*.. . - ..*..'..... *i..-**.**.***** ***-..*. . .... ... *, ..*

6. S. R. Marder, C. B. Gorman, B. G. Tiemann, L.-T.Cheng, Proc. SPIE1775, 19 (1993); C. B. Gormanand S. R. Marder, in preparation.

7. W. Drenth and E. H. Wiebenga, Acta Crystallogr.8, 755 (1955).

8. R. H. Baughman, B. E. Kohler, I. J. Levy, C. W.Spangler, Synth. Methods 11, 37 (1985).

9. P. Groth, Acta Chem. Scand. B 41, 547 (1987).10. F. Chentli-Bechikha, J. P. Declercq, G. Germain,

M. V. Meerssche, Cryst. Struct. Comm. 6, 421(1977).

11. L. G. S. Brooker et al., J. Am. Chem. Soc. 73, 5332(1951).

12. R. Radeglia and S. Dahne, J. Mol. Struct. 5, 399(1970).

13. S. R. Marder et al. J. Am. Chem. Soc. 115, 2525(1993).

14. S. Schneider, Ber. Buns. Ges. 80, 218 (1976).15. H. E. Schaffer, R. R. Chance, R. J. Silbey, K. Knoll,

R. R. Schrock, J. Chem. Phys. 94, 4161 (1991).16. F. Kajzar and J. Messier, Rev. Sci. Instrum. 58,

2081 (1987).17. S. R. Marder, J. W. Perry, F. L. Klavetter, R. H.

Grubbs, in Organic Materials for Nonlinear Op-tics, R. A. Hann and D. Bloor, Eds. (Royal Societyof Chemistry, London, 1989), pp. 288-294.

18. F. Kajzar, in Nonlinear Optics of Organics andSemiconductors, T. Kobayashi, Ed. (Springer-Verlag, Berlin, 1989), pp. 108-119.

19. S. H. Stevenson, D. S. Donald, G. R. Meredith, inNonlinear Optical Properties of Polymers, Mat.Res. Soc. Symp. Proc. 109 (Materials Research

Society, Pittsburgh, 1988), pp. 103-108.20. Y. Marcus, J. Soin. Chem. 20, 929 (1991).21. B. M. Pierce, Proc. SPIE 1560, 148 (1991).22. C. W. Dirk, L.-T. Cheng, M. G. Kuzyk, Int. J. Ouant.

Chem. 43, 27 (1992).23. Enhancement of y by the P2 term has been

discussed previously: A. F. Garito, J. R. Heflin, K.Y. Yong, 0. Zamani-khamiri, Proc. SPIE 971, 2(1988).

24. Increased y in donor-acceptor-conjugated or-ganics relative to centrosymmetric analogs hasbeen observed experimentally: C. W. Spangler, K.0. Havelka, M. W. Becker, T. A. Kelleher, L.-T.Cheng, Proc. SPIE1560, 139 (1991); L.-T. Chenget al., J. Phys. Chem. 95, 10631 (1991); L.-T.Cheng et al., ibid., p. 10643.

25. The research was performed, in part, at the Cen-ter for Space Microelectronics Technology, JetPropulsion Laboratory (JPL), California Institute ofTechnology, and was supported, in part, by theStrategic Defense Initiative Organization-Innova-tive Science and Technology Office, through anagreement with the National Aeronautics andSpace Administration (NASA). Support at theBeckman Institute by the Air Force Office ofScientific Research (grant F49620-92-J-01 77) isalso acknowledged. C.B.G. thanks the JPL direc-tor's office for a postdoctoral fellowship. G.B.thanks the National Research Council and NASAfor a resident research associateship at JPL.

29 January 1993; accepted 19 May 1993

Complex Patterns in a Simple System

John E. PearsonNumerical simulations of a simple reaction-diffusion model reveal a surprising variety ofirregular spatiotemporal patterns. These patterns arise in response to finite-amplitudeperturbations. Some of them resemble the steady irregular patterns recently observed inthin gel reactor experiments. Others consist of spots that grow until they reach a criticalsize, at which time they divide in two. If in some region the spots become overcrowded,all of the spots in that region decay into the uniform background.

