected. In that case (1.1) reduces to the system of ordinary differential equations

dX dY dt = A + X z Y - ( B + I ) X ' d---t = B X - X 2 Z " (2.10)

For this system, the solution (1.4) is stable for

B ~ B0 = I + A z

Moreover (1.4) is the only steady-state solution. When B ~ B0 there is a unique stable autonomous oscillation and the

system converges asymptotically in time to this solution. These, and other results on the system (2.10) are described in Lefever and Nicolis (1971) and in Lavenda, Nicolis and tterschkowitz-Kaufman (1971), J. W. Turner (1974) and J. Boa (1974).

3. Construction of Bifurcating Time-Periodic Solutions. We shall construct time-periodic solutions of the system of equations (1.1) using a method due to E. ttopf. As will be seen, this method even allows us to calculate the periodic solution of the ordinary differential equations (2.10) in a manner that is simpler than the stroboscopic method used by Lefever (1970).

Suppose that when B = Bin, the eigenvalue problem (2.1) has _+ ipm as eigenvahes. Let x = X - A and y = Y - B / A , then our original equations may be rewritten as


cOx~at = DIZkT, q- ( B - 1)x + A 2 y + h ( x , y), (3.1)

~y/~t = D ~ A y - B x - A 2 y - h ( x , y),

where h(x, y) = B A - l x 2 + (x + 2A)xy . (3.2)

The boundary conditions (1.2) become

x = y = 0. (3.3)

Alternatively, if (1.3) holds, it remains unchanged for x and y,

~x ~y - - O . ( 3 . 4 )

~v 8v

We would like to find time-periodic solutions of these equations. Assume they have a periodic solution of period 2u/co and tha t the frequency o~ depends on B. Let z -- o)t, then (3.1) m a y be rewri t ten in operator nota t ion as

(3.5)

y \ y l / \ y 2 / \ y a /

o) = Pro+ e~ol+ e2o~2+ . . . . (3.6)

B = B m + e ? l + e 2 7 2 + . . . .

and rewrite ~3.5) in the form

(;) () (x), z ( h(x, y) (3.7) (9--~z - L m Y = ( B - B I n ) - - +\-h(x, y)/'

where L u = LB,~.

Subst i tut ing (3.6) in (3.7) and equating equal powers of e, one gets the system of equations

Pm k Y n / - L m = - ~ o)~ Yn k=l ~ \ Y ~ - k / k - a n /

Here, L B is the matrix-differential operator defined by

\ y / - D2A - A 2 / \ y ]

and subject to the boundary conditions (3.3) or (3.4). Now we would like to find periodic solutions of period 2u in ~ of equation

(3.5). I n part icular we look for periodic solutions of small ampli tude near B = B m and of frequency near #m- To do this, one assumes an expansion of x, y, co and B in terms of a parameter ~. Le t

332 J .F.G. AUCHMUTY AND G. NICOLIS

Here, n > 1 and an is an expression involving the parameters 71, ~2 , . - - , ?n-1 and the solutions xl, Yl, x2, y~ . . . . . xn-1, Yn-1. The first few expressions are

al = O,

a2 = xl(?l + B m A - l x l + 2Ayl), (3.9)

2 aa = ~2Xl + 71x2 + A-l(2BmxlX2 + 71x 1) + 2A (xly2 + x2yl) + x~yl.

Equation (3.8) is a linear equat ion for Xn and Yn. To obtain its solutions it is simpler to use complex constructions. When n = 1 it becomes

( ) ( xI ) (:) Xl - L m = • (3.10) #m ~ yl Yl

I f (Um, Vm) is the complex eigenfunetion of Lm with eigenvalue ipm one sees tha t (3.10) has the complex solution

(xX) k(um)kvm/e~v (3.11)

where k is an arbi t rary complex constant. .For n > 1, (3.8) in general, is an inhomogeneous linear equation. However

the corresponding homogeneous equat ion has a non-trivial solution and so (3.8) has only a 2~-periodic solution provided a compatibi l i ty condition holds.

The Fredholm al ternat ive requires t ha t the r.h.s, of (3.8) be orthogonal to the non-trivial solutions of

0 _ L m = O, ( 3 . 1 2 ) - #m ~'-~ y

where L * is the adjoint of Lm. For this problem

Lm\ = \ A 2 D 2 A - A 2 J \ v ]

subject to the same boundary conditions as Lm. The solutions of (3.12) are given by

(x,) (,) Um eiV ----]c , y* v m

where

(3.13)

\ v J \ v ~ /

B I F U R C A T I O N ANALYSIS OF R E A C T I O N - D I F F U S I O N E Q U A T I O N S 333

The F redho lm al ternat ive now becomes

f2:~f~ a n ( r ' z ) [ ~ * ( r ' ~ ' ) - ' * ( r ' z ) ] d r d z o

= E drd k = l 0

= - Z k=l 0

where ~ represents the complex conjugate of x and x*, y* are defined b y (3.13). F o r each value of n, (3.14) provides two real equat ions which de termine o~n-1 and 7n-1- Using these values, one can then solve (3.8) to find xn and yn. This i tera t ive me thod can be cont inued to give as m a n y te rms in the series (3.6) as are desired and one can prove t h a t the result ing expansions are con- vergent series in 8 for e sufficiently small.

