modeling reaction--diffusion pattern formation in the couette flow...

Modeling reaction-diffusion pattern formation in the Couette flow reactor J. Elezgaraya) Center for AppIied Mathematics, Cornell University, Ithaca, New York 14853

A. Arneodo’) Centerfor Nonlinear Dynamics and Department of Physics, The University of Texas, Austin, Texas 78712

(Received 3 December 1990; accepted 26 February 1991)

We report on a numerical and theoretical study of spatio-temporal pattern forming phenomena in a one-dimensional reaction-diffusion system with equal diffusion coefficients. When imposing a concentration gradient through the system, this model mimics the sustained stationary and periodically oscillating “front structures” observed in a recent experiment conducted in the Couette flow reactor. Conditions are also found under which oscillations of the nontrivial spatial patterns become chaotic. Singular perturbation techniques are used to study the existence and the linear stability of single-front and multi-front patterns. A nonlinear analysis of bifurcating patterns is carried out using a center manifold/normal form approach. The theoretical predictions of the normal form calculations are found in quantitative agreement with direct simulations of the Hopf bifurcation from steady to oscillating front patterns. The remarkable feature of these sustained spatio-temporal phenomena is the fact that they organize due to the interaction of the diffusion process with a chemical reaction which itself would proceed in a stationary manner if diffusion was negligible. This study clearly demonstrates that complex spatio-temporal patterns do not necessarily result from the coupling of oscillators or nonlinear transport.

I. INTRODUCTION

In recent years there has been increasing interest in pattern forming phenomena in chemical systems.‘-” In the early eighties a great deal of attention has been paid to the dynamics of homogeneous chemical reactions. In well-mixed media, the intrinsic nonlinear nature of chemical kinetics has provided a field of experimentation for the study of low- dimensional dynamical systems.5 When maintained far from thermodynamic equilibrium in a continuously stirred tank reactor (CSTR), chemical reactions have been shown to exhibit a transition from coherent (periodic) temporal patterns to chemical chaos. I3 Among these chemical oscillators, the Belousov-Zhabotinskii (BZ) reactionI has revealed most of the well-known scenarios to chaos including period doubling, intermittency, frequency locking, collapse of tori and crisis phenomena.‘5*16 In contrast, for years there has been only a little progress in the experimental research on sustained spatial and spatio-temporal chemical structures where, in addition, a diffusive transport process com- petes with the local chemical kinetics. Most experiments have been performed in closed systems where the system uncontrollably and irreversibly relaxed to thermodynamic equilibrium. Therefore the applicability of these experiments were limited to the study of transient patterns deve- loping in a rather short time, in practice those resulting from excitability phenomena such as the so-called target patterns and spiral waves.5-7~10*“~‘7-23

In the past two years, however, there has been a rebirth of interest in the formation of dissipative structures in che- mically reacting and diffusing systems. This interest has been mainly sparked by the development of open spatial reactors by groups in Texas2”30 and in Bordeaux.3’-38 Basical-

*) Permanent address: Centre de Recherche Paul Pascal, Universite de Bor- deaux I, Avenue Schweitzer, 33600 Pessac, France.

ly, two types of open reactors are currently operating: (i) the two-dimensional continuously fed unstirred reactors where the transport process is essentially natural molecular diffusion and where the feeding is either uniform (continuously fed unstirred reactorz6.” ) or from the lateral boundaries ( linear3s-37 or annular24V25V38 gel reactors); and (ii) the Couette flow reactor29-34 which provides a practical implementation of an effectively one-dimensional reaction-diffusion system with well-defined boundary conditions and controllable diffusion process. These new available pieces of apparatus have already produced a wealth of genuine sustained chemical dissipative structures. The aim of the present study is to provide theoretical and numerical support for the recent observations of one-dimensional spatio-temporal patterns in the Couette flow reactor with variants of the chlorite-iodide reaction. 39 In this introduction, we briefly describe the corresponding experimental system with special emphasis on the characteristic properties of the observed patterns which require a specific mathematical description. In the conclusion, we will discuss the possibility of generaliz- ing our theoretical approach to two-dimensional reaction- diffusion systems that model self-organization phenomena observed in the gel reactors. For technical details concerning this new generation of open spatial reactors, we refer the reader to the original publications2”39 and to the review arti- cle by Boissonade.40

Nonlinear reaction-diffusion equation models have been widely used to account for pattern forming phenomena in chemical systems maintained far from equilibrium.‘- ‘2*‘8*‘9 From a theoretical point of view, one may distinguish two types of reaction-diffusion structures: (i) global structures resulting from intrinsic symmetry-breaking instabilities, e.g., the Turing structures’-3v4’742 and the phase-wave structures;4 and (ii) localized structures associated with fronts, i.e., steep spatial changes of concentration which ac-

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324 J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation

tually correspond to transitions between two chemical states (e.g., a reduced and a oxidized state) with fast kinetics, e.g., the traveling waves in excitable media.‘-7~18~‘9*4349

Several hindrances have delayed experimental researchs of global dissipative structures. According to theoretical works on model systems, Turing instability4’*42 from a homogeneous state to steady cellular patterns requires the diffusion coefficients of at least two different species to be significantly different.‘V3*5s5s These conditions are met in systems that are governed by the competition between an activator and an inhibitor, when the inhibitor diffuses faster than the activator. This is a common situation in biological systems,3 where many processes are activated by enzymes immobilized in a matrix. This is a generally unrealistic situation for chemical reactions, such as the much-studied BZ reaction,14 involving small size molecules in aqueous solutions with comparable diffusion coefficients D- 10 - 5 cm* s- ‘. As pointed out by Pearson and Horsthemke,59 a way to overcome this difficulty consists in positioning the chemical system in the vicinity of a Takens-Bogdanov point60*6’ where Turing patterns can still organize with nearly equal diffusion coefficients. 58 Unfortunately the research of multicritical points of oscillating chemical reactions turns out to be a very hard experimental task. Diffusive instabilities have also been proposed to account for propagating patterns in homogeneously oscillating systems.4 Near the oscillatory onset, the Ginzburg-Landa@ amplitude equation (Hopf normal form for extended systems) can be reduced to the Kuramoto-Sivashinsky equation,63*64 which describes slow spatial and temporal variations of the phase of the oscillators. This equation has been the center of increasing interest during the past few years.65 Besides regular cellular and propagating solutions, this equation was shown to exhibit chaotic solutions,65-67 a form of weak turbulence usually called “phase turbulence.“4*63 But the Hopf bifurcations identified thus far in chemical systems are for the most part subcritical and lead to large amplitude relaxation oscilla- tions5 Relatively few supercritical Hopf bifurcations have been observed experimentally.68 Therefore, wave patterns observed in chemical systems are not the paradigm for Kura- moto propagative structures as generally believed. Traveling waves in excitable media are the best known example of such propagative patterns, *49 they actually belong to the class of localized structures described just below.

with feeding coming from the boundaries of the system.40 Originally, the basic idea was to localize all the significantly dynamical phenomena inside a narrow stationary reaction front in order to locate the region of strong chemical activity away from the boundaries where the perturbations associated to the feed may disturb the dynamics. Since instabilities can only develop inside the active region, these open reactors provide a very promising experimental support to the study of front pattern formation phenomena in reaction-diffusion systems. The results of preliminary experiments in the gel reactors 24*25*35-38 and the Couette flow reactor29-34239 have confirmed the capability of these apparatus to produce and control sustained chemical front patterns. Because it mimics a one-dimensional reaction-diffusion system with externally adjustable concentration gradient and controllable diffusion rate, the Couette flow reactor is very likely to play a privi- leged role in the experimental approach of spatiotemporal phenomena in nonequilibrium systems.

Localized front structures consist in spatial sequences of abrupt concentration jumps corresponding to rapid switches between steady or quasisteady states. They can originate in initially homogeneous media from a local perturbation giving rise to the well documented propagating waves in excitable media. 5-7,10*11,17-23,4349 Stationary front patterns have been theoretically predicted69-73 but such patterns require the diffusion coefficients of different species to be controlled selectively. Actually they have been mainly observed in the presence of spatial concentration non uniformities.‘p7”77 Localized heterogeneous reacting sites were shown to induce local chemical structures. 78V79 A concentration gradient externally imposed from the boundaries can be used to sustain reaction-diffusion fronts in homogeneous systems.8c-8” Some of the recently developed open reactors were designed

The Couette flow reactor29-34V39 consists of two CSTRs connected by a Couette-Taylor flow, with the inner cylinder rotating and the outer cylinder at rest. Chemicals injected in the CSTRs diffuse and react in the annular region between the two cylinders. At large Reynolds numbers, the Taylor vortices are turbulent enough for the fluid to be well mixed both in radial and azimuthal directions. Under these conditions, the mass transport along the cylinder’s axis was shown to be diffusive over length scales larger than the vortex scale.86 Consequently, the Couette flow reactor can be mod- eled as a one-dimensional array of homogeneous cells, coupled by a diffusion process with a unique diffusion coefficient D for all chemical species. This diffusion coefficient is a tuna- ble parameter which depends mainly on the rotation rate of the inner cylinder. The accessible D values range from 10 - * to 10 cm* s - ‘, i.e., several orders of magnitude larger than molecular diffusion coefficients. This rather wide range of diffusion control makes possible a continuous variation of the structure length scale for a fixed geometry, changing progressively from small to extended system behavior. Let us remark that the number of pairs of vortices is rather low ( -5O-1001, so that this system is neither a continuous system nor a low dimensional system. The role of the two CSTRs is to maintain nonequilibrium boundary conditions, e.g., by imposing a concentration gradient to the system (Bordeaux reactor). 3’-34 The Couette reactor can be fed as well by direct reactant flow (Texas reactor) .29V30

Two different reactions have presently been studied in the Couette flow reactor, namely, the variants of the Belou- sov-Zhabotinskii29-32V34 and chlorite-iodide3’-34V39 reactions. The BZ reaction has revealed a rich variety of steady, periodic, quasiperiodic, frequency-locked, period-doubled, and chaotic spatio-temporal pattems,29*30 well described in terms of the diffusive coupling of oscillating reactor cells, the frequency of which changes continuously along the Couette reactor as the result of the imposed spatial gradient of con- straints. This experimental observation has been successful- ly simulated with a schematic model of the BZ kineticss7 and the recorded bifurcation sequences of patterns resemble that obtained when coupling two nonlinear oscillators.

Much more in the spirit of our prospective theoretical

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study, 83-85 the chlorite-iodide reaction provides a remarkable illustration that spatiotemporal patterns can organize in a chemical system from the diffusive coupling (with equal diffusion coefficients for the different chemical species) of (nonoscillating) steady states reactor cells.31-34 A detailed report of the patterns, state diagrams and details on the chemistry, feed compositions and experimental procedures can be found in Ref 39. The chlorite-iodide reaction88*89 is a bistable reaction, i.e., the reacting medium can be in either a reduced or an oxidized state. Taking advantage of this bistability, stationary nonhomogeneous spatial patterns can be obtained when feeding unsymmetrically at the two ends of the Couette flow reactor, e.g., the left-end (respectively, right-end) CSTR in a reduced (respectively, oxidized) state. Under these conditions, a stationary single-front pattern appears in the Couette reactor: Two rather homogeneous regions corresponding to the reduced and oxidized states, respectively, are spatially separated by a sharp transition front. This somewhat trivial pattern is due to the existence of a switching process in the kinetics of the reaction, which induces a spatial transition between these two states. When varying the chemical input concentrations or the transport rate D as control parameters, bifurcation sequences of patterns have been observed.39 For example, when tuning the chlorite concentration in the feed flow of the right-end CSTR, the stationary single-front pattern bifurcates to a time-dependent state where the position of the front oscil- lates periodically over a finite spatial region in the Couette reactor. When further increasing the chlorite concentration, this oscillating front sweeps a larger and larger domain in the Couette reactor until a new (oscillating) band of oxidized state comes off the original oxidized region resulting in a periodic alternation of a single-front and a three-front pattern. Ultimately, this oscillating pattern stabilizes to produce a multi-front stationary pattern with three spatial switchings from the reduced to the oxidized state. Histori- cally, these multi-peaked spatial concentration profiles were the first experimental evidence for genuine sustained stationary chemical patterns in an isothermal and homogeneous continuous reaction-diffusion system3’ (without external field). Since then, the Bordeaux group has reported the observation of a symmetry-breaking instability leading to a stationary Turing structure in the linear gel reactor35-37 as com- mented in the conclusion. A very rich variety of spatio- temporal patterns has been observed in the Couette flow reactor under either nonsymmetric or symmetric feeding conditions.39 An example of patterns obtained with symmetric feeding are the bursting patterns, where a burst of oxidized state appears periodically in a reduced region imposed from the boundaries. Despite the observation of rather irregular displacements of front patterns, no definite experimental evidence for chaotic spatio-temporal behavior has been obtained thus far.