Patterns occur in nature at scales rangingfrom the developing Drosophila embryo tothe large-scale structure of the universe. Atthe familiar mundane scales we see snow-flakes, cloud streets, and sand ripples. Wesee convective roll patterns in hydrodynamicexperiments. We see regular and almostregular patterns in the concentrations ofchemically reacting and diffusing systems(1). As a consequence of the enormousrange of scales over which pattern formationoccurs, new pattern formation phenomenonis potentially of great scientific interest. Inthis report, I describe patterns recently ob-served in numerical experiments on a simplereaction-diffusion model. These patterns areunlike any that have been previously ob-served in theoretical or numerical studies.

The system is a variant of the autocata-lytic Selkov model of glycolysis (2) and isdue to Gray and Scott (3). A variety ofspatio-temporal patterns form in response

to finite-amplitude perturbations. The re-sponse of this model to such perturbationswas previously studied in one space dimen-sion by Vastano et al. (4), who showed thatsteady spatial patterns could form evenwhen the diffusion coefficients were equal.The response of the system in one spacedimension is nontrivial and depends bothon the control parameters and on the initialperturbation. It will be shown that thepatterns that occur in two dimensions rangefrom the well-known regular hexagons toirregular steady patterns similar to thoserecently observed by Lee et al. (5) to cha-otic spatio-temporal patterns. For the ratioof diffusion coefficients used, there are nostable Turing patterns.

Most work in this field has focused onpattern formation from a spatially uniformstate that is near the transition from linearstability to linear instability. With thisrestriction, standard bifurcation-theoretictools such as amplitude equations havebeen developed and used with considerablesuccess (6). It is unclear whether the pat-

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terns presented in this report will yield tothese now-standard technologies.

The Gray-Scott model corresponds tothe following two reactions:

U + 2V-3V (1)V-P

Both reactions are irreversible, so P is aninert product. A nonequilibrium constraintis represented by a feed term for U. Both Uand V are removed by the feed process. Theresulting reaction-diffusion equations in di-mensionless units are:

au

=DUV2U- UV2+F(1 - U)at

av= DVV2V+ UV2 -(F+ k)V (2)

where k is the dimensionless rate constantof the second reaction and F is the dimen-sionless feed rate. The system size is 2.5 by2.5, and the diffusion coefficients are Du =2 x 10-5 and D, = 10-5. The boundaryconditions are periodic. Before the numer-ical results are presented, consider the be-havior of the reaction kinetics which aredescribed by the ordinary differential equa-tions that result upon dropping the diffusionterms in Eq. 2.

In the phase diagram shown in Fig. 1, atrivial steady-state solution U = 1,V = 0exists and is linearly stable for all positiveF and k. In the region bounded above bythe solid line and below by the dottedline, the system has two stable steadystates. For fixed k, the nontrivial stableuniform solution loses stability throughsaddle-node bifurcation as F is increasedthrough the upper solid line or by Hopfbifurcation to a periodic orbit as F isdecreased through the dotted line. [For adiscussion of bifurcation theory, see chap-ter 3 of (7).] In the case at hand, thebifurcating periodic solution is stable for k< 0.035 and unstable for k > 0.035.There are no periodic orbits for parametervalues outside the region enclosed by thesolid line. Outside this region the system isexcitable. The trivial state is linearly sta-ble and globally attracting. Small pertur-bations decay exponentially but larger per-turbations result in a long excursionthrough phase space before the systemreturns to the trivial state.

The simulations are forward Euler integra-tions of the finite-difference equations result-ing from discretization of the diffusion opera-tor. The spatial mesh consists of 256 by 256grid points. The time step used is 1. Spotchecks made with meshes as large as 1024 by1024 and time steps as small as 0.01 producedno qualitative difference in the results.