4. Calculation of Bifurcating Time-Periodic Solutions. In this section we shall calculate approximate expressions for the bifurcat ing t ime-periodic solutions of (1.1) in the cases where ~ -- [0~ 1] and when either (3.3) or (3.4) are the appropr ia te bounda ry conditions. Also, we shall restr ict a t t en t ion to the first such bifurcat ing branch in each case, as similar analyses hold for the other branches.

A. No flux boundary conditions. Firs t consider the case of no flux bounda ry conditions. In this case when B = B0; _+ iA are eigenvalues of the system (2.1). Take /10 = A, then the eigenvector problem is to find the solutions of

d2u _ d2y D1 "~--r2r2+A2u+A2v = iAu, l)2--~r2r2-(l+A2)u-A2v = iAv.

The solution of this is

v(r)/ \c2/

where Cl, C2 are constants, and cu/cl = ( i - A ) / A . We shall normalize the eigenvector so t h a t cl = 1. Then c2 = p e t° where

p = %/(A2+l)/A, sin 0 = 1 / v ' ( A 2 + I ) and cos 0 = -A/~c/(I+A~). Similarly the normal ized eigcnvector corresponding to the eigenvalue - - iA

is given by (u-(r)~ 1

v-(r)/=(pe-~°) • Thus when m = 0, (3.10) has the complex solution

yl(r, ~) = k Pe~ o e ~r,

334 /[. F . G. A U C I t M U T Y A N D G. N I C O L I S

where k is an a rb i t ra ry complex number . One wants, however, real 2~-periodio solutions of (3.10) and such a solut ion is

(x.,. :,)o( yl(r, cos ( ¢ + 0 + (p ) ]

(4.1)

Here c is an a rb i t ra ry ampl i tude and ~p is a phase factor. Wi thou t loss of general i ty one can take ¢p = 0 and one normalizes e in the expansion (3.6) b y taking c = 1. This defines the first t e rm in the expansion (3.6).

One now finds o21 and 71 b y using the compat ib i l i ty equat ion (3.14). Firstly,

a2(r, z) = 71 cos z + ( I + A 2 ) A -1 cos 2 z + 2 A p cos ( z + 0) cos

and the complex, normalized, solution of (3.12) is given b y

C:)-- (_/,o_,o)o,., where p and 0 are defined as before.

For each value of n, equat ion (3.14) becomes

\ j o

._, (f: ) = ~ i¢01¢ e -~v (xn_ k _ p-1 dOyn_k) dr dz . k=l do

When n = 2, the real and complex par ts yield

(l+p-lcos 0 --sin28 ~(~1) : (;) p-1 sin 0 cos 2 O - 1] (-01 "

(4.2)

(4.3)

This has the unique solution COl = the definition of 0, one sees t h a t sin 0 can never be 0.

Equat ion (3.8) now becomes

a x2 - L o = , A --~ Y~ \Y2 / - a d

where

Vl = 0 unless sin 0 = O. However , from

(4.4)

a2(~) = A- l (1 + A 2) cos2z + A p (cos (2z + 0) + cos 0).

To solve (4.4), one assumes t h a t x2 and Y2 have a Four ier expansion in both r and ~. One easily obtains t ha t there is no non-tr ivial r-dependence. Le t

° ' ' ,

BIFURCATION ANALYSIS OF REACTION-DIFFUSIOIq EQUATIOI~S

Then, subst i tu t ing in (4.4) and solving one finds

Y2(v)/ \bo+b cos (2~+ ~h2)/

where

and

( 1_1)] bo = 2A a ' = 3--A 1 + i A 2A 2

i bel~2 = _ ( l _ - - ~ a e i ~ ~.

\ 2A]

335

This gives us the second-order terms in the expression for (x, y) in (3.6). One now wants to find o~2 and V2. F i r s t ly one uses (3.9) to get the resul t t h a t

a3(r, ~) = ~2 cos ¢+2A-i( l+A2)a cos • cos ( 2 z + ~ i ) + 2 A [ ( b o

+b cos ( 2 z + ~ 2 ) cos "c)+pa cos ( 2 z + ~i) cos ( z + 0)]

+ p eos2z cos (~+ 0).