Our goal is to demonstrate that the experimental spatio- temporal patterns observed in the Couette flow reactor can be described by a reaction-diffusion process and to show that the observations are characteristic of a wide class of systems. More generally, we wish to identify the main ingredients required for pattern formation and to develop a theo-

retical analysis of the bifurcations that produced those dissipative front structures. In Sec. II we define our reaction- diffusion system model; this model is a two-variable Van der Pol-like system with equal diffusion coefficients. In Sec. III we report numerical simulations of this reaction-diffusion model under concentration gradient imposed by either Dir- ichlet or CSTR boundary conditions. A comparative study of the numerical and experimental spatio-temporal patterns is carried out for both asymmetric and symmetric feedings. Section IV is devoted to the theoretical study of the existence and stability of single and multi-front patterns. Our approach of these localized structures is essentially based on singular perturbation techniques.‘9*71.72 Exact analytical results are derived when considering a piecewise linear slow manifold. In Sec. V we perform a nonlinear analysis of bifurcating patterns using center manifold/normal form techniques. 60*6’ Special attention is paid to the Hopf bifurcation from steady to oscillating front patterns. We compare the theoretical predictions of the normal form calculations with the results of direct simulations. We conclude in Sec. VI with a discussion of the possible generalization of this theoretical study to sustained front patterns recently observed in annular and linear gel reactors.

II. A REACTION-DIFFUSION SYSTEM MODEL WITH EQUAL DIFFUSION COEFFICIENTS

In most theoretical studies of chemical systems, pattern forming phenomena as well as the concept of chemical turbulence have been addressed in terms of the linear coupling of spatially distributed nonlinear oscillators.4 According to this prerequisite, the analysis of the partial differential equations which model realistic reaction-diffusion systems has been commonly simplified to the investigation of coupled nonlinear oscillators,4.9s94 coupled nonlinear maps95,96 and cellular automata.97 The aim of the present study is to emphasize that nontrivial regular and irregular spatio-temporal regimes can be attained when coupling (nonoscillatory) steady state reactors with equal diffusion coefficients for the different chemical species, provided a spatial concentration gradient, e.g., a nonhomogeneous feed, is imposed to the system.83-85

As pointed out in the Introduction, the spatially extended open Couette flow reactor29-34,39 provides a practical implementation of an effectively one-dimensional reaction- diffusion system with an external concentration gradient imposed from the boundaries. With the specific motivation to provide theoretical and numerical support for the recent experimental observations of sustained dissipative structures in the Couette flow reactor, we will consider the standard reaction-diffusion equation,

d,C = R(C) + DAC, (1) where C is a concentration vector, D the diffusion matrix, and R( C!) models the reaction process. A faithful modeling of the experimental situation would consist in considering a reaction-diffusion system which meets the experimental conditions and the specific requirements of the chemical kinetics laws of the chlorite-iodide reaction. Here we will adopt a strategy which is much more in the spirit of the

J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation 325

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“normal form approach”:60*61 We will simply consider a formal reaction-diffusion model that will be shown to retain the minimal ingredients necessary to reproduce most of the phenomena associated with the observed front patterns.

A. Reaction process For most of the oscillating reactions, the knowledge of

the reaction mechanism and rate constants is generally very sketchy. Since the details of a particular kinetic model are not relevant close to bifurcation conditions, we will consider the simplest ordinary differential equation model which accounts for the characteristic features of the chlorite-iodide reaction,88.89 namely, bistability, excitability, and relaxation oscillations. Our model of the reaction term is a two-variable Van der Pol-like system98’99 [C = ( U,U) 1,

du -=g dt

- ‘(v -f(u) 1,

dv -= ---II+, (2) dt

where E is a small positive parameter and a a free parameter the role of which will be explained shortly. These equations ensure the existence of a pleated slow manifold v =f( u), on which all trajectories are attracted in a time - 0( 6). The “S” shape of this manifold accounts for the excitable character of the dynamics. The only steady state of the reaction term [u, = a, v, =f( a) ] is necessarily located on the slow manifold; an elementary linear analysis shows that this steady state is stable for a < aL (lower branch), or a > aU (upper branch) while it is unstable for CYLE [ aLL ,a,] as sketched in Fig. 1. The critical values au and aL correspond to Hopf bifurcations leading to oscillatory behavior. According to the specific shapef( U) of the slow manifold, this bifurcation can be either supercritical or subcritical.99 When adding a flux term to this Van der Pol-like equation, bistability can also be recovered. Despite the fact that model (2) does not have all the properties required in a chemical scheme, u and u

FIG. 1. Sketch of the slow manifold u =f( U) = u2 - u’ + us. The unique steadystateofEq.(2):u=cx,u=f(a),isafocus(F)fora; <Ada,+ or a,- < cr < aO+, and a node (N) elsewhere. A solide line indicates a stable steady state (a<~, or cz> a,); a dashed line an unstable steady state (a,<a<a,).

play the role of concentration variables and we will refer to the upper and lower branches of the slow manifold as the analogues of the reduced and oxidized state branches of the chlorite-iodide reaction, respectively. Let us remark that a change of variables of the form ii = u + udr t, = v + vd can be used to ensure these concentration variables to be positive, at the expense of a slight modification of the exact form OfEq. (2).

B. Diffusion process

au -= at

When taking into account the diffusive transport process, the reaction-diffusion model reads

!?!!=Dd-u++ - , (3)


at ax” where XE[ 0, 1 ] is the single space variable; the spatial length of the Couette reactor is resealed to unity for convenience. The cross diffusion terms between the two species u and v are neglected and the diffusion coefficients D, = D, = D are set equal in order to mimic the turbulent mass transport that drives pattern formation in the Couette flow reactor.

C. Boundary conditions 1. Dirichet boundary conditions

In most experimental runs, the volume and feeding flows of the two CSTRs at both ends of the Couette reactor were large enough for their internal state not to be significantly influenced by the dynamics inside the Couette reactor.39 This corresponds mathematically to imposing Dirich- let boundary conditions to our model reaction-diffusion system ( 3). In most of the simulations reported in this paper (Sets. III A 1 and III A 3), the Couette flow reactor is unsymmetrically fed, with the left-end CSTR (x = 0) in a (reduced) upper-branch state while the right-end CSTR (x = 1) is maintained in an (oxidized) lower-branch state,

v(x=O) =f(u,) with u. =u(x=O)>a,,

v(x = 1) =f(u,) with u, = u(x = 1) <a,. (4) Symmetric feeding (u. = u, ) will also be considered in Sec. III A 2. For the sake of simplicity, the value of a in the system (3) is set independent ofx, a > au, so that when switching off the diffusion process, all the intermediate cell points evolve asymptotically to the same stable reduced steady state on the upper branch of the slow manifold. A more realistic model should probably take into account a spatial dependence of a, so that a(x = 0) = u. and a(x = 1) = ul. In related models29*87 of the Couette flow reactor experiments conducted with the BZ system by the Texas group, the role of a is played by a third variable which corresponds to a set of reactants whose concentrations can be considered as being time independent during the experiment, and which act as an effective nonequilibrium constraint at each point of the Couette reactor. It is assumed that their spatial profile is a linear concentration gradient, which explains the linear spatial dependence of a in these models. The situation is not so clear for the chlorite-iodide reaction since such a set of reac-



tants does not seem to exist.lm For the sake of simplicity we will assume in the present study that the nonequilibrium constraint a is constant (zero-order approximation) all the way along the reactor. At this point, let us mention that the spatio-temporal patterns reported in the numerical study in Sec. III are robust against (smooth) spatial perturbations of the a = cst working hypothesis.

2. CSTR boundary conditions

For some feeding conditions, the dynamics inside the Couette reactor has been observed to introduce some feed- back in the ending CSTRs. 34*39 In such conditions the two CSTRs cannot be considered anymore as maintained in a steady state. In order to account for this phenomenon we will also consider the following set of boundary conditions:

d,C=R(C) fk,(C, -C) +Td,C, for x=0,

d,C=R(C) +k,(C, -C) -$d,C, for x= 1,

(5) where R(C) is the reaction term; the inlet concentrations C, = (u,,v, ) and C, = (u, ,u, ) are chosen such that without coupling with the Couette reactor, both CSTRs are set in a stable steady state on either the upper or lower branch of the slow manifold. I is the relative size of the CSTRs with respect to the overall size of the one-dimensional Couette reactor. k, mimics the input flow rate of the two CSTRs. Equation (5) ensures flux conservation at x = 0 and x = 1, respectively. Let us remark that the Dirichlet boundary conditions are recovered in the limit k, + CO.

Ill. FRONT PATTERNS IN ONE-DIMENSIONAL REACTION-DIFFUSION SYSTEMS UNDER CONCENTRATION GRADIENT

The spatio-temporal patterns observed when perform- ing numerical simulations of the reaction-diffusion model (3) are in many respects very similar to those observed experimentally in the Couette flow reactor with the chlorite- iodide reaction.33.83-85 This reaction and its variants provide a remarkable illustration that stationary and oscillating front patterns can organize in a chemical system from the diffusive coupling of steady state reactor cells. The aim of this section is to detail some specific transitions leading to spat&temporal patterns which seem to be generic in both the experiments”’ and the simulations.8”-85 Intuitive argu- ments will be given explaining the pattern formation phenomena through bifurcation mechanisms. A theoretical understanding of these bifurcations based on singular perturbation techniques” and center manifold/normal form calculationsay6’ will be reported in Sets. IV and V.

The partial-differential equations (3) are solved numerically through finite difference approximation for the spatial derivatives and the method of line for time advance- ment. The model medium is represented by a discretized line with a resolution from 50 up to 200 points. The resulting set of ordinary differential equations is integrated with a stiff ODE solver.‘0’ Care is taken to vary the spat&temporal

resolution in order to check the reliability of the reported phenomena.

A. Dirichlet boundary conditions 1. Asymmetric feeding

Let us consider first the situation where the values of the two concentration variables of our reaction-diffusion model, u and U, are kept fixed at the two boundaries x = 0 (reduced upper-branch state) and x = 1 (oxidized lower- branch state) according to Eq. (4). As long as the fronts are located far enough from the two CSTRs at both ends of the Couette reactor, this seems to be a rather good approximation of the experimental situation with asymmetric feeding. 33Y34,39 Let us suppose that E is kept fixed to a (small) positive value.

For D$E- ‘, the diffusion term is predominant, and all the trajectories of the system converge asymptotically to a unique stable steady state. This solution is merely a linear spatial concentration profile linking the two ending concentrations. When D is decreased and becomes - O( 1 ), the reaction term eventually becomes of the order of the diffusion term: the former “diffusion-like” solution is still stable but it develops a sharp front which corresponds to a spatial switching between the two attracting branches of the slow manifold. In other words, in addition to the characteristic size of the system (which has been chosen to be equal to l), a smaller length scale -n (the width of the front) comes into play. When D is further decreased, the transition front becomes sharper and sharper until this single-front solution loses its stability; in parallel, an increasing number of multi- front solutions actually appear, either stable or unstable. This evolution turns out to be generic independently of the specific S shape of the slow manifold. The limit D -+ 0 can be identified to the limit of an extended system: in fact, the effective number of degrees of freedom increases when the diffusion is decreased.‘02s’03 In this limit, we may expect to observe a very rich variety of dynamical behavior. But this limit is far from (i) the current experimental conditions: it would require low rotation rates of the inner cylinder of the Couette reactor for which the transport process could no longer be considered as diffusive; and (ii) the resolution of current numerical simulations. In the present numerical study, we mainly focus on the early bifurcations of front patterns observed on the way to this “extended system” limit and discuss the existence of up to three-front pattern solutions.*’

The numerical patterns shown in Figs. 2 and 3 have been obtained with the following form of the slow manifold:

f(u) = u* - u3 + u5. (6) It is easy to check that a” = 0, a, = - 1, a;

a;d - 1.016 08, a;’ = - 0.982 666, a; = - 0.088 424 5, ao+ = 0.12 1 65 1. This choice makes the Hopf bifurca-

tion in the reaction term subcritical,99 as is the case in most experimental situations.’ The following model parameters are kept unchanged: E = 0.01 and the feed concentration of the right-end CSTR U, = - 1 S. The feed concentration u. of the left-end CSTR, the diffusion coefficient D and the



FIG. 2. A perspective plot of the spatio- temporal variation of the variable u(x.r) as computed with the reaction-diffusion system (3), with Dirichlet boundary conditions (4). the slow manifold (6) and the model parameters: U, = - 1.5,~ = 0.01. (a) Stationary single-front pattern (u,, = 1.1, D=O.l, a=O.Ol); (b) periodically oscillating single-front pattern (u,, = 1.1, D = 0.045, a = 0.01); (c) periodic alternation of a single-front and a three-front pattern via colliding fronts (u. = 1.1, D=O.Ol, a=O.Ol); (d) stationary three-front pattern (uO = 0.5, D = 0.08, a = 0.2); (e) periodically oscillating three-front pattern (u, = 0.5, D = 0.06, a = 0.2); (f) periodic alternation ofa single-front and a three-front pat- tem(u0=0.5,D=0.02,a=0.2).

parameter a are taken as control parameters. In Fig. 2, we characteristic length scale @% of the front decreases until use a three-dimensional space-time representation which il- the steady front solution eventually becomes unstable and lustrates the time evolution of the spatial concentration profile of the u species. The same patterns are illustrated in Fig.

starts oscillating periodically in time, as shown in Figs 2 (b)

3 under a concentration coding similar to the one used to and 3(b). To gain some understanding of these osciZZating

visualize the changes of color in the experimental study. Fig- single-frontpatterns, one may consider a spatially discretized version of our continuous reaction-diffusion system (3). A

ure 3 has to be compared with Fig. 2 in Ref. 3 1 and Fig. 9 in Ref. 39.

straightforward linear calculation shows that, among the one-dimensional array of coupled elementary reactor cells,

For values of u. )a and values of D- 0( 1 ), one ob- serves only steady single-frontpatterns; left to the front, the

the ones that are located at the front zone are driven by

solution is confined to the upper branch of the slow mani- diffusion to an attracting limit cycle because of the presence

fold; right to the front the solution belongs to the lower of a steep concentration gradient. (The physical mechanism

branch [Figs. 2(a) and 3(a) I. When D is decreased, the underlying this oscillatory instability has been identified in the direct diffusive coupling of two unsymmetrically fed

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FIG. 3. The spat&temporal variation of the variable u(x,t) is coded in order to mimic the spatial color profiles observed in the Couette flow reactor (Refs. 33, 34, and 39); 32 shades are used from the left-end upper branch (reduced) state (black) to the right- end lower branch (oxidized) state (white). The numerical spat&temporal patterns in (a)-(f) are the same as in Fig. 2.