Initially, the entire system was placed inthe trivial state (U = 1,V = 0). The 20 by20 mesh point area located symmetrically

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about the center of the grid was thenperturbed to (U = 1/2,V = 1/4). Theseconditions were then perturbed with ± 1%random noise in order to break the squaresymmetry. The system was then integratedfor 200,000 time steps and an image wassaved. In all cases, the initial disturbancepropagated outward from the centralsquare, leaving patterns in its wake, untilthe entire grid was affected by the initialsquare perturbation. The propagation waswave-like, with the leading edge of theperturbation moving with an approximatelyconstant velocity. Depending on the param-eter values, it took on the order of 10,000 to20,000 time steps for the initial perturbationto spread over the entire grid. The propaga-tion velocity of the initial perturbation isthus on the order of 1 x 10-4 space units pertime unit. After the initial period duringwhich the perturbation spread, the systemwent into an asymptotic state that was eithertime-independent or time-dependent, de-pending on the parameter values.

Figures 2 and 3 are phase diagrams; onecan view Fig. 3 as a map and Fig. 2 as the keyto the map. The 12 patterns illustrated inFig. 2 are designated by Greek letters. Thecolor indicates the concentration of U withred representing U = 1 and blue represent-ing U ~ 0.2; yellow is intermediate to redand blue. In Fig. 3, the Greek charactersindicate the pattern found at that point in

0.3

0.2-

0.1

0.0o.00 0.02 0.04 0.06 0.08

kFlg. 1. Phase diagram of the reaction kinetics.Outside the region bounded by the solid line,there is a single spatially uniform state (calledthe trivial state) (U = 1, V = 0) that is stable forall (F, A). Inside the region bounded by the solidline, there are three spatially uniform steadystates. Above the dotted line and below thesolid line, the system is bistable: There are twolinearly stable steady states in this region. As Fis decreased through the dotted line, the non-trivial stable steady state loses stability throughHopf bifurcation. The bifurcating periodic orbitis stable for k < 0.035 and unstable for k >0.035. No periodic orbits exist for parametervalues outside the region bounded by the solidline.

parameter space. There are two additionalsymbols in Fig. 3, R and B, indicatingspatially uniform red and blue states, respec-tively. The red state corresponds to (U =1,V = 0) and the blue state depends on theexact parameter values but correspondsroughly to (U = 0.3,V = 0.25).

Pattern a is time-dependent and consistsof fledgling spirals that are constantly col-liding and annihilating each other: fullspirals never form. Pattern P is time-depen-dent and consists of what is generally called

a a

E

phase turbulence (8), which occurs in thevicinity of a Hopf bifurcation to a stableperiodic orbit. The medium is unable tosynchronize so the phase of the oscillatorsvaries as a function of position. In thepresent case, the small-amplitude periodicorbit that bifurcates is unstable. Pattern y istime-dependent. It consists primarily ofstripes but there are small localized regionsthat oscillate with a relatively high frequen-cy (-10`-). The active regions disappear,but new ones always appear elsewhere. In

'v C

-I

'I

Flg. 2. The key to the map. The patterns shown in the figure are designated by Greek letters, whichare used in Fig. 3 to indicate the pattern found at a given point in parameter space.

0. 08 . a. . . . . . . a. . a a. .. a a a

0.06

0.04

0.02

u.u

0.03 0.04 0.05k

Fig. 3. The map. The Greek lettersindicate the location in parameterspace where the patterns in Fig. 2were found; B and R indicate thatthe system evolved to uniform blueand red states, respectively.

0.06 0.07

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B R-

BB -R-Bt: ic . R

B: ic ji RB

- B...s s RRR-~~~~ R ~ ~ 1 R

e £ R

a R-. R R

. . . . . . . . . . . . . . . . . .

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Fig. 2 there is an active region near the topcenter of pattern y. Pattern 8 consists ofregular hexagons except for apparently sta-ble defects. Pattern ai is time-dependent: afew of the stripes oscillate without apparentdecay, but the remainder of the patternremains time-independent. Pattern L istime-dependent and was observed for only asingle parameter value.

