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Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
Physica D 46 (1990) 10-22 North-Holland
SPATIOTEMPORAL P A T T E R N S IN A ONE-DIMENSIONAL OPEN REACTION-DIFFUSION SYSTEM W.Y. T A M 1 and H a r r y L. S W I N N E Y Center for Nonlinear Dynamics and Department of Physics, The University of Texas, Austin, TX 78712, USA Received 8 March 1990 Accepted 17 April 1990 Communicated by R.M. Westervelt
We have developed a novel spatially extended chemical reactor that can be maintained indefinitely in a well-defined nonequilibrium state. The system is effectively one-dimensional, Reagents of the Belousov-Zhabotinsky reaction are fed at both ends, the oxidizer at one end and the reducer at the other end, but there is no net mass flux in the reactor. The system is designed so that the effective diffusion coefficient, which is the same for all species (typically 0.1 cm2/s), can be varied; spatiotemporal patterns were studied for several values of the diffusion coefficient. The following sequence of well-defined dynamical regimes was observed as the gradient in oxidizer concentration was increased with other control parameters held fixed: steady, periodic, quasiperiodic, frequency-locked, period-doubled, and chaotic. The transitions to different regimes occurred with no observable hysteresis. This is the first observation of a sequence of spatiotemporal regimes in a laboratory reaction-diffusion system. These observations are described qualitatively by a simple reaction-diffusion model with only two species.
I. Introduction Interest in spatial patterns in chemical systems has b e e n growing since Turing's [1] pioneering p a p e r " T h e chemical basis f o r m o r p h o g e n e s i s " (1952) and the observation by Zhabotinsky (1967) of spatial patterns in the B e l o u s o v - Z h a b o t i n s k y reaction [2, 3]. T h e chemical patterns arise from the interplay b e t w e e n diffusion and t h e local chemical kinetics. Until recently, experiments on chemical patterns have b e e n limited to closed systems: chemicals were p o u r e d onto a petri dish or filter paper, and the patterns that e m e r g e d then evolved inexorably towards t h e r m o d y n a m i c equilibrium [4]. This serious restriction has now b e e n lifted by the a d o p t i o n of continuously fed unstirred r e a c t o r s - C F U R s [ 5 - 7 J - w h i c h can be sustained far f r o m equilibrium indefinitely by an external feed of reagents. T h u s asymptotic spaIPresent address: Department of Physics, The University of Arizona, Tucson, AZ 85721, USA.
tiotemporal patterns can be d e t e r m i n e d for welldefined values of the control p a r a m e t e r s (reagent concentrations, feed rate, and temperature), and transitions between different patterns can be studied by varying the control parameters. W e have studied transitions in chemical patterns in a simple, well-controlled spatially ext e n d e d system [6]. T h e reactor, developed in collaboration with O u y a n g et al. [8], is a circular C o u e t t e system: the reaction occurs in an annular region f o r m e d by two concentric cylinders, the inner of which is rotating while the outer one is stationary [6, 8, 9] #1. R e a g e n t s are fed and removed at each end of the annulus; the flow rates are carefully adjusted so that there is no net mass flux in the axial direction. For sufficiently rapid rotation rates of the inner cylinder, the toroidal (Taylor) vortices that encircle the cylinder are #lGrutzner, Patrick, Pellechia and Vera [9] have studied simple acid-base and oxidation reactions in a closed Couette reactor.
0167-2789/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)
w.Y. Tam and H.L. Swinney/Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
turbulent, and their only role in the chemical system is to enhance the transport. Reagents are well mixed in the radial and azimuthal directions on a time scale short compared to the characteristic time for chemical oscillations, while the transport in the axial direction is diffusive [10]. Thus the Couette reactor serves as an effectively one-dimensional reaction-diffusion system. The reagents all have the same effective diffusion coefficient D in the axial direction, and the magnitude of D can be varied simply and reproducibly by varying the cylinder rotation rate [10, 11]; thus D can serve as a control p a r a m e t e r in studies of bifurcations. The value of D is typically 0.1 cm2/s; hence the length scale for spatial patterns is much larger than in a system with molecular diffusion, where D is typically 10 -5 cm2/s. We used a Belousov-Zhabotinsky (BZ) type reaction in our Couette reactor, but instead of using the usual substrate (malonic acid), we replaced it with a dual-substrate of glucose and acetone [12]. A major reason for choosing this substrate is that only steady and periodic states have been observed in the homogeneous system, as fig. 1 illustrates. Thus if any complex spatiotemporal behavior is observed in the extended system, it must arise from an interplay between the simple local dynamics and diffusion. The Couette reactor provides well-defined and controllable conditions with the concentrations imposed at the boundaries of the system. This reactor has been recently examined in theoretical studies by Vastano et al. [13, 14] and Arneodo and Elezgaray [15]. Our Couette reactor with feeds at the ends differs from a Turing system where the initial state is spatially uniform. A supercritical bifurcation from a uniform state to a stationary pattern (a Turing bifurcation) cannot occur in a system in which all species have the same diffusion coefficient [16], but in the Couette reactor, which has axial gradient in the feed chemicals, a supercritical bifurcation from the initial state to a stationary pattern or a timedependent pattern can occur even if all species have the same diffusion coefficient. In fact, we
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Fig. 1. Phase diagram for the glucose-acetone system in a well-stirred continuous flow tank reactor: (e) steady state; (o) periodic state. Two feeds were used in the experiment. The feed rates d 0 and a I are given in ml/h: d 0 contained 0.02 M KBrO3 and 1.0 M H2SO4; al contained 0.1 M glucose, 0.06 M acetone, 0.002 M MnSO4, and 1.0 M H2SO4. The solid line indicates the boundary between the two states and the dashed line shows the path of feed rates used in the Couette reactor. The volume of the well-stirred reactor was 10.5 ml, the stirring rate was 1800 rpm, and the temperature was regulated at 28.0°C. have observed not only the primary transition but also a complex sequence of subsequent supercritical transitions to distinct spatiotemporal regimes, and Ouyang et al. [8], using a chlorite-iodide reaction rather than the B Z reaction, have observed a bifurcation to a stationary multipeaked pattern• We will discuss the experimental system and procedures in section 2, present the experimental results in section 3, and conclude with a discussion in section 4.
2. E x p e r i m e n t a l
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2.1. Couette reactor
A schematic diagram of the Couette reactor is shown in fig. 2. The outer cylinder, m a d e of Plexiglas, was stationary, while the inner cylinder, m a d e of stainless steel with a Teflon sheath, rotated; the inner and outer cylinder radii were 1.093 and 1.270 cm, respectively, giving a radius
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ratio of 0.861. Fixed end rings established the reactor length and eliminated dead space at each end. The active length of the cylindrical annulus formed by the cylinders and by the end rings was 14.4 cm, giving an aspect ratio (ratio of annulus height to the gap between the cylinders) of 81.5 and a volume of 19.0 cm 3. The gap between the cylinders was filled with chemicals that were introduced and removed through holes on the outer cylinder at both ends, as shown in fig. 2. The feed and the removal rates were controlled with highprecision piston pumps (Pharmacia model P500). The rate of removal of chemicals by the draining pump at one end was adjusted to match precisely the feed rate of the feeding pump at that end; thus there was no net axial flow. Sixteen micro bromide-selective electrodes #2 contacted the reacting solution through holes equally spaced along the axial extent of the outer cylinder, as indicated in fig. 2. Bromide ion concentrations were monitored by recording the potential differences between the electrodes and a reference electrode located in one of the feed lines. The electrodes all gave essentially the same response when the reaction was well mixed by rapid rotation of the inner cylinder; however, the electrodes exhibited slow drifts which differed slightly for the different electrodes, and these #2We used a modified version of bromide-selective electrode described in ref. [17].