J. Elezgaray and A. Arneodo: Reaction-diffusion pattern formation

a

~ 1

space -

-- d

space

329

CSTRs in Refs. 104 and 105.) When the diffusive coupling is weak enough, these reactor cells are no longer stabilized by the stable steady state cells located close to the boundaries and the whole system starts oscillating. It is then clear that the amplitude of oscillation is much larger for the cell points located at the front zone: The instability originates in the active region at the front zone and is propagated by the diffusive coupling to the other cell points of the reactor. One may wonder whether this instability persists in the continuous limit. The soundness of an extrapolation to the continuous limit is supported by the analytical study reported in Sec. IV.

When further lowering D, the spatial amplitude of the oscillation of the front pattern increases until a qualitative change occurs in the spatio-temporal evolution of the system as indicated by the reentrance phenomenon shown in the space-time representation in Figs. 2(c) and 3(c). A

three-front profile alternates periodically with a single-front profile. The three-front profile proceeds from the periodic appearance of two traveling fronts. The single-front profile is recovered from the periodic coalescence of one of these two traveling fronts with the originally oscillating front. A similar pattern where the period is about twice the period of the previous one is shown in Figs. 2 ( f) and 3 ( f) . These colliding front patterns have also their experimental counter part.3’- 34*39 They give hints that steady multi-front patterns are very likely to exist.

For u. -a, we have succeeded in freezing asteady three- front pattern involving three spatial switches between the two branches of the slow manifold [Figs. 2 (d) and 3 (d) 1. Again a decrease of the diffusion coefficient induces a transition to a periodically oscillating three-front pattern [Figs. 2(e) and 3 (e) 1. In our numerical simulations, this stafion-

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ary multipeaked structure coexists with the oscillating single-front pattern, in contrast with the experimental situa- tion39 where these two patterns apparently take place in two different regions of the constraint space. However, let us emphasize the striking similarities between the patterns shown in Figs. 2 and 3, and the corresponding patterns identified thus far in the experiments (Fig. 2 in Ref. 3 1 and Fig. 9 in Ref. 39). As discussed in Sec. IV, we have strong indica- tions that the three-front solution originates from a saddle- node bifurcation, i.e., at some distance in phase space from the “natural” single-front solution. As pointed out in Refs. 33 and 39, despite the fact that no hysteresis has been observed thus far, one cannot exclude the presence of multiple stable states in the experiments.

2. Symmetric feeding

In order to mimic sustained patterns observed in the Couette flow reactor with symmetric feeding,39 let us now consider symmetric Dirichlet boundary conditions: u. and U, are for example located on the (oxidized) lower branch of the slow manifold, while a still belongs to the (reduced) upper branch. Eventhough there is no asymmetry in the feeding, there still exists a concentration gradient in the system close to the two boundaries.

When D+ CO, the diffusive transport process dominates and all the trajectories converge asymptotically to a homogeneous solution imposed from the boundary conditions: All the reactor cells are uniformly constrained to the lower branch u(x) = u0 = u, . When D is decreased, the homogeneous solution becomes unstable, and a new stationary solution pops up [Figs. 4(a) and 4(e)]; this new solution displays a spatial profile which involves two fronts separating a central region of reduced states from the two regions of oxidized states close to the two boundaries. As previously observed for the single-front patterns, the steady two-frontpat- tern solution undergoes a Hopf bifurcation when the diffusion D is further decreased, leading to a periodically breathingpattern as illustrated in Figs. 4(b) and 4(f). When lowering D, the amplitude of oscillation of the two fronts increases as shown in Figs. 4(c) and 4( g ), and the breathing pattern transforms into a burstingpattern as observed in the

I

c -U

experiments.39 This bursting phenomenon corresponds to the periodic appearance and coalescence of the two oscillating fronts, i.e., a burst of (reduced) upper-branch state emerges periodically in the Couette reactor from a rather uniform (oxidized) lower-branch state induced by the boundary conditions. Special attention has to be paid, however, to identify unambiguously this bursting phenomenon since the concentration coding used in Fig. 3, as well as in the experiments,“’ can make the distinction between breathing and bursting patterns quite confusing. At this point let us mention that subsequent simulations have revealed more complicated periodic as well as intermittent (chaotic) burst- ings. Moreover, these phenomena seem to be robust with respect to the choice of the boundary conditions; in particular they can be observed in the more surprising situation where the two CSTRs and the Couette reactor steady state are located on the same branch of the slow manifold. A detailed study of bursting patterns in the reaction-diffusion model ( 3 ) will be reported elsewhere. lo6

Actually there are two different types of oscillating modes of the two-front pattern (see Sec. IV). The breathing pattern shown in Figs. 4(b) and 4(f) corresponds to an in- phase (or symmetric) oscillating mode: The reactor cells located at the two front zones oscillate in phase. The wig- glingpattern shown in Figs. 4(d) and 4(h) corresponds to a out-of-phase (or antisymmetric) oscillating mode: The reactor cells at the two-front zone oscillate out of phase. Because the instability of the in-phase mode occurs generally prior to the instability of the out-of-phase mode, the wiggling patterns are usually observed as transient phenomena to either the steady two-front pattern (before the oscillatory instability threshold) or more or less complicated breathing patterns (beyond the oscillatory instability threshold).

3. Diffusion-induced spatio-temporal chaos The whole zoology of patterns reported in Sets. III A 1

and III A 2 are “robust,” in the sense that they can generically be observed in any reaction-diffusion system with a S- shaped slow manifold. In particular, we have reproduced the patterns reported in Figs. 2-4 with the following slow manifolds:

O<u<l-s

f(u) = -fC -u) = -is-‘(1 -U)4f~S-‘(l -U)2- 1 1

1 -&;u(l +a* (7) u-2 u>l +s

In the limit 6-0, this one-parameter family of slow manifolds reduces to a piecewise linear function f which will be used in Sec. IV to derive analytical results. With the slow manifolds (7)) we have found conditions where the oscillating single-front pattern undergoes secondary instabilities leading to more complicated spatio-temporal behavior.*’ In Fig. 5 (a), we show a chaotically oscillating front structure computed with 6 = 10e2 in Eq. (7). The phase portrait re- constructed from the temporal evolution of the variables u and u recorded at an intermediate spatial cell point is shown

I in Fig. 5 (b) . The corresponding Poincare map and 1D map are illustrated in Figs. 5 (c) and 5 (d), respectively. The fact that the Poincare map is not a scattering of points but that all the points lie to a good approximation along a smooth curve indicates that the trajectories lie approximately on a (multi- folded) two-dimensional sheet in the phase space. The well- defined single humped shape of this 1 D map is a clear signa- ture of the low dimensional chaotic nature of these oscillations. It is somewhat puzzling that the phase portraits obtained in our simulations [ Fig. 5 (b) ] are strikingly simi-

J. Chem. Phys., Vol. 95, No. I,1 July 1991 Downloaded 09 Dec 2003 to 128.83.156.151. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp


mace

FIG. 4. A perspective plot of the spatio-temporal variation of the variable u(x,t) as computed with the reaction-diffusion system (3) with Dirichlet boundary conditions, the slow manifold (6), and the model parameters: a,,=~, = -11.5,a=0.2,~=0.01.(a)Sta- tionary two-front pattern (D = 0.05); (b) two in-phase periodically oscillating fronts (D = 0.03); Cc) bursting pattern (D = 0.055); (d) two out of phase periodically oscillating fronts (D = 0.02). Figures (e)- (h) represent the same spatio-temporal patterns as in (a)-(d), respectively, using 32 shades from the lower branch (oxidized ) state (white) to the upper branch (reduced) state (black).

lar to the strange attractors observed in the BZ reaction when conducted in a CSTR. 133’5S16 Moreover, as in the homogeneous BZ reaction, period-doubling bifurcations are observed as precursors to this macroscopic chaos. Let us mention that these chaotic spat&temporal patterns have been obtained when using a spatial discretization ( - 100

intermediate reactor cells) compatible with the number of characteristic diffusion lengths in the Couette flow reactor. We have checked that these chaotic patterns are preserved when increasing spatial resolution. The generality of the observed transition to spatio-temporal chaos extends to a rather large range of values of 6, As pointed out in Ref. 39, be-



-03 GQ-yy-J ”

FIG. 5. Diffusion-induced chaos obtained when integrating the reaction- diffusion system (3) with Dirichlet boundary conditions (4) and the slow manifold (7) (6 = IO-‘). The model parameters are u0 = 2.5, U, = - 3.1, a = 1.1, E = 0.01, D = 0.05. (a) Spatio-temporal variation of the variable u(x,t) coded as in Fig. 3; (b) phase portrait; (c) Poincart map; (d) 1Dmap.

cause of the difficulties in controlling experimental conditions during sufficiently long period of time, there has been thus far no clear demonstration of the chaotic character of the nonperiodic time series recorded in the Couette flow reactor.

6. CSTR boundary conditions In some experiments performed with some variants of

the chlorite-iodide reaction, the oscillating front patterns have been observed to invade one of the ending CSTRS.‘~*~~ Henceforth the two CSTRs cannot be considered in a steady state during the experimental run as before. In order to account for the interplay of the dynamics inside the Couette reactor and in the CSTRs, we have performed subsequent numerical simulationsE3 of our reaction-diffusion model (3) with the “CSTR boundary conditions” defined in Eq. (5). We give here a short description of the patterns observed when considering the slow manifold (6). The following parameters are kept fixed: e = 10 - 2, a = 0.5, u,, = 2, zll = - 4, ui =f( ui ), i = 0,l. D = k, is our control parameter.

As compared to the feeding concentrations used thus far, the left (x = 0) and right (X = 1) end CSTRs are fed on the (reduced) upper and (oxidized) lower branches of the slow manifold, respectively, but the values of u. and 11, considered here induce a very strong gradient of concentration through the reactor. Under Dirichlet boundary conditions, the single-front solution would become unstable at very low values ( - 10w3) of the diffusion coefficient. The situation is quite different with the “CSTR boundary conditions.” At low values of D, almost all the reactor cell points are in a reduced state u = a, except a small region located near x = 1

where the reactor cells are driven in an oxidized state by the right-end CSTR. As shown in Fig. 6, when D is increased, the diffusion process carries further the influence of the right-end CSTR and the steady front moves to the left: In turn more and more reactors switch from the upper to the lower branch of the slow manifold. The displacement of the steady front is found to depend linearly on D. The boundary conditions become much more “Dirichlet like” when the front is located in the central region ofthe Couette reactor; in this stationary situation, the gradient term JC/& does not play any important role in Rq. (5), so that, for reasonable values of k,, one can consider that the two ending CSTRs evolve according to the equation

k=R(C) +k,(Ci -C), i=O,l.

When the front approaches the left-end CSTR (x = 0), it becomes unstable through a Hopf bifurcation, and starts oscillating periodically. At this point, the intuitive picture we gave for the case of Dirichlet boundary conditions breaks down: the steady single-front solution is then distabilized when increasing (instead of decreasing) the diffusion coefficient. This is a direct consequence of the specific type of boundary conditions we are imposing. We suspect that, in this case, the oscillatory instability is not only governed by the interplay of the reaction and diffusion processes at the front zone but is also driven by the dynamics of the left-end CSTR. This is clearly illustrated in Fig. 7; when further increasing D the spatial amplitude of oscillation increases until the front periodically disappears in the left-end CSTR. The system undergoes secondary instabilities, and the behavior of the front inside the Couette reactor become more or less regular as illustrated in Figs. 7 and 8. When looking at the periodic oscillations recorded at different spatial points, the corresponding temporal patterns shown in Figs. 7 (b), 7 (c) and 8 (b), 8 (c) are strikingly similar to the time series obtained in the pioneering experiments on the homogeneous BZ reaction.5*‘5V’6.‘07 Recently, the alternating periodic- chaotic sequences observed in these experiments have been understood in terms of the frequency-locked and chaotic

2 ., .., . .1 *,, I, ., .., ., . * . i

u -

-2 - . . . . . . . . . .

t-

D - 0.01 ---. D - 0.05

D = 0.00

0 1 space

FIG. 6. Displacement of a stationary front when increasing the diffusion coefficient D in the reaction-diffusion model ( 3 ) . The boundary conditions are of CSTR type [ Eq. ( 5 ) 1, the slow manifold is given by Fq ( 6) and the model parameters are e = IO-‘, a = 0.5, u, = 2, u, = - 4.



10 20 JO 40 fIllI*.