When n = 3, (4.3) yields

(1 ~-p-ie~o)(?2+z)-io~2(1 - - e 210) = 0 ,

where z is a wel l -determined complex number . Equa t ing real and imaginary parts of this equat ion, one gets

l + p - i cos 0 - s i n 20 ~( ~2~ = ( ~ i )

p - i sin 0 - 1 + cos 20]\o~2] ~2

where ~i = Re (1 + p-ie~O)z and C2 = I m (1 +p-lei0)z. The solution of this is

sin ( 0 - @ ) = C(cos ~ + A sin ~), o~2 = sin 0

and

- - [sin ~ + p - i sin ( ~ - 0)] = ½~ (A cos 6 - s i n ~), ~2 - 2 sin 2 0

where ~1+i~2 = ~e i~p. One sees t h a t this solution is non-tr ivial provided z # 0. In fact,

z = a e ~ l [ A - i ( 1 + A 2) + 2Ape -i°] + 2A(b0 + be iw-~) + (p/4)(e ~° + 2 cos 0),

and one finds t h a t ~2 is positive, so t h a t one always has supercrit ical bifurcation.

"OAOq~ UOt't~ 0~ 0 pu~ d ozoqax

(.: s! (gI'8) jo uo.~nios poz.q~mzou xoIdmOO oq~ 'Xia~l.~m.tg

st (0I'~) ~o uo.~n[os poz.q~m~ou 'l~Oa oqff, 'sos~ozou T V s~ 0z o~ so~zoAuoo 0 ~q* pu~ ~, ~ 0 ~ ~/~ ~q~ o~o~

• E~ + ~(~q + ~v)]/~ ---- 0 800 (¢'~(/+ g V) -

pu~8

= ~d [~,~ + ~(~%r + ~V)]/~ ¢ + ~(~u~q + ~V)

= 0ut~ It/ I

ozoq~

oa~ suo.t~ounjt~o~.m ~u!ptioclsoz~oo oq£ • oA.t%tsod s.t [~o.tp~z o1~ u.t tlLio~. 0T~qL pop.IAOZCI

O.I0qA~ 'Ir/¢ -T- o.Ye 8on[~Atlo~.to

-~o~to £~u!~m.~ £[oznd s~q (I'~) qo.~q~ ao~ ff ~o on[~A ~s,o[ oq~ L "suo.~.qo~oo LI~punoq pox~j 3o os~o oq~ aopTsuoo ~oK "~uo~.~.puoo t~z~opunoq p~x,.d "~t

"(0I'~) uo.~,~nbo i~!~uozojj.~p £avm.pzo oq~ 3o uo.t~nios ~ osiv s.~ ~.~ ',uopuocloptq -ooeds s.~ ~! oout.s 'pu~ mn.lpom oioq~ oq~ jo uo.t~e[i.~oso ~I[nq * s.~ tlo.~nios s.tq~ L

(W- t - ~r):_ ~,~o~ + V = (~r)o~ q~:~

(0 +leo) soo =

s~ 'aopzo puooos o3 'uogg.w_~ oq X~m o~r = ~r g~ uo.tgn[os i~.tA.t~g oq~ mo~ og~oanj!q qo.tq~ suo.tgnios o.tpoFtocI oqg g~qc~ soos ouo '(9'8) u.t s.tqg ~u.ts~

~V -~r

0~..LIAk 1.I~0 0UO 'O.I0Z .I~80Li ~ .I0~; r

gI'IODIK "D (IXV XE~I~I-ID_0_V "D '~I r 988

+ ~V + ~q~(I

aV-

"o.*°dV} + a-V{ = a7 l °u*

) trf.,g + t~r- I + a~/tGr

3 ° suo!~nios oq~ oa~ a~q '~n oaoq~

Xq pott~op oa~ smao~ aoq3o oq~ L

• suo!~nios oq~ u.te~qo Xem ouo 'oaoq~ ouop s~ s~ ~snf 'pu~ I sod~d~ jo (~I'~) uo.t~nbx s~ om~s oqa, di~o~xo s! s.tq£

~(~-a~/)~/ - ~ ' 0 soo dV+~_V} = o~ l ~-~/~ + a~/

) gq uoAt~ oa~ ~q pu~ ~/v oao H

ppo

• soo oo

s~o~ otto 'z pu~ a q~oq u.t ~t~ pu~ ~x aoj suo.tsu~clxo ,io.t,mo,ff ~u.ts~

-,t:t~u.ts [(0 +z) soo z soo tdV dg+~soo t_Vtff] = (z '~)~v oaoq~

(z '.0~v- 'I -- ~r/

st .~ou ~fi pu~ ~x ao,I uo!~nbo oq~

"0 = to9 X[it~[.~m.~ "oaoz ,IoAotI s.t (0 soo dgv+I) s~ r

0 = tL .IO '0 = I£(0 soodav+I)