drifts prevented our making an accurate determination of the concentration as a function of position for the spatiotemporal patterns. The Couette reactor was mounted horizontally inside an oil bath regulated at 26.5 + 0.5°C. The inner cylinder was driven through a timing belt by a microstepping motor. The speed of the inner cylinder was varied from 6 to 12 Hz, which corresponds to relative Reynolds numbers R / R c (where R c is the critical Reynolds number for the onset of Taylor vortex flow) from 5.97 to 11,95. The effective diffusion coefficient depends not simply on Reynolds number but also on the Reynolds number history [10], which determines the axial and azimuthal wavelengths of the Taylor vortices; therefore, the inner cylinder was carefully accelerated from rest to the final speed using computer controlled ramping procedures and was held at a constant rotation rate during an experiment. At the lowest Reynolds number for which data were obtained ( R / R c = 5.97), the flow consists of waves on the Taylor vortices [18]. At this low Reynolds number the transport on the scale of our apparatus cannot accurately be described simply as a diffusive process - fluid near a vortex boundary is transported much more rapidly than fluid near a vortex center [19]. On the other hand, at the upper end of the Reynolds number range of the present study ( R / R ~ = 12), the flow consists of turbulent Taylor vortices, and the diffusive model provides a fairly accurate description of the transport. (At yet higher R the diffusive model is quite accurate [10].) Extrapolating from previous experiments [10, 11] and using 10 -5 cm2/s for the value of the diffusion coefficient in the limit of that the Reynolds number approaches zero, we estimate diffusion coefficient values of 0.10, 0.15, and 0.20 cmZ/s at R / R ~ = 6, 9, and 12, respectively. One source of uncertainty is the effect of the electrodes on the f l o w - i n order to increase the sensitivity, the tips of the electrodes were extended a few tenths of a millimeter into the 1.8 mm gap between the cylinders. This problem will be circumvented in our future experiments, which will use optical absorp-
W.Y. Tam and H.L. Swinney /Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
tion rather than electrodes to detect the spatiotemporal patterns.
2.2. Chemistry Due to the high sensitivity of the BZ reaction to impurities [20], extra care was taken in preparing the solutions for the experiment. The potassium bromate was recrystallized twice using the method as described in ref. [20]. Stock solutions for the glucose-acetone system were prepared by dissolving reagent grade chemicals in distilled water. The solutions were filtered (0.45 ~ m pore size) and were degassed t o remove gases left during preparation. The glucose-acetone BZ reaction has another advantage in addition to the previously mentioned lack of complex oscillations and chaos in the homogeneous system: The reaction produces an insignificant amount of gas compared to the malonic acid BZ system [12]. Bubbles are highly undesirable in the Couette reactor because they interfere with the flow state and change the diffusion coefficient.
2.3. Experimental procedures The main components of the glucose-acetone system, the oxidizer (bromate) and the reducers (glucose and acetone), were separated into two feeds: bromate was fed at a rate a 0 at one end of the reactor (z = 0) and glucose-acetone was fed at a rate a~ at the other end (z = 1); the concentrations are given in the caption of fig. 1. Bifurcation sequences were studied for cylinder rotation speeds of 6, 9, and 12 Hz. At each rotation rate the bromate feed rate ot0 was varied from 0 to 35 m l / h , while the glucose-acetone feed rate t~1 was held fixed at 10.0 m l / h . For every change in a 0, an equilibration time of 4 - 6 h was allowed for the system to reach an asymptotic state before taking data; to be certain that this was a sufficient waiting period, a wait of 24 h or more was allowed in a few cases, and the observed behavior was the same as after a 4 h
13
wait. The time dependence of the bromide ion potential at the 16 locations of the microbromide electrodes was recorded with a 1 s sampling period. Files of 4096 to 16384 data points were recorded and stored in a computer for data analysis.
Z 4. Errors The speed of the inner cylinder was controlled by a microstepping motor which had high accuracy and negligible drift ( < 0.01%). The dual piston pumps produced an average flow rate reproducible to 0.2%. The periodic switching between the pistons produced pressure pulses of a fraction of a second duration; the effect of these pulses was minimized by using an air trap and a back pressure slightly higher than atmospheric pressure. The values of a 0 corresponding to any particular bifurcation were reproducible to a few tenths of a percent for a given run, but differed by a few percent from run to run, despite the care taken in the system control and in the preparation of the chemicals. One reason for these shifts is probably a difference in the effective diffusion coefficient for different r u n s - t h e diffusion coefficient depends on the vortex size and on the number of azimuthal waves, and it is difficult to produce a flow with exactly the same number of vortices (typically 66) and the same number of waves, even when a computer-controlled ramping rate is used [10].
3. Results
Bromide ion potential time series were recorded at the 16. spatial positions for one hour or more for each value of a0; a0 was typically stepped by 0.5 m l / h between sets of data. Time series data for periodic, quasiperiodic, frequencylocked, and chaotic states are shown in figs. 3-6, respectively. The dynamical behavior was determined for each data set from power spectra, as
14
W. Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
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Time (sec) Fig. 3. Time series of bromide ion potential for a periodic state, at four positions in the Couette reactor. Note the differences in the amplitude of oscillations at the four positions. ( a 0 = 8.0 m l / h ; al = 10.0 m l / h ; rotation rate 6 Hz.)
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Time (sec) Fig. 4. Time series of bromide ion potential for a quasiperiodic state, at four positions in the Couette reactor. Note the differences in the amplitude of oscillations at the four positions. ( a 0 = 9.0 m l / h ; a I = 10.0 m l / h ; rotation rate 6 Hz.)
W. Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
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Time (see) Fig. 5. Time series of bromide ion potential for a frequency-locked state, at four positions in the Couette reactor. Note the differences in the amplitude of oscillations at the four positions. ( a 0 = 10.0 m l / h ; a I = 10.0 m l / h ; rotation rate 6 Hz.)
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T i m e (sec) Fig. 6. Time series of bromide ion potential for a chaotic state, at four positions in the Couette reactor. Note the differences in the amplitude of oscillations at the four positions. ( a 0 = 16.0 m l / h ; a 1 = 10.0 m l / h ; rotation rate 6 Hz.)
16
W.Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
illustrated in fig. 7, and from phase space portraits, as will be discussed in section 4. The bifurcation sequences observed as a function of a 0 at rotation rates of 6, 9, and 12 Hz are shown in figs. 8a-8c. D a t a were obtained with both increasing and decreasing oz0, and no hysteresis was observed. We will first discuss the sequence of bifurcations observed for a rotation rate of 6 Hz. For low a 0 the bromide concentration (i.e. bromide ion potential) varies markedly as a function of position but is time-independent. At a o = 7.5 m l / h there is a transition from the time-independent state to an oscillating state; measurements with increasing and decreasing a 0 did not reveal any hysteresis in the transition. Power spectra for this state contain a single frequency component and harmonics, as illustrated in fig. 7a. The transition from the steady state to periodic oscillations is presumably a supercritical H o p f
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Fig. 7. (a)-(d) Power spectra determined from the time series data for, respectively, periodic (fig. 3), quasiperiodic (fig. 4), frequency-locked (fig. 5) and chaotic (fig. 6) states, obtained in each case from a bromide electrode located at z= 0.2.