-. ~- -. -._ , (c) J = s, 10 :

II t ( ! ; ,i

LO 20 30 40

f,l,ll’

FIG. 7. (a) A periodically invading front obtained with the reaction-diffusion model (3) with the CSTR boundary conditions (5); the parameter valuesare: l=2.13 IO-‘, c= 10 2, D = 0.1, a = 0.5, U0 = 2, 11, = - 4 and the slow manifold is given by Eq. (6); (b) and (c) illustrate the periodic oscillations recorded in two distinct intermediate reactor cells visited by the front.

states issued from the breaking of a T’ torus.68*‘08 But this chaos is a small-scale chaos’5P’6,‘08 which turns out to be extremely difficult to identify as compared to the large-scale chaos encountered in nearly homoclinic conditions.‘09”10 Therefore, in contrast to the macroscopic chaos observed with Dirichlet boundary conditions (Fig. 5)) very high accu- racy numerical simulations are required to distinguish between periodically and chaotically oscillating fronts in our reaction-diffusion model (3) with CSTR boundary conditions. Figure 8(a) illustrates an intermittent front which shows up and disappears apparently in an erratic manner during our finite-time numerical experiment. In a local cell, the possible chaotic nature of the temporal pattern is con- tained in the small amplitude oscillations: Their number may differ from one temporal motif to the next while their amplitude may also fluctuate. We refer the reader to Refs. 68, 108, and 111 for a detailed experimental, numerical, and theoretical study of similar small-scale chaos in the BZ reaction conducted in a CSTR.

IV. EXISTENCE AND STABILITY OF FRONT PATTERNS In this section, we study the existence and stability ofthe

stationary solutions of our reaction-diffusion model (3). Our goal is not to give a full description of all the possible cases, but rather to emphasize some general properties which are common to the class of reaction-diffusion systems given by a set of equations similar to Eq. (3 ) with Dirichlet boundary conditions.“2 In Sec. IV A, we summarize the

,k,(b) ’

FIG. 8. (a) An intermittent front obtained with the reaction-diffusion model (3) with the CSTR boundary conditions (5); the parameter values are E = 10 ‘, D = 0.099, a = 0.5, u(, = 2, U, = - 4 and the slow manifold is given by Eq. (6); (b) and (c) illustrate the oscillations recorded in two distinct intermediate reactor cells visited by the front.

perturbative theory for the existence of stationary solutions proposed by Fife in Ref. 113. We then apply this approach to the particular case of a piecewise linear slow manifold 69,“4*115 and we give some estimates of the number of statllonary solutions. In Sec. IV B, we study the linear stability of these solutions. Analytical calculations of some bifurcation diagrams are detailed in Sec. IV C for a piecewise linear slow manifold.

A. Existence of stationary front patterns

The problem of the existence of stationary solutions for the system (3) consists in solving the stationary problem:

o=d++u-f(u), dx*

O=Dfi-u+a. dx2

(8)

The singular perturbation analysis developed below is a heu- ristic version of the rigorous results derived by Fife.‘13 Ac- cording to this work, the original stationary problem [ Eq. ( 8) ] can in principle be reduced to a more tractable equation. For the specific case of a piecewise linear slow manifold, we will solve the reduced system analytically.“’

1. Perturbafive theory

A key role in the treatment of the stationary problem (8) is played by the small parameter 7 = m. In Eq. (8a),



the second derivative term occurs with a 7’ factor. This implies that two length scales respectively of 0( 1) (the characteristic size of the system) and 0( 77)) characterize the spatial shape of the solutions. A common ansatz when dealing with this kind of equations is a solution of the form

u(x) = U(x) + p(x/~) + h.o.t., u(x) = V(x) + h.o.t. (9)

Namely, one decomposes u(x) in a smooth contribution U(x) , plus a function e, (x/v ) which depends on the resealed variable x/v. The function p(x/v) is only non-negligible in the inner region corresponding to the set of reactor cell points where the solution displays a small-scale behavior characterized by u, - (ED) - ’ (see Fig. 9). The outer region is the complementary set of reactor cell points which are located away from the front zone. In contrast, the function U(X) is expected to display only smooth variations since the coefficient of the second derivative term u,, is - 0( 1) . One can thus deal with each contribution separately.“2

As far as the couple of smooth functions ( U(x), V(x) ) is concerned, one can drop the term EDU,,; one thus gets at leading order

0 = V(x) -fC U(x) 1,

O=D$V-U+a. (10)

Because of the S shape of the graph ( U, f( U) ), Eq. ( 10a) has two solutions:

U=h,(V), for VE] - c~f(u- I[,

U=h,(V), for V~l.fl~+ ),+ CO[, (11)

where u- <u+ are such that f’(u- ) =J’(u+ ) =O. Then Eq. ( lob) becomes

O=D$ V--G(v) +a, (12)

where

for Vcl- co&u- )[, for VEl.f(u+ 1, + CXJ[ ’

(13)

In the following, we will assume that Eq. ( 12) has a solution V(x) satisfying the boundary conditions V(x = 0) = V, and V(x = 1) = u, . Note that this solution V(x) is at least continuously differentiable, but obviously not twice differentiable since, in general, G( v) has some discontinuities. For the sake of simplicity, we will assume that G( v) is only discontinuous at x = x, (x, should be identified with the location ofa front). This implies that U(x) =f( V(x) ) is also discontinuous at x = x, .

The inner contribution CJJ(X/T) will be defined in such a way that U(x) + 9(x/q) will be continuous and differentiable up to the leading order. It is worth introducing the trans- lated function

p, (!3 -pWrl), ic = (x - x, l/q. q~, satisfies the equations

(14)

d2v, -+ V, --f(p, +h,(V,)) =O, for {-CO,

dC2 d2q, -+ V, -fCp, +h,(V,)) =0, for 6~0,

diC2 (15)

U

2

0

-2

u 2

0

-2

v ’

0.5

v-1 = u

-0.5

-1

i -

FIG. 9. Comparison between the exact stationary single- front solution (u(x), u(x)) (full line) and the external approximation of the same solution (U(x), V(x)) (dashed line) of the reaction-diffusion model (3) with Dirichlet boundary conditions; the piecewise linear slow manifold is given by Eq. (25). (a) The functions u(x) and U(x); note that u and U differ significantly in the inner region only; (b) phase portrait in the (u, u) plane: by definition, the points (U(x), V(x)) are located on the slow manifold (dashed line); (c) comparison between the exact solution (full line) and the function U(x) + p, (&) (dashed line); (d) comparison between the exact solution u(x) (full line) and the external function V(x) (dashed line).



2. Phase plane analysis where V, = V(x =x, ) [in the limit e-+0, V(x) becomes a constant when expressed in terms of the resealed variable 5‘: V(x) = V(l@ +x1 ) + V(x, )]. In order to completely determine the function p, , we impose the following boundary conditions:

slim ,I,-m 9, ([) = 0: this condition ensures that q, is negligible outside the inner region.

l h,(V,) +q-+(l=O+) =h,(V,) +a,(S=O-) =~(h,(V,)+h,(V,)) (u,>u+, u, <u-; in the case

u,, < u _ , U, > u + , h, and h, should be exchanged) : this condition guarantees the continuity of U(x) + p, (6) at x = XI. In order to determine V, , we require the differentiability of p1 at 6 = 0,

The standard method (“phase plane analysis”‘16 ) commonly used to solve the boundary problem ( 12) consists in introducing the new variable

w=dv -dx ’

(19)

so that Eq. ( 12) can be rewritten as

dv yp = w,

D*=G(v) -a. dx

(20)

Introducing the function

H(v,w) = + Dw2 - r [G(s) - a]ds, (21)

one can easily see that, if (V(X), w(x)) is a solution of Eq. (20),thenH(v(x),w(x)) =H(v(x=O), w(x=O)) isin- dependent of x; each solution is completely characterized by the quantity H( v(x = O),w(x=O)).Nowifwedenote

F(v) = I

” [G(s) - a]ds, (22)

then, the function

1 ((

@I -g=o-) - y dl )2 ($cE=o+,)3

P _ oc V(h, @I = (V,)+9’,,)-V,>---d~ dl I-+*

+ J o U-W’,)++V,j*dg 45

= -J(V,,,

where we have introduced the function

(16)

J(v) = s

h,(V) (V-f(s))ds. (17)

h,( W It follows that the value of V, is given by the Maxwell equal area condition

J( V, ) = 0. (18) As originally proved by Fife,‘13 3. (18) admits a unique solution.

The above considerations can be easily generalized to the case of N-front solutions. ‘I3 Then, the function U(x) is discontinuous at N points x, , x2 ,...,x,.,, and N inner correc- tions of the form q(si (x - xi )/T) , si = f 1 are necessary to solve the stationary problem. A rigorous proof of the fact that [U(x) -j- Zr= ,q(si(x - x,)/q),V(x)] is close to the exact solution of Eq. (8) is provided in Ref. 113. Smaller e, better is the approximation. Besides regularity assumptions, the only condition required for the theorem by Fife to apply is that the function V(x) should take the value V, onIy at the transition points xi, i = 1 ,...,N. The results reported below show that this condition is even too restrictive.“*

To find an approximate stationary multi-front solution of Eq. (8) we will proceed the following way:

l First we will look for a solution of Eq. ( 12) satisfying V(x,) = V,, i = l,..., N.

l Then we will calculate the inner correction function p, (5) solving Eq. (15). In fact, the problem of the existence of stationary solutions reduces to determining the solutions of Eq. ( 12)) since it can be shown that the function 9 always exists.“3 Let us note that the case N = 0 is also meaningful: it simply means that V(x) never takes on the value V, , i.e., the solution displays no front. Finally, we remark that, owing to the conditions v,, = f( u. > and v, = f( u , ) , the stationary solutions are regular at the boundaries x = 0 and x = 1.

335

r.(uo,u, ,D,H) = D “’

0 -F “0 [H+F(v)]-“‘dv (23)

is such that (we recall that XE [ 0, 1 ] )

duo,~,,D,H) = 1, (24) provided His associated with a continuous trajectory in the (u, w) plane which is a solution of Eq. (20) satisfying v(0) = uo, v( 1) = u, . In the next subsection, we will study the function r(uo,u, ,D,H) for the particular case of a piecewise linear slow manifold. However, it is clear that most of the properties given below are very likely to apply for a large class of reaction-diffusion systems with an S-shaped slow manifold.

3. Piece wise linear s/o w manifold

a. Phaseplane analysis. Let us consider the slow manifold (u, f(u)) givenbyEq. (7) withS=O:

f(u) = -u ]u]<l i

u+2 u( - 1 ( zlower branch) * (25)

u-2 01 ( zupper branch) A straightforward calculation yields J( v) = - 4V, so that V, = 0 from Eq. ( 18). Depending on the branch of the slow manifold visited by the solution, we have to consider two phase-plane diagrams. On the lower branch, the trajectories in the (v, w) plane are given by [see Fig. 10(a)]

w*=p’+D-‘(v*-2(2+a)v), u<l, (26) where p2 = 20 - ‘H is used instead of H for later convenience. On the upper branch, the trajectories are given by [see Fig. 10(b)]

w2=p2+D-‘(v’+2(2-a)~), ~2-1. (27) Note that, in both cases, the trajectories are invariant under the transformations: v-+ u, w-+ - w; this symmetry reflects


336 J. Elezgaray and A. Ameodo: Reaction-diffusion pattern formation

the invariance of the original system (3) under the reflection operation x+ 1 -x, provided the boundary conditions at x = 0 and x = 1 are exchanged. b. Single-front solutions. According to the Dirichlet boundary conditions (4), we consider that the left-end CSTR (no ,v,, ) is located on the upper branch of the slow manifold, while the right-end CSTR (u, ,vi ) stands on the lower branch. Namely, we are looking for trajectories in the (0, W) plane such that (v(x), w(x) ) satisfies Eq. (27) [respectively, Eq. (26)] forxe[O,x,] (respectively,xe[x,,l]). Since His independent of x, one deduces from Eqs (26)) (27) and V(X, ) = V, = 0 that

p=w(x = x, ). (28) Our purpose here is to study the function r( u0 ,u, ,D,p2) for some illustrative cases. In order to make the calculations more transparent, we rewrite Eq. (24) as

r(uo,O,D,~~) + r(O,v, ,D,P’) = 1. (29) Case I: V, >O, U, (0, a~[ 1,2]. For trajectories withp<O,

it can be easily checked that (see Fig. 10)

a - dvo,vI ,D,p2) (0, am

(b) *W

lim r(uo,vI ,D,p’) = 0, IPI- - (30)

For each value of D, one can write

max r(vo,ul ,D,p2) = r(vo,v, ,D,O) = D “2Tuau,, P2

(31)

where Tu,,,, is independent of D. Thus for D> ( Tu,,,,, ) - 2, there exists a unique trajectory such that r( uo,u, ,D,p2) = 1. This solution disappears for D < ( TvoIu, ) - 2. Note that, for D = ( TuD,“, ) - 2, p = dv/dx(x, ) = 0.