Patterns 0, K, and p resemble thoseobserved by Lee et al. (5). When blue wavescollide, they stop, as do those observed byLee et al. In pattern ,±, long stripes grow inlength. The growth is parallel to the stripesand takes place at the tips. If two distinctstripes that are both growing are pointeddirecdy at each other, it is always observedthat when the growing tips reach somecritical separation distance, they alter-theircourse so as not to collide. In pattens 0 andK, the perturbations grow radially outwardwith a velocity normal to the stripes. Inthese cases if two stripes collide, they sim-ply stop, as do those observed by Lee et al.I have also observed, in one space dimen-sion, fronts propagating toward each otherthat stop when they reach a critical separa-tion. This is fundamentally new behaviorfor nonlinear waves that has recently beenobserved in other models as well (9).

Patterns £A, and Xshares ties.They consist ofblue spots on a red or yellowbackground. Pattern ) is time-independentand patterns c and g are time-dependent.Note that spots occur only in regions of

parameter space where the system is excit-able and the sole uniform steady state is thered state (U = 1,V = 0). Thus, the bluespots cannot persist for extended time un-less there is a gradient present. Because thegradient is required for the existence of thespots, they must have a maximum size orthere would be blue regions that were es-sentially gradient-free. Such regions wouldnecessarily decay to the red state. Note thatthese gradients are self-sustaining and arenot imposed externally. After the initialperturbation, the spots increase in numberuntil they fill the system. This process isvisually similar to cell division. After a spothas divided to form two spots, they moveaway from each other. During this period,each spot grows radially outward. Thegrowth is a cons ence of excitability. Asthe spots get furtier apart, they begin toelongate in the direction perpendicular totheir motion. When a critical size isachieved, the gradient is no longer sufficientto maintain he center in the blue state, sothe center decays to red, leaving two bluespots. This process is illustrated in Fig. 4.Figure 4A was made just after the initialsquare perturbation had decayed to leave thefour spots. In Fig. 4B, the spots have movedaway from each other and are beginning toelongate. In Fig. AC, the new spots aredearly visible. In Fig. 4D, the replicationprocess is complete. The subsequent evolu-tion depends on the control parameters.Pattern X remains in a steady state. Pattern ;

remains time-dependent but with long-rangespatial order except for local regions of ac-tivity. The active regions are not stationary.At any one instant, they do not appearqualitatively different from pattern C (Fig. 2)but the location of the red disturbanceschanges with time. Pattern c appears to haveno long-range order either in time or space.Once the system is filled with blue spots,they can die due to overcrowding. Thisoccurs when many spots are crowded togeth-er and the gradient over an extended regionbecomes too weak to support them. Thespots in such a region will collapse nearlysimultaneously to leave an irregular red hole.There are always more spots on the boundaryof any hole, and after a few thousand timesteps no sign of the hole will remain. Thespots on its border will have filled it. Figure5-illustrates this process.

Pattern is chaotic. The Liapunov expo-nent (which determines the rate of separa-tion of nearby trajectories) is positive. TheLiapunov time (the inverse of the Liapunovexponent) is 660 time steps, roughly equal tothe time it takes for a spot to replicate, asshown in Fig. 4. This time period is alsoabout how long it takes for a molecule todiffuse across one of the spots. The timeaverage of pattern £ is constant in space.

All of the patterns presented here arosein response to finite-amplitude p rba-tions. The ratio of diffusion coefficients usedwas 2. It is now well known that Turinginstabilities that lead to spontaneous pattern

Fig. 4 (le. Time evolution of spot multiplication. This figure was FAg 5 (right), lime evolution of pattern e. The images are 250 time unitsproduced in a 256 by 256 simulation with physical dimensions of 0.5 by apart. In the corners (which map to the same point in physical space), one0.5 and a time step of 0.01. The times t at which the figures were taken can see a yellow region in (A) to (C). It has decayed to red in (D). In (A) andare as follows: (A) t = 0; (B) t = 350; (C) t = 510; and (D) t = 650. (B), the center of the left border has a red region that is nearly filled in (D).