bifurcation, but the amplitude of oscillations beyond the onset increases too rapidly to determine (with our resolution) the dependence on a 0. Figs. 3 and 9 show that the amplitude of the oscillations is large near the z = 0 (bromate) end of the reactor, but no oscillations are discernible near the other end. Although the oscillation amplitude is below the level of detectability near the z = 1 end of the reactor, the system is presumably oscillating globally; this is indeed the case for the model [14]. At a 0 = 8.5 m l / h there is a transition from the periodic state to a quasiperiodic state. This transition, like all of the other observed transitions, occurs with no discernible hysteresis. The oscillations of the original periodic state, at a frequency we call to 0, are still dominant near the z = 0 end of the reactor, while a second frequency, to~, is dominant near the z = 1 end of the reactor; see fig. 4. it is difficult to identify the fundamental frequencies from power spectra obtained in the central portion of the reactor because these spectra contain many frequency components that are sums and differences of the two fundamental frequencies; however, it is easy to identify the fundamental frequency components by examining time series or spectra obtained near either end of the reactor, where a single frequency component dominates. The frequencies too and tol are plotted as a function of a0 in fig. 10a and the frequency ratio t o ~ / t o o is plotted in fig. lOd. In the range 8.5 < a o < 9.5 m l / h , tol/to0 apparently varies continuously with a 0, which indicates that the frequencies are incommensurate; hence the system is quasiperiodic. However, at a o = 10.0 the frequencies clearly lock together at the ratio 7: 5; the system is again periodic. In the time required for five large amplitude oscillations on the z = 0 end, there are seven large amplitude oscillations on the z = 1 end, as can be seen in fig. 5. As before, both oscillation frequencies are present throughout the reactor, but too is dominant at the z = 0 end and to1 is dominant at the other end. The frequency-locked region is apparent in the graph
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A O( 10-2 M) Fig. 8. (a)-(c) Bifurcation sequences observed as a function of a 0 for cylinder rotation rates of 6, 9, and 12 Hz, respectively. (d) Bifurcation sequence for a reaction-diffusion model as a function of bromate feed concentration A0 (for D = 0.078 cm2/s) [6, 13, 14]. The labels for the different dynamical regimes are: steady (S), periodic (P), quasiperiodic (QP), frequency-locked ( p / q is the frequency locking ratio, where p and q are integers), period-doubled (PD), and chaotic (C); the region labeled L in (d) corresponds to many different frequency-locked states, and the four long vertical lines in (d) are very narrow windows of frequency-locked states.
of tO1//tO0, fig. 10d; a s p e c t r u m o b t a i n e d in this region is shown in fig. 7c. At a 0 - - 1 1 . 5 m l / h there is a transition to chaotic behavior. T h e chaotic time series data a p p e a r to be almost periodic n e a r the ends of the reactor w h e r e one frequency c o m p o n e n t or the o t h e r dominates, but the irregular c h a r a c t e r of the time series is evident in the central portion of the reactor; see fig. 6. T h e p o w e r spectra contain b r o a d - b a n d noise, as fig. 7d illustrates, and the p h a s e portraits, to be discussed in the following section, are strange attractors.
A further increase in a0 leads to m a n y transitions to different chaotic and frequency-locked states, as can be seen in the d i a g r a m in fig. 8a. T h e frequency-locked regimes are a p p a r e n t as plateaus in the g r a p h of the frequency ratio, fig. 10d. Only the steady state is observed for a 0 > 26.0 m l / h . In one case the transition from a frequencylocked state to chaos was clearly observed to p r o c e e d through per/od doubling. This transition is illustrated in fig. 11. P r e s u m a b l y o t h e r observed transitions from periodic (frequency-locked) states
18
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to chaos also proceed through period-doubling cascades, but the parameter range is too small to detect. The bifurcation sequences observed at cylinder rotation rates of 9 and 12 Hz are shown in figs. 8b and 8c, respectively. The dependence of the frequencies of oscillation on a o is shown in fig. 10 for the three rotation frequencies; in all cases to1 > too and both too and to1 decrease with increasing a0. Steady, periodic, quasiperiodic, and chaotic regimes are observed at cylinder rotation rates of 9 and 12 Hz, as in the experiments at 6 Hz. The observations were all reproducible except for small shifts in the values of a 0 for different runs, as discussed in section 2.4.
4. Discussion Phase space portraits were constructed from t h e time series data by the time delay method using a time delay of 10 s, which is near the optimum time delay calculated by finding the minimum of the mutual information [21]. Phase portraits for periodic, quasiperiodie, frequencylocked, and chaotic states are shown in figs. 12a-12d, respectively. The information dimen-
sion (generalized dimension Dq for q = 1) of the chaotic attractor, calculated by the nearestneighbor method [22], was 2.1 for the attractor in fig. 12d. The Couette reactor has been modeled recently by Vastano et al. [6, 14]. The local chemistry of the Couette reactor was modeled by an Oregonator model with two species, bromous acid and the catalyst of the BZ reaction [23]. The bifurcation sequence found for the model (with D = 0.078 cm2/s for both species) is very similar to the observed bifurcation sequences-compare fig. 8d with figs. 8a-8c. The phase portraits for corresponding regimes of the model (figs. 12e-12h) and the experiment (figs. 12a-12d) are also remarkably similar. In the model, as in the experiment, the chaotic attractors were lowdimensional (d < 3) [14]. The spatial patterns for the model and the laboratory system, like the temporal behavior compared in figs. 8 and 12, exhibit a striking similarity. One way to represent the spatiotemporal behavior is in graphs of the locations of temporal concentration maxima, as shown in fig. 13 for the 7/5 frequency-locked state in the experiments and the model. In both cases, in the time required for 5 oscillations at the z = 0 end, there are 7 oscillations at the z = 1 end; the main difference in the patterns arises simply because the simulations have much higher sensitivity than the experiments. The similarity between the model and the experiment is to some degree fortuitous. The boundary conditions and control parameters are not really the same for the experiment and the model, but, as argued in ref. [14], these differences should be significant mainly at high flow rates (or, in the model, high concentrations); indeed, a good correspondence between model and experiment is found only at low flow rates (low concentrations in the model). Further, the twospecies model of the'chemical kinetics is a greatly over-simplified description of even the wellstudied malonic acid BZ system with a cerium catalyst, and the relation of the model kinetics to
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Fig. 10. (a)-(c) show, for rotation rates of the inner cylinder of 6, 9 and 12 Hz, respectively,the dependence of the frequencies too at z = 0 (o) and to] at z = 1 (o) on the bromate feed rate a 0, and (d)-(f) show the corresponding behavior of the frequency ratio to 1//to 0 .
our variant of the BZ reaction, a glucose-acetone system with a manganese catalyst, is unknown. Another difference between the experiment and the model arises because, as discussed in section 2.1, at low Reynolds numbers the transport in the laboratory system is not described by a diffusive process on the time and length scales of the experiment. The assumption of a diffusive process is correct at sufficiently large length and time scales [10], but work is needed to characterize the transport at low Reynolds numbers. Some measurements were m a d e at R / R c = 3, where the spatial coupling is weaker. Complex patterns were observed, but detailed measurements were
not made because of the unknown character of the transport at this low Reynolds number. In the other limit, large Reynolds number, a few measurements Showed, as expected, that the entire Couette reactor became well mixed and only simple oscillations could be found; there was little variation in the concentration with position. In conclusion, the experiments d e m o n s t r a t e that diffusive coupling of simple chemical kinetics with spatial gradients can generate complex behavior, including spatially induced chaos, in a one-dimensional reaction-diffusion system maintained away from equilibrium. The continuum system has many degrees of freedom (in princi:
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W. Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
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Fig. 11. These data illustrate period doubling of a frequencylocked state (frequency ratio 4:3). The time series in (a), obtained for a 0 = 17.5 ml/h, has three large and one small oscillation per period, while the state in (b), obtained for a 0 = 17.0 ml/h, has a period that is twice as long- successive small-amplitude peaks differ significantly in amplitude. The corresponding power spectra are shown in (c) and (d), respectively. These data were obtained at z = 0.2, where there are three large-amplitude oscillations per period; period-doubling behavior is also evident in the data for large z, where there are four large amplitude oscillations per period. (Rotation rate 6 Hz.)