Let us now consider trajectories with p>O. In this case [see Fig. (IO)]

a - dvo,v, ,D,p2)>0, am

& r(uo,ul ,D,p2)>0. (32)

For each value of D, the particular shape of a single-front solution requires that (i) p[ 0,D - ““J-1 as far as lower branch trajectories (v< 1) are concerned [Fig. lO(a)];and (ii)pc[O,D -“2(2-a)] fortheupperbranch trajectories [the upper bound corresponding to the separatrix in Fig. IO(b) 1. This implies for the whole trajectory that

O<p<D - “2(2 - a). (33) Noting that

From the condition D> ( Tu,,,, > - 2 (respectively, <T,;t ) forp<O (respectivelyp>O), one can conclude that, for each value of D there exists a unique trajectory when v. >O, u, (0 and aE [ 1,2]. Some single-front solutions under conditions “Case 1” are shown in Fig. 11. The examination of the spatial profile of these solutions as compared to the corresponding trajectories in the (u, W) plane is very instruc- tive. Case2:a - 2<vo (0, v, (0, aE[ 1,2]. For such conditions,p is restricted to positive values. Let us consider first the trajectories with w(x = 0) 20. p can take on values in the interval

pc (vo 1 <P<D - “2dm, (36) where the lower bound

lim - “Q2 - a)

Wo,QD,p2) = + co, (34) P-D

p,(vo) =D -I”,/ - (vg +2(2-a)v,) is the positive root of the equation

0 = Dpf + vi + 2(2 - a)v,.

From the inequalities (see Fig. 10) one deduces that there exists a solution only and only if

n$n dvo,vI,D,p2) = ~(uo,u,,D,O)<l, (35)

which is equivalent to saying DG (T,,,, > - 2.

J. Chem. Phys., Vol. 95, No. 1,l July 1991

a - r(O,u, ,D,p2) 20, am a

- dvo,0,D,p2)<o, am

,V

1

FIG. IO. Trajectories in the (u, W) plane of the differential system (20) computed with the piecewise linear slow manifold (25): (a) lower branch; (b) upper branch. The dashed lines represent two separatrix trajectories.

(37)

(38)

(39)

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0

-2

0.5

V

0

-0.5

-1

0.5

V

0

-0.5

-1


---- D=l _--__-_.m_ Dx0.3 - D=O.OZ

r I * 1

0 0.5 x 1

0 0.5 x *

FIG. 11. Stationary single-front solutions belonging to the D Oc branch (see Fig. 17). Case 1: e=O.Ol, a= 1.1, u,, =2.5, 0,=0.5, U, = -3.1, U, = - 1.1. (a) u(x) profile; (b) u(x) profile; (c) phase portrait u(u).

one cannot conclude about the sign of d/&p’) (r(vo,v, ,D,p2)). In fact, it can be shown (numerically) that the function r( vo,v, ,D,p*) presents a single minimum with respect to p2. However, the following remark holds in general: owing to the fact that Dp’ belongs to a bounded closed interval where the function ~(u~,u,,D,p~) = D “*Tuo ,“,, DpL, since T “,,, v,,D1lz is a bounded

positive function depending only on Dp*, this implies that solutions with w(x = 0) 20 can only exist in a bounded interval of values of D not containing D = 0. The picture we get is thus the following. As D-+ CO, no solution with w(x = 0) 20 does exist. For D less than some critical value, one couple of solutions appears (because of the existence of a minimum for 7). As D is further decreased, these two branches of solutions behave differently. The solution with the higher value ofp evolves in such a way that Dp2 -, 3 + 2a, andx, -0, i.e., the location of the front moves to the left-end boundary (see Fig. 12). Eventually, this solution fails to exist. We will see in Sec. IV B that this phenomenon can be related to a saddle-node bifurcation resulting in the collision of this solution with a “zero-front” solution. The solution corresponding to the lower value ofp is such that p +pc (v. ) as D is decreased, which implies that w(x = 0) +O. This solution disappears exactly at the point where w(x = 0) = 0 (see below ) .

Let us now consider trajectories with w(x = O)<O. From the relations

a - r(uo,u, ,D,p*bO, a(p*) $ r(uo>v, ,Qp2)>0, (40)

VD, lim dvo,u,,D,p2) = + co, P-D ‘92 -a)

one can deduce that a solution exists provided D< D,, where D, is given by the equation

~(uo,u,,Dc,~~(uo)) = 1.

From the expression = D k/2T

(41) of r(vo,v,,D,p~(vo))

u”,ul7P:(u,,) ’ where the quantity TUo,U,,PfcUo, is indepen-

dent of D, one concludes that Eq. (41) has a unique solution. Obviously, D, corresponds exactly to the value where the second branch solution with w(x = 0) )O disappears.

To summarize the situation for conditions “Case 2,” when D goes to infinity, solutions fail to exist. When D is decreased, one couple of solutions appears. One of them [satisfying UI( x = 0) > 0] disappears for D below some critical value (Fig. 12). The other branch [w(x = O)gO] persists as D goes to zero, and is such that p - D - v2 (2 - a), i.e., most of the solution is close to a - 2, and the front is located close to the right-end boundary x, - 1 (Fig. 13). Remark that, without the knowledge of the existence of the minimum of r( v. ,v I ,D,p’), one can still draw the conclusion that for conditions “Case 2,” solutions fail to exist in the limit D- CO, and that only one solution persists in the limit D-+0. These two properties seem to depend essentially upon the topology of the phase space of Eq. (20)) which is completely general for any S-shaped slow manifold, independently of its exact form. Case 3: - 10, <a - 2, v, ~0, ae [ 1,2]. As in the preceding case, one can easily compute bounds for allowed p values,

D - “2(2 - a) <p<D - “2,/w. (42)

In the limit p-D - “2( 2 - a), one recovers the previous result



V 1

0

-1

V 1

0

-1

-I * - - - t -. - - 1

- - - - - D=O. 135 ___e___. D~0.2

- D=0.28

- D=0.28 ------mm. Dz0.2 ----. D=O. 135

0 0.5 x l

,-----I

-mm----- Jjs0.2

/ . , . . . , . . .

----- D~0.135

-2 0 U

FIG. 12. Stationary single-front solutions belonging to the D - branch (see FIG. 13. Stationary single-front solutions belonging to the D + branch (see Fig. 17). Case 2: E= 0.01, (z = 1.1, u0 = 1.5, u, = - 0.5, uI = - 3.1, Fig. 17). Case 2: .z=O.Ol, a= 1.1, u0 = 1.5, u,, = -0.5, u, = -3.1, u, = - 1.1. (a) u(x) profile; (b) v(x) profile; (c) phase portrait u(u). u, = - 1.1. (a) u(x) profile; (b) u(x) profile; (c) phase portrait u(u).

VD, lim ~(Q.&,D,p2) = + co. (43) P-D 92 -a)

In addition, it can be proved that, VU~E[ - 1,a - 2 ] and Vu, (0, one has

a - r(vo,v, ,D,p2 = D - ‘(3 + 2a)) a(p2)

U

0

-2

V

0

-0.5

-1

V

0

-0.5

-1

I~-:----:----~~-~.,~- ~ ,, -;, --__ -------t,-

\

I , I , 1 t 8 , ,

: , 1 I , I

I 1 ( > a \

Y--l I I I I I I

II2;; y,/

I . 1 0 0.5

x l

‘- D=0.28 ‘- D=0.28 I ._-----. DzO.1 ._-----. DzO.1 /

---- D-0.02 ---- D-0.02

-2 0 U

1

( 2-a -

(6a -a2 - 1)“2 Jw +a$

>I 20 if ae[1,2].

From the limit (4)

J. Chem. Phys., Vol. 95, NO. 1,1 July 1991 Downloaded 09 Dec 2003 to 128.83.156.151. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

J. Elezgaray and A. Arneodo: ReactiotMiffusion pattern formation 339

lim a --7(uo,v,,D,p2) = - co, (45) P--D -1/2(2-n) a($)

and Eq. (44), one can deduce that r( v,,v, ,D,p2), as a function of p2, has at least one minimum. It follows that for D small enough, at least one couple of solutions does exist, As D-0, only one solution persists, which corresponds to p 4 D - “2 (2 - a) and x, -+ 1, i.e., the front is located close to the right-end boundary. The other solutions eventually disappear, in particular the one which corresponds to p-D - “‘dm and x, -+O. As numerically discussed in Sec. IV B, it turns out that only one couple of solutions appears when D is decreased; in the limit D-+0 the solution x, -PO collides with the “zero-front” solution, and then disappears in a saddle-node bifurcation, as for “Case 2.” c. Multi-front solutions. In the limit D+ 0, conditions can be found such that the reaction-diffusion system (3) displays an increasing number of stationary solutions. To illustrate this tendency, let us investigate the existence of (2N + 1) - front solutions, for conditions similar to “Case 2” previously considered for single-front solutions. A key remark in this study follows from the invariance of the trajectories in the (v,w) plane under the transformation (v -* v,w + - w) . At each switch from one branch to the other of the slow manifold, Iw(x = xi ) ( is independent of i and one can use from now on the notation

p=:]w(x=x;)), k[1,2N+ 13.

As for single-front solutions, we have

(46)

lim “I(2 - a)

duo,u,,D,p2) = + 00, (47) P-D

and that independently of N. Again it follows that VD<D,, a (2N + 1 )-front solution exists, where D, is some critical value. This solution is unique in the limit D-0. But a large number of (2N + 1 )-front solutions seem to exist for D not too small, as suggested by numerical simulations. It is likely that these solutions disappear in the limit D-0 via saddle- node collisions with 2N-front solutions, perturbed at x = 0 or x = 1. To summarize, in the limit D-O, our reaction- diffusion system generates an increasing number of (2N + 1 )-front solutions; for each value of N, a unique solution persists in the limit D-+0. Note that, if v. (a - 2, only single-front solutions exist (see Fig. 10). Let us emphasize that we have only used very general considerations to show the existence of (2N + 1 )-front solutions.

B. Linear stability analysis of stationary front patterns

In Sec. IV A, we have provided some evidence for the existence of saddle-node bifurcations, as a (local) mechanism accounting for the creation of pairs of solutions of our reaction-diffusion system (3) with Dirichlet boundary conditions. As the normal form theory@‘p6’ shows, at least one of these two solutions is unstable. From a physical point of view, one can guess that, in the limit D+ CO, any stationary solution should be stable. However, as D is decreased, or equivalently, as the size of the system is increased, the system

becomes approximately translationally invariant, and the stationary solutions are very likely to be unstable. As shown in this section, this transition involves not only (stationary) saddle-node bifurcations, but (oscillatory) Hopf bifurcations as well.’ l2 In Sec. IV B 1, we give the general setting of the perturbative calculation of the spectrum of the linearized operator around a single-front solution. We follow closely the approach developed in Refs. 7 1 and 72 for a similar problem with zero-flux boundary conditions. We discuss respectively the existence of (unstable) real and complex eigenvalues. In Sec. IV B 2, we generalize these calculations to the case of N-front solutions.

1. Singular limit of the eigenvalue problem for single- front solutions

a. The e-+0 limit eigenvalueproblem. The linear stability of a stationary solution (u, (x) ,v, (x) ) is determined by the eigenvalue problem:

d’IJ” = DEY!& + ‘I’“-f’(u,(x))Y”,

AY” = DY;, - Y”, (48) where /2 is the (a priori complex) eigenvalue, and (Y”,Y”) are the (u,v) components of the eigenvector. We introduce the notations

cm, 7&a, (49) which allow us to rewrite Eq. (48) as

?plY” = q”l/;, -I- Y” -f’(u,(x))YU,

AY” = DY& - ‘4”. (50) In order to make the calculations more transparent, we will consider the three quantities q,r and D as independent; we will restore the relation r = ~0 - ’ only at the end of the calculations. We will be mainly interested in the limit e-+0, which is equivalent to v-+0. Note that, due to the relation r = m, if E = 0, then r = 0. In fact, we will see that (E = 0,r = 0) is a singular limit, in the sense that one of the eigenvalues goes to infinity. In order to avoid this singular- ity, we will keep r away from zero.

Let us introduce the operator

L,Y = $Y,, -f’(u,(x))Y. (51) This operator is self-adjoint, so that its spectrum is real. We denote {Co >5, > ***>cn > * * * } the set of its eigenvalues, and 4, the associated eigenfunctions,

Ls+n =cnqS,, n=O,l,... . (52) . We can then write the inverse of (L, - a), a being any complex number not included in the spectrum of L,, by applying the eigenfunction expansion

M,,*) (L7 -a)-‘= 2 ___ nip 5, - a “.

This allows us to rewrite Eq. (50a) as

(53)

y”= -c G?L ,‘r/“M, nZ~ gn - A77 ’

(54)

while Eq. (50b) becomes

W, t G, - /in/g - ‘YV = /iY. (55)



Since the operator L, is a keystone of our analysis, we quote here some of its relevant properties, refering the reader to the Ref. 71 for further details and rigorous proofs of these results. (a) The positive eigenvector & (x) is a localized function which behaves, in the limit ~‘0, as

(56)

where p, ({) is the inner part of u, (x), and

(57)

The minus sign in Eq. (56) is needed as @, /dg < 0, because U(x; ) 7 V(x,+ ) when imposing the boundary condition (4). In addition, one can show that

lim L f& (x) = S(x - x1 )cO, g-o J;i

(58)

where co = K* 1 V(x,+ ) - U(x, ) 1 is a positive constant. (b) In the limit 7 -+ 0, it is also possible to give an estimate of the eigenvalue co,

Co = do + exp(v), (59) where

lo =CoK*~ (x =x, ) = CO/r*&

and exp( 7) is exponentially small as 7-O. We recall that (V(x), V(x)) is the outer part of the stationary solution. Since in the limit D+ CO, p is negative (see Sec. IV A), one gets

A f. ~0, for D--+ CO. (61)

However, in the limit D-O, one gets the opposite result

lo 70, for D-+0. (62) This remark will play an important role in the sequel. (c) &, - c, is bounded below when v-+0.