~q~ soos ouo s~:t~d I~Oa ~ctt~nb X

"0 = (~dr~av- I)tC°. ~- (o~.-odav + I) tL}

oa~ soonpo~t s.~q~ 'g = u uoq~B_

o I o C t=~ (9'~) "~p ,tp .,u u.ts (~-ufio.,_od~v-~-~x) ~.,_o ~co~. ~ = I3 ~zg$ I-u

o I of x p ~p .~,, m.s [o.~_od~v + i](z '~)u~ ~_o tJ ~ J

'somoooq (~['~) uo.~nbo £~.q.tq.t~clmoo oq~ pu~

[(0 +z) soo z soo d~ffVg+Zasoo ~_V~ff],tuams+,tu ms z soo z£ = (z '.Oar

~ocu.t~ s!qff,

/~g8 ~KOI&VflOH KOISflaIaIlff-XOI,l, OVH~I ~IO SlSX~vxv KoI,l,vo~Ifl~II~

338 J . F . G . A U C H M U T Y A_N'D G. N I C O L I S

One now would like to find (o2 and 72. The calculations now become very long. To start with

a~(r, z) = 72 cos z sin ur+Blp cose~ cos ( z + 0) sin37:r+2ABl[A-2cos

+ p cos ( ~ + O)] sin u r ( ~ [akTS/6 cos (2~ + Ok)J sin k~r ) 16 odd

+ 2 A cos z sin ur ~ [b/6+Sk (cos (2v+~/6)] sin/cur. /6 odd

The compatibili ty equation (4.6) reduces to

1(1 + A2pe-iO)~2 - ico2(1 - Blp2A 2) = z,

where z is a complicated expression, depending primari ly on the solutions x2 and y2.

Equating real and imaginary parts, one finds o~2 and ~ :

~2 -- 2z1(1 +A2p cos 0) -1,

(o2 = (Blp2A 2 - 1)-1(1 + A2p cos O)-l[z2+ A2p(zl sin 0+z2 cos 0)],

where zl and z2 are the real and imaginary parts of z. Just as before one can use reversion of power series to get an explicit expres-

sion for the bifurcating time-periodic solutions near the bifurcation point. How- ever, it is probably just as useful to keep the expression in the implicit form of (3.6). Then one has, to second order,

( co.o, ) y(r,t) = ~ Blp cos (o~t+ O) sin ~r

÷e 2 ~ (a/6-kakc°s(2°)t~-~k)~sin/c~r (4.7) k=l \b/6~-/~k cos (2mt~-~0k)] /6 odd

where ~o = #1 + ~2w2 and B = B1 + ~u~2. Similar calculations m a y be made when B is in the neighborhood of any

other value Bm for which there are a simple pair of purely imaginary eigenvalues of the stability operator. Also similar analyses hold ff one uses other geometries. In each case, the eigenvalues and eigcnfunctions are different so the detailed calculations are somewhat changed.

5. Qualitative Properties of Time-Periodic Dissipative Structures A. Synchronization via diffusion coupling. In the absence of diffusion, and

when B ~ B0, the concentrations in the tr imolecular model oscillate at each point in space. When diffusion is introduced, these local oscillations become coupled and can produce synchronized oscillations. These oscillations arise as the asymptotic behaviour in t ime of the system. When they are time-periodic


the system maintains sharp, reproducible, phase relations between oscillations at different points in the system.

This is a consequence of having a bounded medium. In an unbounded medium, diffusion does not couple the oscillations in the same way and the solutions do not necessarily converge to a well-defined asymptotic state. Moreover, the size of the bounded medium has an important effect on the oscillations obtained as is described in paper I, Section 6e, and one sees that taking larger and larger systems is equivalent to making the diffusion coupling smaller and smaller.

These solutions are quite different from the plane wave solutions discussed by a number of authors; Kopell and Howard (1973), Howard (1974) and Ortoleva and Ross (]973, 1974). Our solutions are not given by 1-parameter families but are periodic oscillations with a characteristic frequency, amplitude and wavelength which axe determined by the kinetics and the size of the system and are largely independent of its initial excitation. The stable space-dependent patterns we have obtained can emerge from initially uniform systems because of an instability which is triggered by diffusion. Depending on the distance of the bifurcation point from its critical value, these new patterns can either be of small amplitude or appear similar to relaxation oscillations--although now they are space-dependent as well as oscillatory in times.

These latter patterns are similar to the trigger waves described by Winfree (1973, 1974) and by Kopell and Howard (1973), but they axe quite different from the leinematic or pseudo-waves discussed by the same authors. These later waves were regarded as being the result of a loose coupling of local oscillators of the limit cycle type. In unbounded domains, however, there seems to be a question about the mechanism for synchronizing these oscillations to produce a coherent regime. In contrast in bounded domains, the boundary conditions effect the synchronization and in such systems the kinematic waves can only represent transient states which eventually will evolve to the asymptotically stable solutions described above.

B. The existence of a propagation velocity. The expressions (4.7) for our time- periodic solutions describe a superposition of standing waves. The first-order terms have frequency o~, the second order terms have frequency 20) and it may be observed that higher-order terms have frequencies which involve higher multiples of w. Each term corresponds to an oscillatory chemical concentration similar in form to a "vibrating string". The superposition of these oscillations gives rise to the appearance of propagating fronts which, however, only subsist during part of the overall period. This situation is depicted below in Figure l.