pie, infinite), y e t t h e c h a o s r e m a i n s low d i m e n s i o n a l ( d < 3) t h r o u g h o u t t h e p a r a m e t e r r e g i o n s t u d i e d ; i n fact, t h e d y n a m i c s c a n b e d e s c r i b e d b y a s i m p l e f o r c e d n o n l i n e a r o s c i l l a t o r [14]. T h e o b s e r v e d s e q u e n c e o f b i f u r c a t i o n s is t h e first o b served in an experiment on a reaction-diffusion systems, b u t w i t h t h e n e w l y d e v e l o p e d C F U R s it s h o u l d n o w b e s t r a i g h t f o r w a r d to s t u d y b i f u r c a t i o n s e q u e n c e s for a v a r i e t y o f o t h e r g e o m e t r i e s a n d b o u n d a r y c o n d i t i o n s . T h e p r e s e n t experim e n t s w e r e c o n d u c t e d for a p a r t i c u l a r B Z c h e m istry, b u t it is c o n j e c t u r e d t h a t o t h e r c h e m i s t r i e s
Fig. 12. (a)-(d) These attractors were constructed from the laboratory time series data for the periodic, quasiperiodic, frequency-locked, and chaotic states shown in figs. 3-6 (with z = 0.2), respectively. (In (a) the amplitude is 10 times smaller than (b)-(d), and the spread of the trajectory is due to experimental noise.) (e)-(h) These attractors were constructed from time series for periodic, quasiperiodic, frequency-locked, and chaotic states, respectively, obtained in a numerical simulation of a two-species reaction-diffusion model [6, 13, 14]; in the model A 0 = 0.020, 0.025, 0.050, and 0.065 M, respectively, and D = 0.078 cm2/s for both species (z = 0.25). The attractors in (a) and (e) are limit cycles; (b) and (f) are 2-tori; (c) and (g) are both frequency locked at a 7 : 5 ratio; and (d) and (h) are strange attractors. For both the laboratory experiment and the model the temporal behavior is the same at all sites, but the amplitude varies markedly with position.
~E.Y. Tam and H.L. Swinney /Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
olllIIll llIIlllIlllll 0
1000
t~
0
Time (s)
600
Fig. 13. Location of local temporal maxima of a 7/5 frequency-locked state deduced from (a) experiments for a 0 = 10.0 m l / h (rotation rate, 6 Hz) and (b) the reaction-diffusion model for A 0 = 0.050 M. These diagrams and similar ones for the other states show that the oscillations are neither traveling waves nor standing waves-the phase varies slightly as a function of z, but the variation is not monotonic. (The small peak near z = 0 in the experimental data is an artifact due to drift in an electrode; see section 2.1.)
in a one-dimensional reactor with opposing oxidation and reduction gradients will yield similar sequences.
Acknowledgements We thank T. Russo, K. Pope, and E. Kostelich for assistance. This research is supported by BP Venture Research and was conducted in collaboration with P. DeKepper, J.C. Roux, Qi Ouyang, and J. Boissonade of the Centre de Recherche Paul Pascal, Bordeaux.
References [1] A.M. Turing, Phil. Trans. R. Soc. B 237 (1952) 37. [2] A.M. Zhabotinskii, in: Oscillatory Processes in Biological and Chemical Systems, ed. G.M. Frank (Nauka, Moscow, 1967).
21
[3] B.P. Belousov, in: Sbornik Referatov po Radiatsionni Meditsine (Medgiz, Moscow, 1958); A.M. Zhabotinskii, Biofizika 9 (1964) 306. [4l J. Ross, S.C. Miiller and C. Vidal, Science 240 (1988) 460; K.I. Agladze, A.V. Panfilov and A.N. Rudenko, Physica D 29 (1988) 409; C. Vidal, J. Stat. Phys. 48 (1987) 1017; S.C. Muller, T. Plesser and B. Hess, Science 230 (1985) 661; B.J. Welsh, J. Gomatan and A.E. Burgess, Nature 304 (1983) 611; A.T. Winfree, Science 175 (1972) 634. [4] F. Argoul, A. Arneodo, P. Richetti, J.C. Roux and H.L. Swinney, Accounts Chem. Res. 20 (1987) 436. [5] Z. Noszticzius, W. Horsthemke, W.D. McCormick, H.L. Swinney and W.Y. Tam, Nature 329 (1987) 619; W.Y. Tam, W. Horsthemke, Z. Noszticzius and H.L. Swinney, J. Chem. Phys. 88 (1988) 3395. [6] W.Y. Tam, J.A. Vastano, H.L. Swinney and W. Horsthemke, Phys. Rev. Lett. 61 (1988) 2163. [7] N. Kreisberg, W.D. McCormick and H.L. Swinney, J. Chem. Phys. 91 (1989) 6532. [8] Q. Ouyang, J. Boissonade, J.C. Roux and P. DeKepper, Phys. Lett. A 134 (1989) 282; Q. Ouyang, Ph.D. Thesis, Universit6 de Bordeaux (1989). [9] J.B. Grutzner, E.A. Patrick, P.J. Pellechia and M. Vera, J. Am. Chem. Soc. 110 (1988) 726. [10] W.Y. Tam and H.L. Swinney, Phys. Rev. A 36 (1987) 1374. [11] Y. Enokida, K. Nakata and A. Suzuki, AIChE J. 35 (1989) 1211. [12] Q. Ouyang, W.Y. Tam, P. DeKepper, W.D. McCormick, Z. Noszticzius and H.L. Swinney, J. Phys. Chem. 91 (1987) 2181. [13] J.A. Vastano, Ph.D. Dissertation, The University of Texas at Austin (1988). [14] J.A. Vastano, T. Russo and H.L. Swinney, Bifurcation to spatiotemporal chaos in a reaction-diffusion system, Physica D 46 (1990) 23. [15] A. Arneodo and J. Elezgaray, in: Spatial Inhomogeneities and Transient Behavior in Chemical Kinetics, eds. P. Gray, G. Nicolis, F. Baras, P. Borckmans and S.K. Scott (Manchester Univ. Press, Manchester, 1990); in: New Trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibrium, eds. P. Coullet and P. Huerre (Plenum, New York, 1989); Physica D, to appear. [16] J.A. Vastano, J.E. Pearson, W. Horsthemke and H.L. Swinney, Phys. Lett. A 124 (1987) 320; J. Chem. Phys. 88 (1988) 6175; J.E. Pearson and W. Horsthemke, J. Chem. Phys. 90 (1989) 1588; P.B. Monk and H.G. Othmer, Wave propagation in aggregation fields of the cellular slim mold Dictyostelium discoideum, Phil. Trans. R. Soc. London, submitted for publication.
22
W.Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
[17] Z. Noszticzius, M. Wittmann and P. Stirling, 4th Symposium of Ion-Selective Electrodes, Matrafured 38 (1984). [18] M. Gorman and H.L. Swinney, J. Fluid Mech. 117 (1982) 123. [19] F. Sagues and W. Horsthemke, Phys. Rev. A 34 (1986) 4136; B.I. Shraiman, Phys. Rev. A 36 (1987) 261; M.N. Rosenbluth, H.L. Berk, I. Doxas and W. Horton, Phys. Fluids 30 (1987) 2636. [20] Z. Noszticzius, W.D. McCormick and H.L. Swinney,
J. Phys. Chem. 91 (1987) 5129; J. Phys. Chem. 93 (1989) 2796. [21] A.M. Fraser and H.L. Swinney, Phys. Rev. A 33 (1986) 1134; A.M. Fraser, IEEE Trans. Information Theory 35 (1989) 245. [22] R. Badii and A. Politi, J. Stat. Phys. 40 (1985) 725; E.J. Kostelich and H.L. Swinney, Physica Scripta 40 (1989) 436. [23] J.J. Tyson and P.C. Fife, J. Chem. Phys. 73 (1980) 2224.