To proceed further, let us apply the eigenfunction expansion (53) to Eq. (55),

Dy” + (4O~~“MO +c

(A~‘u”M, = ay” xx go - ar?j n>l g, - a77 *

(63)

It can be shown” that

lim C MI~~“M, = _ Y”(x) ‘)-on>1 g, --/znl f’(Wx)) .

(64)

Let us remark that in the limit v-+0, <, - /2~7 remains bounded away from zero.

Note thatf’(U(x))#O, Vxe[O,l]; in fact,f’(u) =0 has only two solutions (u + and u _ ). The function V(x) never takes one of these two values. Using Eq. (58), we rewrite the limiting form of the second term of Eq. (63) as

lim (40,Y”Mo (xl 1 v-0 go -aTp-

= n Y”(x, )c&qx - x, ). 50 -2-r

(65) Let us introduce the operator T,

lY= -D’P,f ’ f’( U(x) ) y(x).

Tis also a self-adjoint operator. Now, we can write the limiting form of Eq. (63) (in the limit v-+0),

- (T+A)‘J’“(x) + &G

c~Y”(x, )S(x -x1 ) = 0.

(67) We next define the operator K, ,

KA = (T+A) -‘. (68) Then, combining Eqs (67) and (68), one gets

Y”(x) = & c: y”(x, X, (6(x - x1 1). (69)

Taking the scalar product of Eq. (69) with 6(x - x, ) yields

(Y”(x),&x - Xl 1) = Y”(x, 1 1

= - c;Y”(x, )(K, (6(x -x1 )),&x -x, )), co --r

(70) which can be reexpressed as a scalar complex equation for/z (in the limit V-O),

lo -;17=c,:(KA(G(x-x,)),S(x-xX1)). (71) In order to go further in our discussion, we have to de-

rive an explicit expression for the r.h.s. of Eq. (71). Let us note {Y,,) the set of (real) eigenvectors of the operator T [ Eq. (66) 1, and {r,, } the set of associated (positive) eigenvalues. Then the operator KA [ Eq. (68) ] takes the form,

K,=C- w,,*> y nil yn +R

IIt

so that

WA (6(x -x1 1),6(x -x, ,) = nGl 5.

To make the notation less awkward, we denote

A(A,,A,) = 4 c Yn +a, .>J:+(y,+R,)*

yf, (x1 1,

W,,~,) =d c 1

.,1a: + (yn fR,12 ‘y”, (x, 1,

with R, = Re(/l), R, = Im(R).

(72)

(73)

(74)

(75)

(76) Taking the real and the imaginary parts of Eq. (71), we obtain the final equations for /2 in the limit 740:

to - d;I, =A(A-,,R,L a,[B(a,,a,) -71 =o. (77)

b. Unstable real eigenvalues (stationary bifurcations). We first discuss the existence of positive real solutions il=il R > 0 to the reduced eigenvalue problem Eq. (77). Note that, when R, = 0, Eq. (77) reduces to the single equation

lo - ‘2, =A@,,01 = c; n;, ;:;;I, . (78)

Using the fact that y,, 20, Vn> 1, and R, >O, we see that the rhs. of Eq. (78) is a positive decreasing function of il,, which vanishes in the limit /2, -t + CO. Moreover,


-c (A(A,,O)) 70. d/2 ‘,


(79) infinity. As suggested by the numerical simulations of Sec. III, the single front solution actually becomes unstable through a Hopf bifurcation. From the above discussion, a natural guess is to suppose that, for some D, > D, (r), a couple of complex eigenvalues appears in the spectrum, eventually becoming real for D = D, (7). A finer analysis of Eq. (77) around co = 5, shows (see, for instance, Ref. 72) that this is actually what happens. c. Unstable complex eigenvalues (Hopf bifurcation). In order to demonstrate the existence of a Hopf bifurcation of the single-front solutions, one must prove the existence of a solution of Eq. (77) with ;1, = 0. Namely, one has to solve

ito =AW,), 7 = B(OJ,). (80)

From the explicit expression of B [Eq. (75) 1, straightforward calculations yield

Now, let us suppose that r is fixed to a (small) positive value. For large values of D, lo < 0 as previously discussed [ Eq. (61) ] and consequently Eq. (78) has no positive solution as illustrated in Fig. 14(a).

As D is decreased, co becomes positive [Fig. 14(a) 1. As shown in Fig. 14(b), for jb = CC > 0, a positive solution of Eq. (78) appears; this solution splits into two different positive solutions as illustrated in Fig. 14(c). We see that the only role of r in the problem is to control the magnitude of the positive eigenvalues, but not their existence. That is, for any small r, Eq. (78) has (at least) one positive solution for D<D,(r). Note that, as r+O, DC(r)-+DGO, where D, is such that lo (D, ) = 0. Then, it is clear from Fig. 14(c) that, in the limit r-+0, one of the two positive eigenvalues goes to

1 z (4 c 2 . . . . . . . . . . . to _ rXR T

o I... . . . . . < ,).,, , ] 0 5 10

(4

341

B(O,A,) > 0, $- BW,) ~0, d/zz ‘* B(O,AI) 70. I I

(81) It follows that, for each small enough value of r, there exists a single solution to the equation r = B(O,R, ), say /2,(r), such that ];1, (7) 1 -+ CO as r--r 0, and dA,/dr < 0. If we consider a value of D, close enough to D,, then Eq. (80a) has a solution ;1, (D) which behaves like 2, (D) + + CO when D-D,, and Eq. (80b) determines a single value of r(D) = B(O,A,(D)), such that r(D)+0 when D+D,. In other words, for each value of D ( < D, > close enough to D,, one gets a value of r = r(D) such that Eq. (80) admits one solution. This establishes the proof of the existence of a Hopf bifurcation for single-front solutions. Let us remark that this value r(D) can be made as small as necessary.

We can thus summarize the global picture for single- front solutions as follows:“2 when D is such that co < 0, the single-front solution is stable. For values of D<D, (r), a Hopf bifurcation occurs, and the stationary solution becomes unstable. Eventually, the couple of complex eigenvalues becomes real for D = D, (r) < D, ( r) .

2. Linear stability of a general N-front solution

XR 10

In Sec. IV B 1, we have shown that, in the e--r0 limit, the eigenvalue problem obtained from the linearization around a single-front solution, reduces (when considering only the most unstable modes) to a system of coupled nonlinear equations [ Eq. (77) 1. The generalization of this result to the case of an N-front solution is straightforward (at least formally ) . One can again simplify the original eigenvalue problem to a set of two coupled nonlinear equations; but their highly nonlinear character makes these equations more difficult to solve in general. We will focus here on the formal derivation of these two equations.

FIG. 14. The two functions &, - TAG (dotted line) and A(A,,O) (full line) plottedvsR, (r> 0 is fixed) for three different valuesof d,: (a) 2,) <&, Eq. (78) has no positive solution; (b) L$ = &, a positive solution of Eq. (78) appears; (c) j,, > g,, this positive solution splits into two different solutions (see text).

Let us note {x,,i = l,..., N] the locations of the N fronts: the outer u component U(x) of the solution is discontinuous at each xi, and V(x, ) = I’, . It can be shown that the operator L, [Eq. (5111 has N eigenvalues {gf’ = q$r),i = l,...,N) going to zero as 77 goes to zero, where

J. Chem. Phys., Vol. 95, No. I,1 July 1991 Downloaded 09 Dec 2003 to 128.83.156.151. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp


lim 5:) = T-0

-~(x=xi)(K*)2(u(xi+) - U(x,-)).

(82) Let us recall that

z(x=xi)i =p, Vk[l,N] (83)

due to the invariance of the trajectories given by Eqs. (26) and (27) under the transformation (u-u, w+ - w); consequently, in the limit q-0, lt’ is independent of i [up to 0( 7) terms]. The associated positive eigenfunctions ~$2’ are picked functions, each centered around the corresponding xi; in the limit q-+0, they read

lim -&y’(x) =6(x-x,)c,. 7-o 67

(84)

The straightforward generalizations of Eqs. (54) and (55) are, respectively,

(85)

and

Dyh + 2 (&?9Y”>4v + c i= 1 56” -A,T1j)

(~/f~y”)~~ =;1yv” (86) * tl>l gn --l-q

When 77 + 0, this equation becomes

N - (T+A)Y”(x) + c: c YYX, )6(x - x, ) = 0, i=I 52’-/27 (87)

where Tis defined in Eq. (66). Using the operator Kl introduced in Eq. (68), one gets

Y”(x, )S(x - x, )

l;i) -Ar I * (88)

Let us introduce the N * numbers

A(ij) =c; (S(x-x,),K~S(x-xj)) f?) -j/r ’

i&[ l,N]. (89)

Taking the scalar product of Eq. (88) with 6(x - xj ), one gets a set of N complex equations,

Y”(xj) = 2 A(j,i)Y”(x,). (90) i= 1

This amounts to solve the dispersion relation det(fiiJ - A(ij)) = 0, (91)

which is the analogue of Eq. (71). At this point, one must emphasize that the nonlinear complexity of the reduced problem (91) increases with the number N of fronts.

In order to illustrate a solution of Eq. (9 1 ), let us report on numerical results obtained in the case N= 2 with the slow manifold defined in Eq. (6) for parameters values close to the threshold of a Hopf bifurcation. It turns out that, under these conditions, Eq. (91) has only two (complex) solutions il, and /2,, with Re(/2, ) > Re(A2 ). The u components of the associated eigenvectors Yy and Y,U are shown in Figs. 15 and 16, respectively. Both are approximately given by linear combinations of &,” and +A*‘, in agreement with

1

u

0

-1

0.3 =3

i-2 02

0. I

0

(a

-:; space 1

-0.2 I 0 0.2 0.4 0.6 0.8 1

X

FIG. 15. Periodically oscillating two-front solution (breathing pattern) of the reaction-diffusion system (3) with Dirichlet boundary conditions, the slow manifold (6) and the model parameters: u0 = a, = - 1.5, a = 0.2, l = 0.01, D = 0.03. (a) Unstable stationary profile u(x); (b) spatio-temporal variation of u(x,t) using the same coding as in Fig. 3; (c) real part and (d) imaginary part of the u component of the critical in phase Hopf eigenmode.

Eq. (85). The in-phase mode (Yy,Yy ) is less stable than the out-of-phase mode (Yu,U,Y; ), so that the Hopf bifurcation observed numerically is associated with the in-phase mode as illustrated in Figs. 4(b) and 4(f). However, the presence

0 0.2 0.4 0.6 O.8x I 0 space 1

0.4

0.2

$0 d

-0.2

@ )I

f -i

0.02

0.01

B E O

-0.01

-0.02

-0.4 0 0.2 0.4 0.6 0.8 x 1 0 0.2 0.4 0.6 0.8 x I

FIG. 16. Periodically oscillating two-front solution (wiggling pattern) of the reaction-diffusion system (3) with Dirichlet boundary conditions, the slow manifold (6) and the model parameters: u,, = u, = - 1.5, a = 0.2, E = 0.01, D = 0.02. (a) Unstable stationary profile u(x); (b) spat&tern- poral variation of u(x,t) using the samecoding as in Fig. 3; (c) real part and (d) imaginary part of the u component of the critical out of phase Hopf



of the out-of-phase mode can be identified as a transitory damped out of phase oscillation, when the initial condition is enhanced in the (Y;,Y; ) direction as shown in Figs. 4(d) and 4(h).

C. Piecewise linear slow manifold 1. Exact single-front solutions

The two preceding sections provide qualitative results concerning the existence and linear stability of the stationary solutions of our reaction-diffusion system (3) with Dirich- let boundary conditions. With the specific piecewise linear form (25) of the slow manifold, it is actually not difficult to find (numerically) the exact stationary solutions of Eq. (3) and to determine their stability. Let us first discuss the existence and stability of single-front solutions. The algorithm we use to numerically determine these exact solutions as- sumes the existence of two constants (actually, the un- knowns of the problem), O<x- <x+ < 1, such that the stationary solution satisfies

u(x)>l, for xe[O,x- 3, - la(x)Gl, for xE[x- ,x+ 1,

u(x)< - 1, for xE[x+ ,l]. (92)

In each of these three regions, Eq. (8) reduces to a set of two second-order linear ODES, which can be solved explicitly. Requiring the solution to be three times differentiable, we obtain a set of coupled nonlinear equations, which is solved by a standard Newton method, and which completely deter- minesx- andx,. In order to track “all” the stationary solutions, we randomly generated the initial conditions for the Newton method. The same method is used to solve the eigenvalue problem (48). Figure 9 (a) shows a typical profile for the u component of the exact solution, compared to the outer solution U(x), which is discontinuous at x = x1 . The comparison of the exact solution (u(x), u(x) ) with the zeroth-order approximation ( U(x) + p, (6)) V(x) ) in Figs. 9(c) and 9(d) shows that this is indeed a rather good approximation.