These solutions axe not, however, propagating waves. To see this, one cal- culates the rate of change of concentration at a point. From (4.7) one sees that

--~- (r, t) = - ~co sin o~t sin u r - 2e u ~ co5~ sin (2cot+ ~hk) sin/cur. odd

340 J . F . G . AUCHMUTY AND G. I~ICOLIS

I f the c o n c e n t r a t i o n o f X were d e s c r i b e d b y a w a v e w h o s e p r o p a g a t i o n

ve loc i ty is v, t h e n

~x ~x v

' ~, ( i }

I"

(ii)

(iii) I"

Figure 1. Characteristic steps of evolution of the spatial profile of X during one period.

(i) and (ii) : the modes m -- 1 and m = 3 predominate. (iii) The m --- 3 mode vanishes and the one with m = 5 takes over. Comparing (ii) and (iii) we see that, effectively, the two fronts denoted by the dot ted lines have propagated to the middle. Thus, a wave-like act ivi ty appears in the system during a par t

of the overall period

Using the expres s ion (4.7) for x one sees t h a t t h i s v e l o c i t y is a c o m p l i c a t e d

funct ion of r a n d t. A t t h e e n d p o i n t s one h a s v(0, t ) = v(1, t) = 0, wh i l e a t t h e

center, to second order , one h a s (~x/ar)(½, t) = 0, so v(½, t) = _+ oo.

Thus the usua l i deas a s s o c i a t e d w i t h p r o p a g a t i n g w a v e s in e l e c t r o m a g n e t i s m ,

fluid d y n a m i c s a n d e l s ewhe re do n o t d e s c r i b e t h e s e so lu t ions . T h e t e r m

"chemica l w a v e s " s h o u l d t h e r e f o r e b e u s e d w i t h c a u t i o n . T h e s e so lu t i ons a r e

more s imi lar to t h e s y n c h r o n o u s o sc i l l a t i o ns e n c o u n t e r e d in m a n y e l ec t r i ca l

engineering p r o b l e m s ( A n d r o n o v , V i t t a n d Cha ik in , 1966).

B I F U R C A T I O N A N A L Y SIS OF R E A C T I O N - D I F F U S I O N E Q U A T I O N S 341

Such results are to be expected from the mathematicM formulation. A reaction-diffusion equation propagates information with ilffinite velocity in that the concentration at a given point at a particular time depends on the concentration at every point of the system at all previous times. This is different from the behaviour of the classical wave equation or of hyperbo]ic equations where there is a well-defined domain of influence and a characteristi c velocity of propagation of information. Also our time-periodic oscillations are obtained as the asymptotic behaviour in time of our system for a wide range of initial conditions. In contrast, for the usual wave equations, one gets different waves for different initial conditions as, for example, is well known for electro- magnetic wave propagation.

Nevertheless, the chemical oscillations described in the preceding sections are capable of transporting matter. To see this, one computes the diffusion current of the chemical X at a given point r

~X (B-BI~I /9" J x(r, t) = - D1 ~ -~ Dire - - cos cot cos ~r

\ ~2 /

+DI~(B-B------I) ~ m(am+~m COS (2oot+Om)) \ ~)2 / m odd

x cos m u r + o ( ] B - B l [ ) .

Averaging over a period, one sees tha t

(B- ]x ( r ) =Dlrc ~ ~ mare cos rarer.

~2 m odd

Thus, one has

Jz( ) = o

and ( B - B1)

Jx(0) -- - i x ( l ) -- D l r c - - ~ mare. ~)2 modd

In this system, matter is transported by the diffusion of the chemicals in tho medium.

C. Othe r qualitative effects. I t is of interest to compare the bulk properties of these temporal dissipative structures with those of the thermodynamic solution. From equation (4.7) one observes that the mean

f 20s/m ~(r) = x(r, t) dt = e 2 ~ a~ sin k~r /c odd

and similarly for ~(r). Thus the change in mean concentration is only a second- order correction but it is spatially dependent.

342 J . F . G. AUCH_MUTY AN]:) G. hTICOLIS

One can also evaluate the spatial averages:

(x(t)) = x(r, t) dr = - - cos ~ t + - - ~ [a~ + ~ cos (2~ot+ ~ ) ] 7C ]~=I

This is not even conserved to the dominant order. Similar results hold for (y(t)>. Under appropriate conditions, there can be a number of branches of time-

periodic solutions which bifurcate from the thermodynamic branch. Moreover there might also be secondary b~furcation of more complicated solutions from these branches of time-periodic solutions in a manner similar to that for steady-state solutions (Mahar and Matkowsky (1975)).

D. Comparison with computer simulation. The numerical solution of these equations reported in Erneux and Herschkowitz-Kaufman (1975) also predicts the existence of time-periodic solutions exhibiting shaxp wave fronts. During each period, a wave front originates near each boundary and propagates into the system. Eventually the fronts collide, the wave character is annihilated and the concentrations then change slowly until the wave fronts form again. The entire phenomena repeat themselves with a well-determined period. There is quite a striking agreement with the evolution depicted in Figure 1.