SPATIOTEMPORAL P A T T E R N S IN A ONE-DIMENSIONAL OPEN REACTION-DIFFUSION SYSTEM W.Y. T A M 1 and H a r r y L. S W I N N E Y Center for Nonlinear Dynamics and Department of Physics, The University of Texas, Austin, TX 78712, USA Received 8 March 1990 Accepted 17 April 1990 Communicated by R.M. Westervelt
We have developed a novel spatially extended chemical reactor that can be maintained indefinitely in a well-defined nonequilibrium state. The system is effectively one-dimensional, Reagents of the Belousov-Zhabotinsky reaction are fed at both ends, the oxidizer at one end and the reducer at the other end, but there is no net mass flux in the reactor. The system is designed so that the effective diffusion coefficient, which is the same for all species (typically 0.1 cm2/s), can be varied; spatiotemporal patterns were studied for several values of the diffusion coefficient. The following sequence of well-defined dynamical regimes was observed as the gradient in oxidizer concentration was increased with other control parameters held fixed: steady, periodic, quasiperiodic, frequency-locked, period-doubled, and chaotic. The transitions to different regimes occurred with no observable hysteresis. This is the first observation of a sequence of spatiotemporal regimes in a laboratory reaction-diffusion system. These observations are described qualitatively by a simple reaction-diffusion model with only two species.
I. Introduction Interest in spatial patterns in chemical systems has b e e n growing since Turing's [1] pioneering p a p e r " T h e chemical basis f o r m o r p h o g e n e s i s " (1952) and the observation by Zhabotinsky (1967) of spatial patterns in the B e l o u s o v - Z h a b o t i n s k y reaction [2, 3]. T h e chemical patterns arise from the interplay b e t w e e n diffusion and t h e local chemical kinetics. Until recently, experiments on chemical patterns have b e e n limited to closed systems: chemicals were p o u r e d onto a petri dish or filter paper, and the patterns that e m e r g e d then evolved inexorably towards t h e r m o d y n a m i c equilibrium [4]. This serious restriction has now b e e n lifted by the a d o p t i o n of continuously fed unstirred r e a c t o r s - C F U R s [ 5 - 7 J - w h i c h can be sustained far f r o m equilibrium indefinitely by an external feed of reagents. T h u s asymptotic spaIPresent address: Department of Physics, The University of Arizona, Tucson, AZ 85721, USA.
tiotemporal patterns can be d e t e r m i n e d for welldefined values of the control p a r a m e t e r s (reagent concentrations, feed rate, and temperature), and transitions between different patterns can be studied by varying the control parameters. W e have studied transitions in chemical patterns in a simple, well-controlled spatially ext e n d e d system [6]. T h e reactor, developed in collaboration with O u y a n g et al. [8], is a circular C o u e t t e system: the reaction occurs in an annular region f o r m e d by two concentric cylinders, the inner of which is rotating while the outer one is stationary [6, 8, 9] #1. R e a g e n t s are fed and removed at each end of the annulus; the flow rates are carefully adjusted so that there is no net mass flux in the axial direction. For sufficiently rapid rotation rates of the inner cylinder, the toroidal (Taylor) vortices that encircle the cylinder are #lGrutzner, Patrick, Pellechia and Vera [9] have studied simple acid-base and oxidation reactions in a closed Couette reactor.
0167-2789/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)
w.Y. Tam and H.L. Swinney/Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
turbulent, and their only role in the chemical system is to enhance the transport. Reagents are well mixed in the radial and azimuthal directions on a time scale short compared to the characteristic time for chemical oscillations, while the transport in the axial direction is diffusive [10]. Thus the Couette reactor serves as an effectively one-dimensional reaction-diffusion system. The reagents all have the same effective diffusion coefficient D in the axial direction, and the magnitude of D can be varied simply and reproducibly by varying the cylinder rotation rate [10, 11]; thus D can serve as a control p a r a m e t e r in studies of bifurcations. The value of D is typically 0.1 cm2/s; hence the length scale for spatial patterns is much larger than in a system with molecular diffusion, where D is typically 10 -5 cm2/s. We used a Belousov-Zhabotinsky (BZ) type reaction in our Couette reactor, but instead of using the usual substrate (malonic acid), we replaced it with a dual-substrate of glucose and acetone [12]. A major reason for choosing this substrate is that only steady and periodic states have been observed in the homogeneous system, as fig. 1 illustrates. Thus if any complex spatiotemporal behavior is observed in the extended system, it must arise from an interplay between the simple local dynamics and diffusion. The Couette reactor provides well-defined and controllable conditions with the concentrations imposed at the boundaries of the system. This reactor has been recently examined in theoretical studies by Vastano et al. [13, 14] and Arneodo and Elezgaray [15]. Our Couette reactor with feeds at the ends differs from a Turing system where the initial state is spatially uniform. A supercritical bifurcation from a uniform state to a stationary pattern (a Turing bifurcation) cannot occur in a system in which all species have the same diffusion coefficient [16], but in the Couette reactor, which has axial gradient in the feed chemicals, a supercritical bifurcation from the initial state to a stationary pattern or a timedependent pattern can occur even if all species have the same diffusion coefficient. In fact, we
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Fig. 1. Phase diagram for the glucose-acetone system in a well-stirred continuous flow tank reactor: (e) steady state; (o) periodic state. Two feeds were used in the experiment. The feed rates d 0 and a I are given in ml/h: d 0 contained 0.02 M KBrO3 and 1.0 M H2SO4; al contained 0.1 M glucose, 0.06 M acetone, 0.002 M MnSO4, and 1.0 M H2SO4. The solid line indicates the boundary between the two states and the dashed line shows the path of feed rates used in the Couette reactor. The volume of the well-stirred reactor was 10.5 ml, the stirring rate was 1800 rpm, and the temperature was regulated at 28.0°C. have observed not only the primary transition but also a complex sequence of subsequent supercritical transitions to distinct spatiotemporal regimes, and Ouyang et al. [8], using a chlorite-iodide reaction rather than the B Z reaction, have observed a bifurcation to a stationary multipeaked pattern• We will discuss the experimental system and procedures in section 2, present the experimental results in section 3, and conclude with a discussion in section 4.
2. E x p e r i m e n t a l
methods
2.1. Couette reactor
A schematic diagram of the Couette reactor is shown in fig. 2. The outer cylinder, m a d e of Plexiglas, was stationary, while the inner cylinder, m a d e of stainless steel with a Teflon sheath, rotated; the inner and outer cylinder radii were 1.093 and 1.270 cm, respectively, giving a radius
1~.Y. Tam and H.L. Swinney /Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
12
i
I
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iJ 5 1 Fig. 2. The Couette reactor. The reagent concentrations in the feeds a0 and a I at z = 0 and z = 1, respectively, are given in the caption for fig. 1; a 0 was varied from 0 to 35 m l / h while a I was kept fixed at 10.0 m l / h . The vertical lines show the locations of the 16 evenly spaced electrodes.
ratio of 0.861. Fixed end rings established the reactor length and eliminated dead space at each end. The active length of the cylindrical annulus formed by the cylinders and by the end rings was 14.4 cm, giving an aspect ratio (ratio of annulus height to the gap between the cylinders) of 81.5 and a volume of 19.0 cm 3. The gap between the cylinders was filled with chemicals that were introduced and removed through holes on the outer cylinder at both ends, as shown in fig. 2. The feed and the removal rates were controlled with highprecision piston pumps (Pharmacia model P500). The rate of removal of chemicals by the draining pump at one end was adjusted to match precisely the feed rate of the feeding pump at that end; thus there was no net axial flow. Sixteen micro bromide-selective electrodes #2 contacted the reacting solution through holes equally spaced along the axial extent of the outer cylinder, as indicated in fig. 2. Bromide ion concentrations were monitored by recording the potential differences between the electrodes and a reference electrode located in one of the feed lines. The electrodes all gave essentially the same response when the reaction was well mixed by rapid rotation of the inner cylinder; however, the electrodes exhibited slow drifts which differed slightly for the different electrodes, and these #2We used a modified version of bromide-selective electrode described in ref. [17].