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formation cannot occur in systems in whichall diffusion coefficients are equal. [For acomprehensive discussion of these issues, seePearson and co-workers (10, 1 1); for a dis-cussion of Turing instabilities in the modelat hand, see Vastano et al. (12).] The onlyTuring patterns that can occur bifurcate offthe nontrivial steady uniform state (the bluestate). Most of the patterns discussed in thisreport occur for parameter values such thatthe nontrivial steady state does not exist.With the ratio of diffusion coefficients usedhere, Turing patterns occur only in a narrowparameter region in the vicinity of F = k =0.0625, where the line of saddle-node bifur-cations coalesces with the line of Hopf bi-furcations. In the vicinity of this point, thebranch of small-amplitude Turing patterns isunstable (12).

With equal diffusion coefficients, no pat-terns formed in which small asymmetries inthe initial conditions were amplified by thedynamics. This observation can probably beunderstood in terms of the following fact:Nonlinear plane waves in two dimensionscannot be destabilized by diffusion in thecase that all diffusion coefficients are equal(13). During the initial stages of the evolu-tion, the corners of the square perturbationare rounded off. The perturbation thenevolves as a radial wave, either inward oroutward depending on the parameter values.Such a wave cannot undergo spontaneoussymmetry breaking unless the diffusion coef-ficients are unequal. However, I found sym-metry breaking over a wide range of param-eter values for a ratio of diffusion coefficientsof 2. Such a ratio is physically reasonableeven for small molecules in aqueous solu-tion. Given this diffusion ratio and the widerange of parameters over which the replicat-ing spot patterns exist, it is likely that theywill soon be observed experimentally.

Recently Hasslacher et al. have demon-strated the plausibility of subcellular chem-ical patterns through lattice-gas simulationsof the Selkov model (14). The patternsdiscussed in the present article can also befound in lattice-gas simulations of theSelkov model and in simulations carried outin three space dimensions. Perhaps they arerelated to dynamical processes in the cellsuch as centrosome replication.

REFERENCES AND NOTES1. G. Nicolis and 1. Prigogine, Self-Organization in

Non-Equilibrium Systems (Wiley, New York, 1977).2. E. E. Selkov, Eur. J. Biochem. 4, 79 (1968).3. P. Gray and S. K. Scott, Chem. Eng. Sci. 38, 29

(1983); ibid. 39, 1087 (1984); J. Phys. Chem. 89,22 (1985).

4. J. A. Vastano, J. E. Pearson, W. Horsthemke, H. L.Swinney, Phys. Lett. A 124, 6 (1987). ibid., p. 7;ibid., p. 320.

5. K. J. Lee, W. D. McCormick, 0. Ouyang, H. LSwinney, Science 261, 192 (1993).

6. P. Hohenberg and M. Cross, Rev. Mod. Phys. 65,3 (1993).

192

7. J. Guckenheimer and P. Holmes, Nonlinear Oscilla-tions, Dynamical Systems, and Bifurcations of Vec-tor Fields (Springer-Verlag, Berlin, 1983), chap. 3.

8. Y. Kuramoto, Chemical Oscillations, Waves, andTurbulence (Springer-Verlag, Berlin, 1984).

9. A. Kawczynski, W. Comstock, R. Field, Physica D54, 220 (1992); A. Hagberg and E. Meron, Uni-versity of Arizona preprint.