The simulations also indicate the existence of a number of time-periodic dissipative structures when B > BH. In particular, with zero flux boundary conditions, one can find, in addition to the uniform limit cycle solution described previously, space-dependent solutions which persist for more than 60 periods without decaying to homogeneous oscillations. These solutions also featured a propagating front which propagated in a preferred direction. Under certain conditions, wave trains were also found.

6. Travelling Waves in Periodic Geometries. In the previous sections we have studied the oscillatory solutions of (1.1) subject to boundary conditions given by either (1.2) or (1.3). There is another type of boundary condition which is of considerable interest, in biological problems as well as elsewhere.

Such boundary conditions arise when one solves the equations in a medium which can be represented by a closed curve in two/dimensional space, or a .closed surface in three-dimensional space. The simplest cases are a circle or ring 5n two dimensions, or the surface of a sphere in three dimensions.

Here we shall s tudy the case of a ring of length 2u in two-dimensions. Then ~he boundary conditions are periodic boundary conditions

~X aX X(0, t) = X(2~, t), aO (0, t) = --~ (2n, t)

X(O, t) = Y(2n, t), aY aY ao (o, t) = --~ (2~, t)

(6.1)


Obviously,

X(O, t) ~ A , :Y(O, t) - B / A (6.2)

is still a solution of the system (1.1) and one would like to see what other types of solutions bifurcate from this branch of solutions.

The surprising result is that one can now get travelling wave solutions. To see this, let ~ = 0 - vt and look for solutions X , Y which are functions of ~ only.

Then equations (1.1) become

D 1 X " - ( B + I ) X + X 2 Y + A = - v X ' , D 2 Y " + B X - X 2 Y = - v Y "

(6.3)

subject to X(~) ~ X(2=+ ~), Y(~) =_ Y(2=+ 4)for 0 < ~ < 2= and where the primes denote differentiation with respect to 4.

One would like to find those values of v for which there are non-trivial solutions X and Y of (6.3). The basic soIutions of (6.3) are given as usual by (6.2).

Let x = X - A , y = Y - B / A , then (6.3) becomes

Dlx" + ( B - 1)x + A2y + h(x, y) = - v x ' , D 2 y " - B x - A 2 y - h ( x , y) = - v y ' ,

h(x, y) = B A - l x 2 + xy(2A + x). (6.4)

One wants to find 2u-periodic solutions of this equation. Now x = 0 and y ~ 0 is a solution of this equation for each value of B and the only points where new solutions may bifurcate from the trivial solution are at those values of B for which the linear equation

Dlx" + ( B - 1 ) x + A 2 y = - v x ' , D 2 y " - B x - A U y = - v y '

has non-trivial 2=-periodic solutions. Any such solution must have the form

(:)o,o° y(~)/

and there is such a solution if and only ff m ¢ 0 and

B = Bm = I + A 2+mz(DI+D2) .

For this value of B, the corresponding value of v obeys

v 2 = vm2 = A~m-2+ (Dl_D~)A2_D22m 2.

Thus one sees that vm is real provided

D~m ~ <= A 2 m - 9 " + ( D 1 - D 2 ) A 2. (6.5)

In particular, there can only be a finite number of possible points o f bifur-

344 J .F .G. AUCHMUTY AND G. NICOLI$

cation for travelling wave solutions. There will be no such bifurcation points ff (6.5) does not hold with m = 1. That is, if

D2 > ½A21%/(1 +A-2(1 +D1) ) - 1]. (6.6)

In general there will be M possible bifurcation points, given by B1, B2 . . . . . BM where M is the largest integer less than

# = ½A2D~l[O-l+x/((O-1)2+4A-2)] , and 0 = D1/D2.

The actual construction of these travelling wave solutions proceeds in a manner similar to that of the Hopf bifurcation. As will be shown, at each allowable Bin, the eigenspace is two-dimensional, but for this type of bifurcation one also has to find two unknowns so, in general, the equations are soluble. For these problems, the velocity of the travelling wave plays a role analogous to the frequency of the periodic solution in the Hopf bifurcation.

We shall assume (6.6) holds and describe the calculations near B = B1 and v = vl. Similar analyses hold near any other bifurcation point. Firstly one can rewrite (6.4) as

Here

t L x = l ( y ) (V--Vl)(~,)-k (B--B1)(x'~[_xj+~__h(x,y))h(x, y) '~

Li(Z~ : ( -n lx"-v lx ' - (Ul-1)x-A2y~. \ y / \ - D2y" - vly' -k Blx-[- A2y ]

(6.7)

T o find solutions near x = y = 0 a n d B = B1, v = vl, we shall expand everything in powers of a parameter s. Let

\ Y l / \Y2/ \Ya/

B ---- Bl-[-sT1-]-~2~2-~ . . . ,

v -- vlq- e/tlq- ~2#2-b . . . .