drifts prevented our making an accurate determination of the concentration as a function of position for the spatiotemporal patterns. The Couette reactor was mounted horizontally inside an oil bath regulated at 26.5 + 0.5°C. The inner cylinder was driven through a timing belt by a microstepping motor. The speed of the inner cylinder was varied from 6 to 12 Hz, which corresponds to relative Reynolds numbers R / R c (where R c is the critical Reynolds number for the onset of Taylor vortex flow) from 5.97 to 11,95. The effective diffusion coefficient depends not simply on Reynolds number but also on the Reynolds number history [10], which determines the axial and azimuthal wavelengths of the Taylor vortices; therefore, the inner cylinder was carefully accelerated from rest to the final speed using computer controlled ramping procedures and was held at a constant rotation rate during an experiment. At the lowest Reynolds number for which data were obtained ( R / R c = 5.97), the flow consists of waves on the Taylor vortices [18]. At this low Reynolds number the transport on the scale of our apparatus cannot accurately be described simply as a diffusive process - fluid near a vortex boundary is transported much more rapidly than fluid near a vortex center [19]. On the other hand, at the upper end of the Reynolds number range of the present study ( R / R ~ = 12), the flow consists of turbulent Taylor vortices, and the diffusive model provides a fairly accurate description of the transport. (At yet higher R the diffusive model is quite accurate [10].) Extrapolating from previous experiments [10, 11] and using 10 -5 cm2/s for the value of the diffusion coefficient in the limit of that the Reynolds number approaches zero, we estimate diffusion coefficient values of 0.10, 0.15, and 0.20 cmZ/s at R / R ~ = 6, 9, and 12, respectively. One source of uncertainty is the effect of the electrodes on the f l o w - i n order to increase the sensitivity, the tips of the electrodes were extended a few tenths of a millimeter into the 1.8 mm gap between the cylinders. This problem will be circumvented in our future experiments, which will use optical absorp-
W.Y. Tam and H.L. Swinney /Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
tion rather than electrodes to detect the spatiotemporal patterns.
2.2. Chemistry Due to the high sensitivity of the BZ reaction to impurities [20], extra care was taken in preparing the solutions for the experiment. The potassium bromate was recrystallized twice using the method as described in ref. [20]. Stock solutions for the glucose-acetone system were prepared by dissolving reagent grade chemicals in distilled water. The solutions were filtered (0.45 ~ m pore size) and were degassed t o remove gases left during preparation. The glucose-acetone BZ reaction has another advantage in addition to the previously mentioned lack of complex oscillations and chaos in the homogeneous system: The reaction produces an insignificant amount of gas compared to the malonic acid BZ system [12]. Bubbles are highly undesirable in the Couette reactor because they interfere with the flow state and change the diffusion coefficient.
2.3. Experimental procedures The main components of the glucose-acetone system, the oxidizer (bromate) and the reducers (glucose and acetone), were separated into two feeds: bromate was fed at a rate a 0 at one end of the reactor (z = 0) and glucose-acetone was fed at a rate a~ at the other end (z = 1); the concentrations are given in the caption of fig. 1. Bifurcation sequences were studied for cylinder rotation speeds of 6, 9, and 12 Hz. At each rotation rate the bromate feed rate ot0 was varied from 0 to 35 m l / h , while the glucose-acetone feed rate t~1 was held fixed at 10.0 m l / h . For every change in a 0, an equilibration time of 4 - 6 h was allowed for the system to reach an asymptotic state before taking data; to be certain that this was a sufficient waiting period, a wait of 24 h or more was allowed in a few cases, and the observed behavior was the same as after a 4 h
13
wait. The time dependence of the bromide ion potential at the 16 locations of the microbromide electrodes was recorded with a 1 s sampling period. Files of 4096 to 16384 data points were recorded and stored in a computer for data analysis.
Z 4. Errors The speed of the inner cylinder was controlled by a microstepping motor which had high accuracy and negligible drift ( < 0.01%). The dual piston pumps produced an average flow rate reproducible to 0.2%. The periodic switching between the pistons produced pressure pulses of a fraction of a second duration; the effect of these pulses was minimized by using an air trap and a back pressure slightly higher than atmospheric pressure. The values of a 0 corresponding to any particular bifurcation were reproducible to a few tenths of a percent for a given run, but differed by a few percent from run to run, despite the care taken in the system control and in the preparation of the chemicals. One reason for these shifts is probably a difference in the effective diffusion coefficient for different r u n s - t h e diffusion coefficient depends on the vortex size and on the number of azimuthal waves, and it is difficult to produce a flow with exactly the same number of vortices (typically 66) and the same number of waves, even when a computer-controlled ramping rate is used [10].
3. Results
Bromide ion potential time series were recorded at the 16. spatial positions for one hour or more for each value of a0; a0 was typically stepped by 0.5 m l / h between sets of data. Time series data for periodic, quasiperiodic, frequencylocked, and chaotic states are shown in figs. 3-6, respectively. The dynamical behavior was determined for each data set from power spectra, as
14
W. Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
265
250 o~
285 .d
z=0.4
'~-. e.. 283 ~
285
o
z=0.6
"~ 283 289
z=0.8 286 0
300
600
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Time (sec) Fig. 3. Time series of bromide ion potential for a periodic state, at four positions in the Couette reactor. Note the differences in the amplitude of oscillations at the four positions. ( a 0 = 8.0 m l / h ; al = 10.0 m l / h ; rotation rate 6 Hz.)
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Time (sec) Fig. 4. Time series of bromide ion potential for a quasiperiodic state, at four positions in the Couette reactor. Note the differences in the amplitude of oscillations at the four positions. ( a 0 = 9.0 m l / h ; a I = 10.0 m l / h ; rotation rate 6 Hz.)
W. Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
15
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Time (see) Fig. 5. Time series of bromide ion potential for a frequency-locked state, at four positions in the Couette reactor. Note the differences in the amplitude of oscillations at the four positions. ( a 0 = 10.0 m l / h ; a I = 10.0 m l / h ; rotation rate 6 Hz.)
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T i m e (sec) Fig. 6. Time series of bromide ion potential for a chaotic state, at four positions in the Couette reactor. Note the differences in the amplitude of oscillations at the four positions. ( a 0 = 16.0 m l / h ; a 1 = 10.0 m l / h ; rotation rate 6 Hz.)