10. J. E. Pearson and W. Horsthemke, J. Chem. Phys.90, 1588 (1989).

11. J. E. Pearson and W. J. Bruno, Chaos 2, 4 (1992);ibid., p. 513.

12. J. A. Vastano, J. E. Pearson, W. Horsthemke, H. L.Swinney, J. Chem. Phys. 88, 6175 (1988).

13. J. E. Pearson, Los Alamos Publ. LAUR 93-1758(Los Alamos National Laboratory, Los Alamos,NM, 1993).

14. B. Hasslacher, R. Kapral, A. Lawniczak, Chaos 3,1 (1993).

15. am happy to acknowledge useful conversationswith S. Ponce-Dawson, W. Horsthemke, K. Lee, L.Segel, H. Swinney, B. Reynolds, and J. Theiler.also thank the Los Alamos Advanced ComputingLaboratory for the use of the Connection Machineand A. Chapman, C. Hansen, and P. Hinker fortheir ever-cheerful assistance with the figures.7 April 1993; accepted 13 May 1993

Pattern Formation by Interacting Chemical Fronts

Kyoung J. Lee, W. D. McCormick, Qi Ouyang, Harry L. Swinney*Experiments on a bistable chemical reaction in a continuously fed thin gel layer reveal anew type of spatiotemporal pattern, one in which fronts propagate at a constant speed untilthey reach a critical separation (typically 0.4 millimeter) and stop. The resulting asymptoticstate is a highly irregular stationary pattern that contrasts with the regular patterns suchas hexagons, squares, and stripes that have been observed in many nonequilibriumsystems. The observed patterns are initiated by a finite amplitude perturbation rather thanthrough spontaneous symmetry breaking.

In recent years, pattern formation has be-come a very active area of research, moti-vated in part by the realization that thereare many common aspects of patternsformed by diverse physical, chemical, andbiological systems and by cellular automataand differential equation models. In exper-iments on a chemical system, we havediscovered a new type of pattern that differsqualitatively from the previously studiedchemical waves [rotating spirals (1)1, sta-tionary "Turing" patterns (2-4), and cha-otic patterns (5). These new patterns formonly in response to large-amplitude pertur-bations-small-amplitude perturbations de-cay. A large perturbation evolves into anirregular pattern that is stationary (time-independent) (Fig. 1). The patterns have alength scale determined by the interactionof the chemical fronts, which propagatetoward one another at constant speed untilthey reach a critical distance and stop, asFig. 2 illustrates. The growth of these frontpatterns is markedly different from Turingpatterns: The front patterns develop locallyand spread to fill space, as in crystal growth,whereas Turing patterns emerge spontane-ously everywhere when the critical value ofa control parameter is exceeded.

The front patterns are highly irregular,in contrast with Turing patterns, whichemerge as a regular array of stripes or hexa-gons (in two-dimensional systems) at the

Center for Nonlinear Dynamics and the Department ofPhysics, University of Texas at Austin, Austin, TX78712.*To whom correspondence should be addressed.

SCIENCE * VOL. 261 * 9 JULY 1993

transition from a uniform state (4). Theinteraction of fronts illustrated in Fig. 2 alsocontrasts with the behavior in excitablechemical media, where colliding fronts an-nihilate one another (1), and with solitons,where nonlinear waves pass through oneanother (6).

Our experiments have been conductedusing an iodate-ferrocyanide-sulfite reac-tion, which is known to exhibit bistabilityand large oscillations in pH in stirred flowreactors (7). The other reactions that yieldstationary chemical patterns are the well-studied chlorite-iodide-malonic acid reac-tion (3-5) and a variant reaction (8) thatuses chlorine dioxide instead of chlorite.We chose the iodate-ferrocyanide-sulfite re-action as a new candidate for studies ofpattern formation because a pH indicatorcould be used to visualize patterns thatmight form.

The following experiments illustrate thedifferences between our patterns and thosepreviously observed in reaction-diffusionsystems. A diagram of the gel disc reactor isshown in Fig. 3. Gel-filled reactors weredeveloped several years ago (9) to studyreaction-diffusion systems maintained inwell-defined states far from equilibrium.These reactors are now widely used forstudying sustained patterns that arise solelyfrom the interplay of diffusion and chemicalkinetics-the gel prevents convective mo-tion. A thin polyacrylamide gel layer (0.2mm thick, 22 mm in diameter) is feddiffusively by a continuously refreshed res-ervoir of chemicals (10). There are twothin membranes between the polyacrylam-

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