We summarize in Fig. 17 the evolution of xfront = 4(x _ + x + ) as a function of D, for several values of the model parameter u. , the parameters a = 1.1, E = 0.01, u, = - 1.1 being kept fixed. Let us mention that the main characteristics of this diagram remain mainly unchanged when varying the parameters a and u, ( CO). We thus be- lieve that the following discussion is of general relevance.“*

In agreement with our previous results in Sec. IV A 3, when u. > 0, only one solution exists for each value of D. In the sequel, we will denote by D n; the unique branch of single front solutions existing in the limit D- to. When u, > 0, D m is actually the branch of stationary solutions predicted by our phase-plane analysis. As D-+0, xfront -+ 1, and the u component of the solution displays a large region where u-a [see Fig. 11 (a) 1. The situation changes when u. < u~,~, where u,., - - 0.25 for the parameter values investigated here. As illustrated in Fig. 17, for large values of D, D m is still the unique solution branch. But, as shown in Fig. 18 (a), the spatial profile of the solution changes dramatically; in fact, except a small region located near x = 0, the solution

0.6 -

0.4 -

0.2 -

O- 0 0.2 0.4 0.6 0.6

II ’

FIG. 17. Evolution of the position of the (single) front xlront = 4(x _ + x + ) as a function of D, for several values of u,,; the model parameters are E = 0.01, a = 1.1, u, = - 1.1. The calculations were performed with the reaction-diffusion system (3) with Dirichlet boundary conditions and the piecewise linear slow manifold (25).

belongs to the lower branch, and is better described as being a zero front solution perturbed at x = 0.

The perturbative analysis carried out in Sec. IV B 1 pre- dicts the existence of a Hopf bifurcation when D gets close to D, . The exact results obtained in Fig. 17 confirm this asser- tion. More precisely, as D-+0, and u0 > u~,~, the single-front stationary solution D m becomes always unstable through a Hopf bifurcation. Figure 19 represents the spatial profile of the u component of the Hopf eigenmode Y” at the bifurcation threshold. From Eq. (54), Y” is, to the zeroth order, proportiona to 4. (x), i.e., the Hopf eigenmode is a localized mode with a characteristic length scale -O(a); Y” decreases exponentially away from the front location. The choice of phase made in Fig. 19 is such that Im yU(x-xrront ) = 0, which explains the small amplitude of Im YU(x).

As previously mentioned, when u, < uO,, , D cc is no longer described by our perturbative analysis, so that the argu- ments in Sec. IV B 1 concerning the linear stability do not apply to this branch ofsolutions. It turns out that this branch remains stable until it collides another branch denoted D - in Fig. 17. Similarly, our stability analysis does not account for the saddle-node bifurcation giving rise to the couple of solution branches D + and D -. In fact, the validity of our analysis requires the existence of a stationary solution; this implicit assumption obviously fails in the vicinity of a saddle-node bifurcation. The main point is that a zero eigenvalue appears in the spectrum of the operator T defined in Eq. (66). A perturbative calculation of this bifurcation should necessarily take into account this difficulty. Let us point out that both branches D + and D - have been found unstable for all the values of D studied so far.

2. Exact three-front solutions

Our strategy for tracking the three-front solutions as- sumes the existence of six positive numbers x,, ic [ 1,6], with XiE[O,l] andxi <Xi+ 1, such that the solution satisfies