(6.s)

If one substitutes these series in (6.7) and equates equal powers of ~ one obtains the family of equations

\y~/ \b~(~)/

Here an and bn are expressions involving the parameters ~1, T2 . . . . . ~n-1 and


#1, P2, • • . , #n-1 together with the solutions Xl, Yl, x2, Y2 up to xn- z , Yn-1. The first few expressions are

a1(4) = b1(4) = 0,

r a2(4) = #1% + ],1xl + B iA-Zx~ + 2 A x l y l ,

52(4) = PlY'I - 71xl - B I A - l x ~ - 2 A x l y l ,

aa(~) = ~1x'2 + ~x'~ + ~ ( 4 ) ,

b3(4) -- #1Y2 + #2Yl- 3(4),

where 53(4) is formally the same as the expression for as in (3.9). The equat ion

has two linearly independent solutions

(x(1)(~!~ ( a l ) ~x(2)(~)~ ( a 2 ) = cos 4 and = sin 4.

\y(1)(~)/ bl \y (2) (~) / b2

These can be taken to be the real and imaginary parts of

1

\9(4)/ where

and

p-2 = A2 (A2+ 1 + Dz + D2)+ D e ( D 2 - 1)

t an 6 - V l and 0 < ~ < n A2 + D 2 2

Equat ion (6.9) has a solution for Xn, Yn if and only if a compatibi l i ty condition holds. Namely the r.h.s, must be orthogonal to the null space of L~. The equat ion

has two linearly independent solutions which can be taken to be the real and imaginary parts of

Y*(4)] A2pe - ~

where ~ and p are given above.

346 J . F . G . AUCHMUTY AND G. NICOLIS

Two linearly independent solutions are

(AZp c°s4 ~ and (A2pSinC ~. cos ( 4 - ~ ) / sin ( 4 - ~ ) ]

The compatibil i ty conditions for (6.9) now become

f an [an(C) cos C +bn(~) A2 cos ( 4 - $)] d4 = 0 P 0

and (6.10)

~ [an(C) sin 4+A 2 pbn(~) sin ( 4 - ~)] d4 = 0.

In complex form, these can be wr i t t en

Xs [an(~)e-'¢ + A 2 pbn(~)e -'(~-v)] d~ = O. (6.11)

For each value of n greater t h a n 1, these equat ions determine #n-1 and 7n-1. When these terms are known, one can then solve (6.8) for the funct ions Xn(~) and Yn(~).

When n = 1, (6.8) reduces to

(Xl~ (:) .L1 = . \Y l /

The general solution of this is any l inear combinat ion of the two l inearly independent solutions given above. We shall take

cos ) Yl(C)] - -B lp COS ( ~ + ~ )

This choice determines bo th a phase factor and the normal iza t ion of e. With this expression for xl and Yl, one finds t h a t

• a~(~) = - #1 sin 4 + 71 cos 4 + B1A-1 cos24- 2ABlp cos 4 cos (C + ~),

b~(~) = plBlp sin (~ + ~) - 71 cos 4 - B1A -1 cos2C + 2ABlp cos 4 cos (~ + ~h).

After a calculation, one finds t h a t the compat ibi l i ty equat ions (6.11) become, when n --- 2,

( 1 - A ~ p e~V)71+i(1 -B1A2p2)#l = O.

Equating real parts , one gets

(1 - -A 2 p cos ~b)71 = 0

which implies 71 = 0. Thus/~1 = 0 as p-~ ~ B1A 2 provided D2 is nei ther 0 nor ].

B I F U R C A T I O N ANALYSIS OF R E A C T I O N - D I F F U S I O N E Q U A T I O N S 347

Now, xs a n d y~ a re s o l u t i o n s o f

- D l x ' ~ - v lx ' 2 - ( B 1 - 1 ) x s - A 2 y 2 = B 1 A -~ c o s g ' ~ - 2 A B l p cos ~ cos (4 + ~)

-Dsy '~ - vly'2 +-Blx~ + A ~Y2 = - B I A -~ cos ~ ~ + 2 A B ~ p cos ~ cos (4 + ~).

To f ind x~ a n d Ys one uses a F o u r i e r e x p a n s i o n :

y2(~) / ~=o d~

J u s t as be fo re , one f inds t h a t a l l t h e coeff ic ients a re zero e x c e p t t h o s e corres-

p o n d i n g t o l = 0 a n d _+ 2. W h e n 1 = 0, one ge t s

B~ A ~" / \ C l o / - ~o '

where ao = ½(BIA - ~ - 2 A B ~ p cos ~).

T h u s Co = 0 a n d do = B ~ A - ~ ( A 2 p cos ~ - ½ ) .