16
W.Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
illustrated in fig. 7, and from phase space portraits, as will be discussed in section 4. The bifurcation sequences observed as a function of a 0 at rotation rates of 6, 9, and 12 Hz are shown in figs. 8a-8c. D a t a were obtained with both increasing and decreasing oz0, and no hysteresis was observed. We will first discuss the sequence of bifurcations observed for a rotation rate of 6 Hz. For low a 0 the bromide concentration (i.e. bromide ion potential) varies markedly as a function of position but is time-independent. At a o = 7.5 m l / h there is a transition from the time-independent state to an oscillating state; measurements with increasing and decreasing a 0 did not reveal any hysteresis in the transition. Power spectra for this state contain a single frequency component and harmonics, as illustrated in fig. 7a. The transition from the steady state to periodic oscillations is presumably a supercritical H o p f
0
0.05 Frequency ff.Lz)
O.1-
Fig. 7. (a)-(d) Power spectra determined from the time series data for, respectively, periodic (fig. 3), quasiperiodic (fig. 4), frequency-locked (fig. 5) and chaotic (fig. 6) states, obtained in each case from a bromide electrode located at z= 0.2.
bifurcation, but the amplitude of oscillations beyond the onset increases too rapidly to determine (with our resolution) the dependence on a 0. Figs. 3 and 9 show that the amplitude of the oscillations is large near the z = 0 (bromate) end of the reactor, but no oscillations are discernible near the other end. Although the oscillation amplitude is below the level of detectability near the z = 1 end of the reactor, the system is presumably oscillating globally; this is indeed the case for the model [14]. At a 0 = 8.5 m l / h there is a transition from the periodic state to a quasiperiodic state. This transition, like all of the other observed transitions, occurs with no discernible hysteresis. The oscillations of the original periodic state, at a frequency we call to 0, are still dominant near the z = 0 end of the reactor, while a second frequency, to~, is dominant near the z = 1 end of the reactor; see fig. 4. it is difficult to identify the fundamental frequencies from power spectra obtained in the central portion of the reactor because these spectra contain many frequency components that are sums and differences of the two fundamental frequencies; however, it is easy to identify the fundamental frequency components by examining time series or spectra obtained near either end of the reactor, where a single frequency component dominates. The frequencies too and tol are plotted as a function of a0 in fig. 10a and the frequency ratio t o ~ / t o o is plotted in fig. lOd. In the range 8.5 < a o < 9.5 m l / h , tol/to0 apparently varies continuously with a 0, which indicates that the frequencies are incommensurate; hence the system is quasiperiodic. However, at a o = 10.0 the frequencies clearly lock together at the ratio 7: 5; the system is again periodic. In the time required for five large amplitude oscillations on the z = 0 end, there are seven large amplitude oscillations on the z = 1 end, as can be seen in fig. 5. As before, both oscillation frequencies are present throughout the reactor, but too is dominant at the z = 0 end and to1 is dominant at the other end. The frequency-locked region is apparent in the graph
W. Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
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10
A O( 10-2 M) Fig. 8. (a)-(c) Bifurcation sequences observed as a function of a 0 for cylinder rotation rates of 6, 9, and 12 Hz, respectively. (d) Bifurcation sequence for a reaction-diffusion model as a function of bromate feed concentration A0 (for D = 0.078 cm2/s) [6, 13, 14]. The labels for the different dynamical regimes are: steady (S), periodic (P), quasiperiodic (QP), frequency-locked ( p / q is the frequency locking ratio, where p and q are integers), period-doubled (PD), and chaotic (C); the region labeled L in (d) corresponds to many different frequency-locked states, and the four long vertical lines in (d) are very narrow windows of frequency-locked states.
of tO1//tO0, fig. 10d; a s p e c t r u m o b t a i n e d in this region is shown in fig. 7c. At a 0 - - 1 1 . 5 m l / h there is a transition to chaotic behavior. T h e chaotic time series data a p p e a r to be almost periodic n e a r the ends of the reactor w h e r e one frequency c o m p o n e n t or the o t h e r dominates, but the irregular c h a r a c t e r of the time series is evident in the central portion of the reactor; see fig. 6. T h e p o w e r spectra contain b r o a d - b a n d noise, as fig. 7d illustrates, and the p h a s e portraits, to be discussed in the following section, are strange attractors.
A further increase in a0 leads to m a n y transitions to different chaotic and frequency-locked states, as can be seen in the d i a g r a m in fig. 8a. T h e frequency-locked regimes are a p p a r e n t as plateaus in the g r a p h of the frequency ratio, fig. 10d. Only the steady state is observed for a 0 > 26.0 m l / h . In one case the transition from a frequencylocked state to chaos was clearly observed to p r o c e e d through per/od doubling. This transition is illustrated in fig. 11. P r e s u m a b l y o t h e r observed transitions from periodic (frequency-locked) states
18
W.Y. Tam and H.L. Swinney /Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
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to chaos also proceed through period-doubling cascades, but the parameter range is too small to detect. The bifurcation sequences observed at cylinder rotation rates of 9 and 12 Hz are shown in figs. 8b and 8c, respectively. The dependence of the frequencies of oscillation on a o is shown in fig. 10 for the three rotation frequencies; in all cases to1 > too and both too and to1 decrease with increasing a0. Steady, periodic, quasiperiodic, and chaotic regimes are observed at cylinder rotation rates of 9 and 12 Hz, as in the experiments at 6 Hz. The observations were all reproducible except for small shifts in the values of a 0 for different runs, as discussed in section 2.4.
4. Discussion Phase space portraits were constructed from t h e time series data by the time delay method using a time delay of 10 s, which is near the optimum time delay calculated by finding the minimum of the mutual information [21]. Phase portraits for periodic, quasiperiodie, frequencylocked, and chaotic states are shown in figs. 12a-12d, respectively. The information dimen-
sion (generalized dimension Dq for q = 1) of the chaotic attractor, calculated by the nearestneighbor method [22], was 2.1 for the attractor in fig. 12d. The Couette reactor has been modeled recently by Vastano et al. [6, 14]. The local chemistry of the Couette reactor was modeled by an Oregonator model with two species, bromous acid and the catalyst of the BZ reaction [23]. The bifurcation sequence found for the model (with D = 0.078 cm2/s for both species) is very similar to the observed bifurcation sequences-compare fig. 8d with figs. 8a-8c. The phase portraits for corresponding regimes of the model (figs. 12e-12h) and the experiment (figs. 12a-12d) are also remarkably similar. In the model, as in the experiment, the chaotic attractors were lowdimensional (d < 3) [14]. The spatial patterns for the model and the laboratory system, like the temporal behavior compared in figs. 8 and 12, exhibit a striking similarity. One way to represent the spatiotemporal behavior is in graphs of the locations of temporal concentration maxima, as shown in fig. 13 for the 7/5 frequency-locked state in the experiments and the model. In both cases, in the time required for 5 oscillations at the z = 0 end, there are 7 oscillations at the z = 1 end; the main difference in the patterns arises simply because the simulations have much higher sensitivity than the experiments. The similarity between the model and the experiment is to some degree fortuitous. The boundary conditions and control parameters are not really the same for the experiment and the model, but, as argued in ref. [14], these differences should be significant mainly at high flow rates (or, in the model, high concentrations); indeed, a good correspondence between model and experiment is found only at low flow rates (low concentrations in the model). Further, the twospecies model of the'chemical kinetics is a greatly over-simplified description of even the wellstudied malonic acid BZ system with a cerium catalyst, and the relation of the model kinetics to
W.Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system i
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Fig. 10. (a)-(c) show, for rotation rates of the inner cylinder of 6, 9 and 12 Hz, respectively,the dependence of the frequencies too at z = 0 (o) and to] at z = 1 (o) on the bromate feed rate a 0, and (d)-(f) show the corresponding behavior of the frequency ratio to 1//to 0 .
our variant of the BZ reaction, a glucose-acetone system with a manganese catalyst, is unknown. Another difference between the experiment and the model arises because, as discussed in section 2.1, at low Reynolds numbers the transport in the laboratory system is not described by a diffusive process on the time and length scales of the experiment. The assumption of a diffusive process is correct at sufficiently large length and time scales [10], but work is needed to characterize the transport at low Reynolds numbers. Some measurements were m a d e at R / R c = 3, where the spatial coupling is weaker. Complex patterns were observed, but detailed measurements were
not made because of the unknown character of the transport at this low Reynolds number. In the other limit, large Reynolds number, a few measurements Showed, as expected, that the entire Couette reactor became well mixed and only simple oscillations could be found; there was little variation in the concentration with position. In conclusion, the experiments d e m o n s t r a t e that diffusive coupling of simple chemical kinetics with spatial gradients can generate complex behavior, including spatially induced chaos, in a one-dimensional reaction-diffusion system maintained away from equilibrium. The continuum system has many degrees of freedom (in princi:
20
W. Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
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Fig. 11. These data illustrate period doubling of a frequencylocked state (frequency ratio 4:3). The time series in (a), obtained for a 0 = 17.5 ml/h, has three large and one small oscillation per period, while the state in (b), obtained for a 0 = 17.0 ml/h, has a period that is twice as long- successive small-amplitude peaks differ significantly in amplitude. The corresponding power spectra are shown in (c) and (d), respectively. These data were obtained at z = 0.2, where there are three large-amplitude oscillations per period; period-doubling behavior is also evident in the data for large z, where there are four large amplitude oscillations per period. (Rotation rate 6 Hz.)