344 J. Elezgaray and A. Arneodo: Reaction4iffusion pattern formation

-2

I - . . 1

- b1.0 .-_----_ DzO.5 ---- DzO.2

I . I I 0 0.5 x 1

v t

0.5 -

O-

-0.5 -

-1 - I

,/----. / . 0 \ (b) 0 \

/ \ / \ / \

/’ \ \

~~~

x *

I I 1 -2 0 U

FIG. 18. Stationary single-front solutions belonging to the D z branch (see Fig. 17).Case2:~=0.01,a=1.1,u0=1.5,u,= -0.5, u, = -3.1, a, = - 1.1. (a) U(X) profile; (b) u(x) profile; (c) phase portrait u(u).

W)>l, for .=[OJ, ] u [%Js], - l<UCC)<l, for XE[X,,XZ] u [X,9&] u [X5&],

(93)

u(x)< - 1, for XE[X~,X~] U [x6,1]. As before, when restricted to one of these three subsets of [0, I], Eq. (8 1 reduces to a set of two coupled ODES, which can be solved explicitly. We require the solution to be three

2 u

0

-2

-4

0.4

5 2

0.2

0

0 0.5 1

0.02

3 so

-0.02

-0.04

0 0.5 1 X

FIG. 19. (a) Spatial profile of the u component of a single-front solution of Eq. (3), with Dirichlet boundary conditions, the slow manifoldf( u) given by Eq. (6), and the model parameters a0 = 2.5, U, = - 3.1, a = 1.1, e=O.Ol, D=D uopl = 0.011. (b) Spatial profile of Re P” (full line), as compared to its zeroth-order approximation given by the function e$ (x) (dotted line). (c) Spatial profile of Im Y” (full line) as compared to its zeroth-order approximation q5” (x) (dotted line).

times differentiable, which provides the necessary equations to completely determine the xi%.

In Figs. 20 (b ) and 20 ( c 1, we represent the II and u components of a typical spatial profile obtained using this algorithm. It is clear from Fig. 20(b) that this solution should be considered as a two-front solution perturbed at x = 0 rather than a three-front solution. In terms of the Fife’s perturbative approach, ’ I3 the outer part of the zeroth-order approximation presents two (rather than three) points of discontinuities, an inner correction being also necessary at x = 0, in order to satisfy the boundary conditions. So far, we have not found any of the three-front solution branches predicted theoretically. The reasons for this failure are mainly numerical: for “reasonable” values ofD( - 10 - *), the distance between the first and second front is estimated - O( 10 - 2). How- ever, the typical scale for the front zone is -m - 10 - 2 for E = 10 - 2. Therefore, very small values of E & 10 - ’ are required to provide evidence for the existence of these three- front solutions. Such values are not accessible with respect to

J. Chem. Phys., Vol. 95, No. 1, 1 July 1991

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2 ”

0

-2

0.6

7 L 2

0.4

- vo=-0.3 -. - - I _. _ _ vo=-0.33

(4

./- +-p;-.-.. _ _ _ _ ._. -.-. -.- .- . .

3

-0 0.1

0 0.5 z 1 0 0.3 2 1

FIG. 20. (a) Evolution of XFron, = 4(x,, + x, ) as a function of D, for two values of u,,; the model parameters are 6 = 0.01, a = 1.1, u, = - 1.1. The calculations were performed with Dirichlet boundary conditions and the piecewise linear slow manifold (25); (b) u(x) profile and (c) u(x) profile of a three-front pattern.

the resolution of our numerical simulations. Figure 20(a) shows the evolution ofxfront = i(x, + x4 )

as a function of D, for two different values of u,, while ~=O.Ol,cr= l.l,andu, = - l.larekeptfixed.Forthisset of parameter values, only two branches (noted D 3+ and D ; ) of stationary three-front solutions exist. They appear through a saddle-node bifurcation and exist for values of D less than a critical value D,. The D : branch persists as D goes to zero; originally, for values of D close to D,, this branch is stable. However, when D is decreased, it becomes unstable via a Hopf bifurcation. The D ; branch is always unstable.

V. NONLINEAR ANALYSIS: HOPF NORMAL FORM REDUCTION

The reaction-diffusion system (3) can be viewed as a dynamical system in an infinite dimensional space. More pictorially, if the solution is decomposed onto some basis (e.g., the Fourier basis), the original PDEs can be written as an infinite number of coupled ODES, the variables being the coefficients of this decomposition. However, due to the dissipative character of this type of PDEs, all the solutions are attracted exponentially fast to some finite dimensional manifold (embeded in the infinite dimensional space of reference), called the inertial manifold of the system.‘02*‘03 Con- sequently, the dynamics of Eq. (3) is likely to be finite dimensional asymptotically. To our knowledge, there exists in the literature only very poor estimates of this finite dimen- sion which in general turn out to be rather large numbers.“‘.“”

In this section, we will focus on the dynamics in the neighborhood of a Hopf bifurcation of a single-front pattern.

It can be proved that, generically, in a small neighborhood of any bifurcation point, the local structure of the inertial manifold can be thoroughly described in terms of a low-dimensional system of coupled ODES, called the normal form of the bifurcation. For the specific case of the Hopf bifurcation, 60*617117 this set of equations is given by

i = (,u + iw)z + ~z(zJ* + h.o.t., (94) where z is a complex amplitude, and p + iw is the complex critical eigenvalue which crosses the imaginary axis (p = 0) at the bifurcation point. We will assume that the real parts of all the other eigenvalues remain negative (at least in some neighborhood of the bifurcation point) and correspond to stable modes. We recall that, according to the sign of Re K, the Hopf bifurcation is either subcritical (Re K > 0) or supercritical (Re K < 0).

In Sec. V A, we carry out the normal form calculation of the Hopf bifurcation of single-front patterns in our reaction- diffusion system (3). In Sec. V B we compare the normal form predictions with the results of direct simulations of the PDEs system (3). Besides the intrinsic interest of the knowledge of the character of the bifurcation (particularly its relation to the character of the Hopf bifurcation associated with the reaction term), the following calculations are of general importance since the instabilities of highly localized structures are generally very difficult to work out theoretically. This point will become apparent at the end of Sec. V B.

A. Normal form calculation

This paragraph gives a short description of the normal form reduction method proposed by Coullet and Spiegel”* (see, also, Ref. 119). Let us first introduce some notations. We will write the original PDEs in the general form

d,X = MX +4’-(X), (95) where M is a linear operator and L,4’(X) accounts for the nonlinear terms. For the sake of simplicity, we will restrict ourselves to the computation of the normal form at the bifurcation point, so that we will drop the dependence of the operators M and ,,j/‘ upon the unfolding parameters. Let X, be the stationary solution which undergoes the bifurcation. It is convenient to introduce the change of variables U = X - X,; then Eq. (95) can be readily put into the form

a,U = MU + N(U), (96) where the transformed nonlinear term NW =&‘“(X,+U) -+4’-(U)vanishesforU=O(N(O) =O>.

As stated before, we will assume that the spectrum of M splits into two sets: the critical eigenvalues {A f}, such that Re il r = 0, and the stable eigenvalues, {;1 s}, with Re il T < 0. Let us note Qj the corresponding critical eigenvectors,

MQi = ;1 ;Qi. (97) The Hilbert space of reference Hcan be decomposed into the direct sum of the critical space H, spanned by the Qi’s and its orthogonal complement H ,‘,

H=H,eHf. (98) The operator J is defined as the restriction of M to H,. In general, M restricted to Ht is not diagonalizable. We will



only assume the existence of a basis for this subspace,

H’ = span&S,}. (99) The method proposed by Coullet and Spiege1’*8P”9 is

based on the following ansatz (the convention of implicit summation over repeated indices is adopted in the sequel) :

U(x,t) =Ai (t)Qi (~1 + V(x,A), (100)

hi(t) =JgAj(t) + Gi(A(t)), (101) where V and G are nonlinear functions of A to be determined. Indeed, Eq. ( 101) is the normal form (or amplitude equation) we want to calculate. Substituting Eqs. ( 100) and (101) into Eq. (96), one obtains

k,$ (x) + k,~Y~,v(x,A)

= [JgAj + G,(A)] [*i(x) + a~,v(x,A)]

=AiJg@j(X) + MV(x,A) + N(Ai@, + V(x,A)), which can be rewritten as

(L - M)V(x,A) = N(A,@, + V(x,A))

- Gj (A)d,,V(x,A) - G,(A)+,. (102)

L is the following operator: L=J,,A,d,. (103)

In order to solve Eq. (102), we expand both V and G as Taylor series in the amplitude A,

V(x,A) = 1 V,‘k’(A)@, (xl + C Wjk’(A)Si(x), k>2 k>2

(104)

G;(A)rC Gjk’(A), k>2

(105)

where I’,! k), W,( ‘), and G j k, are linear combinations of mon- omials of degree k in the amplitude vector A. We solve Eq. ( 102) in successive orders.

At the k th order, Eq. (102) reduces to the following nonhomogeneous linear system defined in the tensor space H 8 Pk, where Pk is the linear space of homogeneous polyno- mials of degree k:

(L - M)( vjk)(A)<Pi(x) + W;“‘(A)S,(x))

=I;k’(A)@i(x) + Hjk’(A)Si(x) - Gjk’(A)Q,(x). (106)

The two first terms of the rhs of Eq. ( 106) are known at each order k, since I j k, (A) and H j li) (A) depend only upon quantities ( Wjk”(A), Vjk’)(A), Glk”(A) with k’<k) that are known from computations at previous orders.

We are interested here in solving Eq. ( 106) in the particular case of a Hopf bifurcation. Therefore, H, is a two-dimensional subspace, spanned by the complex eigenmodes {Q, =@,Q2 =m>, respectively associated with the eigenvalue {iw, - iw}. The corresponding amplitude vector is noted A = (z,Z) . The operator J reads

J = iw(d/dz - d/d?). (107)

It can be easily checked that cf,., = zk - n F?,n = O,...,k} is a basis for Ph, and that the operator L - M restricted to H, 8 Pk is diagonal in the basis {Q ~f,,~ ,G 8 fn,k}. With this

notation, Eq. ( 106) can be expressed in the matrix form,

(

L?Lk’ 0

zf) (2;::) = ((‘:“~~~~p)@i),

(108)

where Yrk) (respectively, 3jk’) is the restriction of L - M to H, 0 Pk (respectively, Hi 0 Pk ), and Y,,,, stands for the set of matrix elements {(a @f,,& 1 (L - M) (Si @fm,& I), (m~f,,,I(L-M)(Sief,,k))).Notethat~,,,, =OifM is self-adjoint, which is not the case in our problem.

The operator 3’ik) is always invertible, so that the pro- jection of Eq. ( 106) onto Hi can be formally solved,

Wjk)(A)S,(x) = (YSk’) -‘(Hjk’(A)Si(x)). (109)

On the opposite, Y:k) is invertible only for certain values of k. If this is the case, we will choose Gck’ = 0, and the projec- tion of Eq. (106) onto H, can be readily solved,

Vik’(A)@,(x) = (aY:k’) -‘(Ijk’(A)Qi(x)

- cY:,“d,, Wjk)(A)Si(x)). (110)

Note that other choices for Gck’ are also possible; they lead to different versions of the normal form equation. However, all these normal forms are related by nonlinear transformations, and the dynamics they describe are then equivalent. 6oV61V1’8+119 If k is such that 3:k’ is not invertible, Gck) will be chosen in order to satisfy the solvability conditions,

(a; ,(l:k’(A) - Gik’(A) )Qj - Y’:,“d,, Wjk’(A)Si) = 0, (111)

where the a,+ ‘s are the null eigenvectors of the adjoint of zrk’.

Let us consider in some detail the first two orders of this expansion. At second order k = 2, the representation of the operator 21” in the basis {a 8fo,2,@ ef,,2 ,..., G af2,2 1 reads

Tr2’ = diag(iw, - iw, - 3iw,3iw,iw, - iw). (112) In this case, -!Y12) is invertible; thus Gc2’ = 0 which explains the absence of quadratic terms in the Hopf normal form (94). Then one can calculate the solution at second order according to Eqs. ( 109) and ( 1 lo),

Wj2’S; = (YS2’) -‘C(N(AjQ,),Si Bf,,z)Si ef,,zI, 3) (1 1

and

vj2’@, = (yi2’) - ‘C<NCAj(a,),@L afn,z )@i @fn,2 - A?;;& Wj2’Si). (1 1

At third order k = 3, in the b ‘a: 4) sis

C@ @fo., ,Q, @fi.‘,..., & 8 f3,3 1, the operator 2:‘) is diagonal,

9:” = diag(2iw,O, - 2iw, - 4iw,4iw,2io,O, - 2iw). (115)

Thus, the adjoint of 3’:‘) has two null eigenvectors, @ 8 f,,3 and 5 8 f2,3. The solvability conditions can now be written under the more explicit form,

(@%fi,‘,W” - G;“‘)@, - eY& W:“‘S;) = 0, (116)

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J. Elezgaray and A. Ameodo: Reaction-diffusion pattern formation 347

(5@fz,3,u!3’ - G j3’)@, - c.Ci!‘::Jss W;3'Si) = 0. (117)

Note that Eq. ( 117) is the complex conjugate of Eq. ( 116); it follows that G I” = c.c.( Gi3’). Therefore we recover the Hopf normal form defined in Eq. (94) with

K= Gi3’. (118) To complete the calculation of the coefficients K in front of the cubic term of the Hopf normal form, we use Eq. ( 109) to get Wi3’,

w;~)s, = (2j3)) - ‘{(N(A~*~ + Y;~)Q~

+ wJ2’sj>Si @fn,3 jsi @f,Jl* (119) Then from the solvability condition ( 116) one gets the final expression,

K= (@~f,,,,(N(Aj@j + Vj”@j + Wj”Sj),@i Bfn,3)

@pi @fn,3 - 2y& w j3’Sj)

= (N(A,Qj + vyDj + w;%,,,m4fi,‘)

- (* @fi,39~~~cjsss w13’si). (120)

B. Normal form predictions versus direct simulations The compact equations. obtained in Sec. V A hide an

extremely difficult computational problem. Namely, the normal form calculation requires

(i) the choice of a basis for H :: this can be achieved according to the following construction.

If {Y, ,n) 1) is an orthogonal basis of H, then

s, = Y, - (@,Y”)@ - (co,Yn)&, ?I>,3 (121) is a good candidate. Although, in general, the S, are not orthogonal, it is not difficult to decompose any vector of H onto the basis {Q,&,S,, (n)3)}. (ii) The representation of the operators Ybk) and Y:,kd,, on this basis: it turns out that the (infinite) matrix obtained in this way is a full matrix (the same is true whether the S, are orthogonal or not). This is mainly due to the existence of localized strong gradients in the stationary front solutions we are interested in, which implies that both Yjk) and Y$ are differential operators with rapidly varying coefficients.

The strategy adopted in this work takes into account the fact that only an approximation of the exact stationary solution is available from the numerical calculation. In order to make the whole computation coherent, we use the approxi- mations of the operators Yjk) and Yz,“d,, provided by the finite-difference scheme. To this order of approximation, these operators are finite matrices which can be inverted numerically.

Our purpose is to compare the theoretical prediction Eq. ( 120) for the value of Re K, with the measurement of the same quantity from direct simulations of the reaction-diffusion system (3). The numerical estimate is easily obtained from the amplitude of oscillation of the single-front solution in the <p direction. If we write z = peie, the real part of Eq. (94) yields

i, = ,up + Re K,O~ + h.o.t.,

so that the amplitude of oscillation is given by (122)

p=JV. (123) We have carried out this comparative analysis for differ-

ent conditions and various choices of the slow manifold. When using the slow manifold given by Eq. (6)) all the Hopf bifurcations considered have been found subcritical. Let us recall that the Hopf bifurcation of the reaction term is also subcritical.99 On the opposite, when using the slow manifold

f(u) = 23 + u3, (124)

which ensures the existence of a supercritical Hopf bifurcation for the reaction term,99 all the Hopf bifurcations of single-front solutions investigated thus far turn out to be also supercritical. Figure 2 1 (a) illustrates the numerical deter- mination of the scaling exponent (0 = l/2) of the amplitude of the Hopf mode as a function of the distancep to the bifurcation point [ Eq. ( 123) 1. Figure 21 (b) shows the numerical values of the coefficient Re K extracted from the p

5

z $

4.5

4

4 5 6

N -0.71

$

; -0.72

c2

-0.73

I I

0 0.005 0.0 1 0.015

P

FIG. 21. Direct simulations of the reaction-diffusion system (3) with Dir- ichlet boundary conditions vs Hopf normal form predictions. Iz( = ( - /./Re K) I” is the amplitude of the Hopf critical mode. (a) Deter- mination of the critical exponent /? = l/2; (b) measurement of Re K: the dashed line corresponds to the normal form predictions on the critical sur- face (/J = 0): ReK= - 0.7301. Model parameters: f(u) = U* + Us, u, = 1.5, u, = - 1.5,a=0.001, ~=O.Ol,D,,,,, = 1.537x10-*.



dependence of the amplitude of the limit cycle as compared to the theoretical normal form prediction given by Eq. ( 120). The numerical points range on a straight line whose intercept at p = 0 is in remarkable agreement with the theoretical value Re K(,U = 0) = - 0.7301. Let us emphasize that the linear behavior observed numerically can be understood (quantitatively) by pushing the normal form calculation at next order in perturbation; a small deviation from the critical situation introduces a correction term of O(p) in the expression of Re K (Refs. 60, 61, and 117) which again can be calculated analytically.

To conclude, let us point out that this theoretical nonlinear analysis is likely to be tractable in the more general case of N-front solutions.

Vi. CONCLUSION In this paper, we have shown that the existence of an S-

shaped slow manifold in the reaction term is the main ingre- dient required for our model reaction-diffusion equation system to reproduce the sustained spatial and spatio-temporal chemical patterns recently observed in the Couette flow reactor with either asymmetric or symmetric external feeding. Our numerical results provide a remarkable illustration that regular steady patterns can organize in a one-dimensional extended medium in the limit of equal diffusion coefficients for the different chemical species, provided a concentration gradient is imposed to the system. We have used singular perturbation techniques to study the existence and the linear stability of single-front and multi-front patterns. A nonlinear treatment of the primary Hopf instability leading to periodically oscillating front patterns has been achieved with normal form reduction techniques and the theoretical predictions of the normal form calculations have been found in remarkable agreement with direct simulations of our reaction-diffusion model. Sequences of bifurcations leading to more complex spat&-temporal patterns have been numerically identified. A striking result is that these front patterns can disorganize into diffusion-induced chaos, which displays the same “strange attractor” characteristics as the so- called chemical chaos arising from the nonlinear complexity of the chemical kinetics of the BZ reaction. In a forthcoming paper, lo6 we will elaborate on the observation of intermittent bursting patterns which are likely to be understood in terms of homoclinic chaos.60F61~‘09~“0~“9 The remarkable feature of the wide variety of sustained spat&temporal patterns reported in the present work is the fact that they organize due to the interaction of the diffusion process with a chemical reaction which itself would proceed in a stationary manner if diffusion was negligible. To some extent, this study clearly demonstrates that spatiotemporal complexity do not necessarily result from the coupling of oscillators or nonlinear transport.

daries. These reactors have a natural tendency to produce narrow front (linear or circular) structures away from the boundaries.40 Stationary single-front and multi-front patterns have been observed experimentally. According to the Hopf bifurcation mechanism reported in Sec. V, these front patterns are expected to destabilize into periodically oscillating structures. Since the Hopf mode is likely to be condensed in the active region at the front zones, the reaction-diffusion system will thus reduce, at the onset of the bifurcation, to one or several very narrow strips of oscillating intermediate reactor cells coupled by diffusion and uniformly fed, i.e., a situation so far unattained in bench experiments. In principle, all the phenomena found in the one-dimensional Ginzburg- Landau62 and Kuramoto-Sivashinsky4,63-67 equations such as standing waves, traveling waves, phase turbulence,... should be observable in such systems. Rotating waves have been already observed in the annular gel reactor24*25*38.M but as induced by an external perturbation in an excitable regime and not originating from an intrinsic phase instability.

A similar theoretical analysis can be reproduced for higher dimensional systems.73 In a prospective paper,‘“’ in collaboration with Pearson and Russo, we have reported preliminary results of a study of two-dimensional reaction- diffusion systems that model sustained front patterns observed in gel reactors. The linear35-37 and annular24v25*38 gel reactors are strips of gel that are fed from the lateral boun-

Along the line of the pioneering work by Boissonade,82 the Bordeaux group has recently reported the observation of a symmetry breaking instability leading to a stationary Tur- ing structure along the front pattern, i.e., transverse to the external concentration gradient, in a very promising experiment in gel reactors. 35-37 As suggested by numerical simulations of reaction-diffusion systems, this Turing bifurcation generally occur in the neighborhood of the Hopf bifurcation of the homogeneous front pattern. In Ref. 120, we have pointed out the intimate relationship between the characteristic wavelength il of the Turing pattern and the frequency of oscillation 0 of the homogeneous front pattern [ il - 274 D /w ) “2] as a possible experimental test of the intrinsic nature of the observed spatial structure. The proxim- ity of these two bifurcations naturally raises the issue of their interaction. In the case of spatially confined geometry, i.e., when the length of the system is small relative to the intrinsic chemical wavelengths, the mathematical technique of center manifold/normal form reduction60p61 can be used to calculate the normal form of this bifurcation problem according to the symmetry of the system. In the annular gel reactor configuration [ 0( 2) symmetry], this normal form is likely to produce secondary bifurcations leading to standing waves (e.g., breathing of the Turing pattern) or traveling waves (e.g., drift of the Turing pattern) and ultimately to spatio- temporal complexity.‘20 In the case of spatially extended systems, i.e., when the size of the system is large as compared to the intrinsic chemical wavelengths (which is the case in the Bordeaux experiment where the observed Turing patterns are about 100-500 wavelength long), then one can expect to encounter rich dynamics from the onset of either primary bifurcation: (i) phase instabilities leading to standing waves, traveling waves, and phase turbulence at the onset of the Hopfbifurcation;4v63-67 (ii) breathing, vacillation, and drift instabilities’2’-‘24 of the basic cellular pattern at the onset of the Turing bifurcation where the occurrence of localized solutions and defects may result in the destruction of the spatial order.‘22

Those increasingly complex dynamics are thus very likely to be observed in the active region of the open gel


reactors which is confined to the front, e.g., in a narrow linear (linear gel reactors) or circular (annular gel reactors) strip. The understanding of these dynamics provides a very exciting theoretical and experimental challenge. Prospective numerical simulations are currently in progress. In addition to nearly one-dimensional phenomena, the geometry of the open gel reactor may prove useful in the investigations of the outstanding issue of the transition to defect mediated’25-‘29 chemical turbulence in two-dimensional reacting media.12’ The new set of recently designed open spatial reactors is likely to bring some answers to the most difficult questions concerning self-organization phenomena in chemical systems.

ACKNOWLEDGMENTS

We would like to acknowledge fruitful discussions with J. Boissonade, P. De Kepper, J. Pearson, Q. Ouyang, and H. L. Swinney. This work is supported by the Direction des Recherches Etudes et Techniques (DRET) under Contract (No. 89/196). During his visit to the Center for Nonlinear Dynamics at the University of Texas (Austin), A.A. was also supported by DOE under Contract No. DE-FGOS- 88ER1382 1. J.E. is also partly supported by the U.S. Army Research Office through the M.S.I. of Cornell University.

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modeling reaction--diffusion pattern formation in the couette flow...

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