W h e n 1 = - 2, t h e e q u a t i o n s b e c o m e

(..,_.1÷1÷.,v, B1 4Ds + A 2 + 2 i v l ] \ d - 2 ] - a-s~

where a - s = B l [ ( 4 A ) - l - ½ A p e i W ] , a n d t h e s e h a v e a u n i q u e so lu t ion for

C-2, d--2 •

S i m i l a r l y , w h e n l = 2 one ge t s

B1 A 2 + 4D2 - 2iv1,/ \d2] \ - ~ -2 /

where a-2 as a b o v e . T h u s one sees t h a t c2 = 5-2, d2 = d - s .

I f one w r i t e s c2 = p l e i ~ d2 = p2 ei~2, t h e n

Ys(~)) \ d o + 2 p 2 cos ( 2 ~ + ~ 2 ) 1

N e x t one w o u l d l i ke to f ind p2 a n d ~2- Since/~1 = yl = 0 one has

a~(~) = - #2 s in ~ + ~s cos ~ + h s ( ~ ) ,

ba({) = #2P s in ( ~ + @ ) - Y s cos ~ - h s ( ~ ) ,

where hs(~) is a c o m p l i c a t e d exp re s s ion .

The c o m p a t i b i l i t y e q u a t i o n (6.11) w i t h n = 3, b e c o m e s

( 1 - A S p ei~)Ts + i ( 1 - B 1 A S p 2 ) # s = a e *~',

where

e e~ °, = ~ - l ( A 2 p e - ~ - 1) hs(~) e - ~ d ~ . o

(6.12)

348 J, F. G. AUCH_MUT¥ ~ G. NICOLIS

In general a ~ 0, so (6.12) has a unique, non-zero solution for #2 and ?'2.

When these values are substituted back in (6.8). one has an expression for a first approximation to the bifurcating solution as, if T2 > 0, then

( B - B : ~ 1/2, s "~ +_ - - v(B) = v l + ( B - B 1 ) #2

\ T2 / Y2 and

X(~) = + COS ~ '~-2 P l COS (2~ '~ -~1 ) , - \ 72 / \ - - ~ 2 /

together with a similar expression for y(~). Thus for B near B1, one has tha t there is a propagating wave solution of this

system of equations whose amplitude is proportional to %/(B-B1) and whose velocity tends to Vl as B converges to B1.

7. Conclusions. In this paper we have described certain oscillatory solutions of a system of reaction-diffusion equations. Such solutions exist for a wide range of values of the parameters describing the system, and may be calculated using the techniques of bifurcation theory.

The character of the solutions depends greatly on the boundary conditions. When one imposes no flux boundary conditions, one can get homogeneous bulk oscillations of the system. When one requires that the concentrations be kept fixed at the boundary, the solutions are space-dependent as well as time- periodic oscillations. While if one solves the equations in a ring, one finds propagating waves of a characteristic velocity. For each of these systems some of these time-periodic solutions arise as the asymptotic behaviour in time of these equations and often this is largely independent of the initial conditions of the system. So far, we have not carried out a detailed study of the stability of these oscillations but preliminary numerical solutions indicate tha t some of these oscillations are very stable.

These time-periodic solutions have a number of remarkable physico-chemical properties. They can produce appreciable quantities of chemicals in a limited region of space and at regular time intervals, thus endowing these chemical oscillators with powerful regulatory properties. Also the wave fronts can have a very high velocity of propagation--it can be orders of magnitude larger than the "velocity" of spread of a concentration front undergoing ordinary diffusion. Thus these waves are an efficient way of transmitting information over macroscopic distances and during macroscopic time intervals.

There are still many important unanswered questions about the oscillatory behavionr of these and other, similar, reaction diffusion equations. Firstly one would like to be able to describe the transient behaviour of these systems as they evolve to the asymptotic solutions described in the previous sections. Secondly, one would like to have a better description of oscillations in two-and three-


dimensional systems. The experimental observations on the Belousov-Zhabo-

tinskii reactions as well as the analysis of model systems (Winfree,

1974; Stanshine, 1975) suggest the existence of ro ta t ing wave solutions of

reaction-diffusion equations. Such solutions seem to have leading centers, t h a t is, there are regions in space which do not oscillate despite the fact t ha t

surrounding regions undergo oscillations and one would like to explain this

phenomenon.

Nevertheless the solutions described in this paper seem to be qual i ta t ively

similar to some of the observed experiments and we believe the methods

employed here can be of great use in m a n y reaction-diffusion systems where

oscillatory phenomena arise.

The work of the first au thor was par t ia l ly supported by the Nat ional Science

Foundat ion grant GP44008 and was done while visiting the Ins t i tu ts In ter -

nat ionaux de Chimie et de Physique fond~e par E. Solvay in Brussels, Belgium. We would also like to t h a n k E. Hopf, I. Prigogine, R. Lefever, M. Herschkowitz-

Kaufman, A. Bab loyan tz and Th. Erneux for their comments and assistance.

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Auchmuty, J.F.G.; Nicolis, G. - Bifurcation Analysis of Reaction-diffusion Eqs. (1976)

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Transcript of Auchmuty, J.F.G.; Nicolis, G. - Bifurcation Analysis of Reaction-diffusion Eqs. (1976)