pie, infinite), y e t t h e c h a o s r e m a i n s low d i m e n s i o n a l ( d < 3) t h r o u g h o u t t h e p a r a m e t e r r e g i o n s t u d i e d ; i n fact, t h e d y n a m i c s c a n b e d e s c r i b e d b y a s i m p l e f o r c e d n o n l i n e a r o s c i l l a t o r [14]. T h e o b s e r v e d s e q u e n c e o f b i f u r c a t i o n s is t h e first o b served in an experiment on a reaction-diffusion systems, b u t w i t h t h e n e w l y d e v e l o p e d C F U R s it s h o u l d n o w b e s t r a i g h t f o r w a r d to s t u d y b i f u r c a t i o n s e q u e n c e s for a v a r i e t y o f o t h e r g e o m e t r i e s a n d b o u n d a r y c o n d i t i o n s . T h e p r e s e n t experim e n t s w e r e c o n d u c t e d for a p a r t i c u l a r B Z c h e m istry, b u t it is c o n j e c t u r e d t h a t o t h e r c h e m i s t r i e s
Fig. 12. (a)-(d) These attractors were constructed from the laboratory time series data for the periodic, quasiperiodic, frequency-locked, and chaotic states shown in figs. 3-6 (with z = 0.2), respectively. (In (a) the amplitude is 10 times smaller than (b)-(d), and the spread of the trajectory is due to experimental noise.) (e)-(h) These attractors were constructed from time series for periodic, quasiperiodic, frequency-locked, and chaotic states, respectively, obtained in a numerical simulation of a two-species reaction-diffusion model [6, 13, 14]; in the model A 0 = 0.020, 0.025, 0.050, and 0.065 M, respectively, and D = 0.078 cm2/s for both species (z = 0.25). The attractors in (a) and (e) are limit cycles; (b) and (f) are 2-tori; (c) and (g) are both frequency locked at a 7 : 5 ratio; and (d) and (h) are strange attractors. For both the laboratory experiment and the model the temporal behavior is the same at all sites, but the amplitude varies markedly with position.
~E.Y. Tam and H.L. Swinney /Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
olllIIll llIIlllIlllll 0
1000
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600
Fig. 13. Location of local temporal maxima of a 7/5 frequency-locked state deduced from (a) experiments for a 0 = 10.0 m l / h (rotation rate, 6 Hz) and (b) the reaction-diffusion model for A 0 = 0.050 M. These diagrams and similar ones for the other states show that the oscillations are neither traveling waves nor standing waves-the phase varies slightly as a function of z, but the variation is not monotonic. (The small peak near z = 0 in the experimental data is an artifact due to drift in an electrode; see section 2.1.)
in a one-dimensional reactor with opposing oxidation and reduction gradients will yield similar sequences.
Acknowledgements We thank T. Russo, K. Pope, and E. Kostelich for assistance. This research is supported by BP Venture Research and was conducted in collaboration with P. DeKepper, J.C. Roux, Qi Ouyang, and J. Boissonade of the Centre de Recherche Paul Pascal, Bordeaux.
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[3] B.P. Belousov, in: Sbornik Referatov po Radiatsionni Meditsine (Medgiz, Moscow, 1958); A.M. Zhabotinskii, Biofizika 9 (1964) 306. [4l J. Ross, S.C. Miiller and C. Vidal, Science 240 (1988) 460; K.I. Agladze, A.V. Panfilov and A.N. Rudenko, Physica D 29 (1988) 409; C. Vidal, J. Stat. Phys. 48 (1987) 1017; S.C. Muller, T. Plesser and B. Hess, Science 230 (1985) 661; B.J. Welsh, J. Gomatan and A.E. Burgess, Nature 304 (1983) 611; A.T. Winfree, Science 175 (1972) 634. [4] F. Argoul, A. Arneodo, P. Richetti, J.C. Roux and H.L. Swinney, Accounts Chem. Res. 20 (1987) 436. [5] Z. Noszticzius, W. Horsthemke, W.D. McCormick, H.L. Swinney and W.Y. Tam, Nature 329 (1987) 619; W.Y. Tam, W. Horsthemke, Z. Noszticzius and H.L. Swinney, J. Chem. Phys. 88 (1988) 3395. [6] W.Y. Tam, J.A. Vastano, H.L. Swinney and W. Horsthemke, Phys. Rev. Lett. 61 (1988) 2163. [7] N. Kreisberg, W.D. McCormick and H.L. Swinney, J. Chem. Phys. 91 (1989) 6532. [8] Q. Ouyang, J. Boissonade, J.C. Roux and P. DeKepper, Phys. Lett. A 134 (1989) 282; Q. Ouyang, Ph.D. Thesis, Universit6 de Bordeaux (1989). [9] J.B. Grutzner, E.A. Patrick, P.J. Pellechia and M. Vera, J. Am. Chem. Soc. 110 (1988) 726. [10] W.Y. Tam and H.L. Swinney, Phys. Rev. A 36 (1987) 1374. [11] Y. Enokida, K. Nakata and A. Suzuki, AIChE J. 35 (1989) 1211. [12] Q. Ouyang, W.Y. Tam, P. DeKepper, W.D. McCormick, Z. Noszticzius and H.L. Swinney, J. Phys. Chem. 91 (1987) 2181. [13] J.A. Vastano, Ph.D. Dissertation, The University of Texas at Austin (1988). [14] J.A. Vastano, T. Russo and H.L. Swinney, Bifurcation to spatiotemporal chaos in a reaction-diffusion system, Physica D 46 (1990) 23. [15] A. Arneodo and J. Elezgaray, in: Spatial Inhomogeneities and Transient Behavior in Chemical Kinetics, eds. P. Gray, G. Nicolis, F. Baras, P. Borckmans and S.K. Scott (Manchester Univ. Press, Manchester, 1990); in: New Trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibrium, eds. P. Coullet and P. Huerre (Plenum, New York, 1989); Physica D, to appear. [16] J.A. Vastano, J.E. Pearson, W. Horsthemke and H.L. Swinney, Phys. Lett. A 124 (1987) 320; J. Chem. Phys. 88 (1988) 6175; J.E. Pearson and W. Horsthemke, J. Chem. Phys. 90 (1989) 1588; P.B. Monk and H.G. Othmer, Wave propagation in aggregation fields of the cellular slim mold Dictyostelium discoideum, Phil. Trans. R. Soc. London, submitted for publication.
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W.Y. Tam and H.L. Swinney / Spatiotemporal patterns in a one-dimensional open reaction-diffusion system
[17] Z. Noszticzius, M. Wittmann and P. Stirling, 4th Symposium of Ion-Selective Electrodes, Matrafured 38 (1984). [18] M. Gorman and H.L. Swinney, J. Fluid Mech. 117 (1982) 123. [19] F. Sagues and W. Horsthemke, Phys. Rev. A 34 (1986) 4136; B.I. Shraiman, Phys. Rev. A 36 (1987) 261; M.N. Rosenbluth, H.L. Berk, I. Doxas and W. Horton, Phys. Fluids 30 (1987) 2636. [20] Z. Noszticzius, W.D. McCormick and H.L. Swinney,
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