J. Phys. Chem. 1992, 96, 8915-8919 (3) Suzuki, H. Bull. Chem. Soc. Jpn. 1960, 33, 389,944. (4) Higuchi, J.; Ito, T.; Kanehisa, 0. Chem. P h p . Lett. 1973, 23, 440. (5) Hoshi, T.; Ota, K.; Yoshino, J.; Murofushi, K.;Tanizaki, Y. Chem. Letr. 1977, 357. (6) Suzuki, H.; Koyano, K.; Shida, T.; Kira, A. Bull. Chem. Soc. Jpn. 1982, 55, 3690. (7) Ota, K.; Murofushi. M. Terrahedron Lett. 1974, 15, 1431. (8) Cooky, C. J.; Courtneidge. J. L.; Davies, A. G.; Evans, J. C.; Gregory, P. S.;Rowlands, C. C. J . Chem. SOC.,Chem. Commun. 1986,549. (9) Stephens R. D.; Castro, C. E. J. Org. Chem. 1963, 28, 3313. (10) Castro, C. E.; Gaughan, E. J.; Oweley, D. C. J. Org. Chem. 1966,31, 407 1. (11) Baldwin, J. E.; Barden, T. C. J. Am. Chem. Soc. 1984, 106, 5312. (12) Hashimoto, S.; Shimojima, A,; Yuzawa, T.; Hiura, H.; A h , J.; Takahashi, H. J. Mol. Srrucr. 1991, 242, 1.
8915
(13) Kelsall, B. J.; Arlinghaus, R. T.; Andrews, L. High Temp. Sci. 1984, 17, 155.
(14) Wilson, E. B., Jr. Phys. Rev. 1934,45, 706. (15) Baranovic, G.; Colombo, L.; Skare, D. J. Mol. Srrucr. 1986, 147, 275. (16) Shimojima, A.; Takahashi, H. To be published. (17) Kruppa, A. I.; Leshina T. L.; Sagdeev, R. Z. Chem. Phys. Lerr. 1985, 121, 386. (18) Murov, S.L. Handbook ofPhorochemisrry; Marcel Dekker: New York, 1973. (19) Turro, N . J. Modern Molecular Photochemistry: Benjamin/Cummings: New York, 1978. ( 2 0 ) Roth, H. D.; Schilling, M. L. M.; Gassman, P. G.; Smith, J. L. J. Am. Chem. Soc. 1984. 106. 271 I. (21) Guerry-Butty,'E.; Haselbach, E.; Pasquier, C.; Suppan, P. Helu. Chim. Acta 1985, 68, 912.
Mutually Coupled Oscillators with Time Delay J. Weiner, R. Holz, F. W. Schneider,* Institut fiir Physikalische Chemie der Universitat Wiirzburg, Marcusstr. 9- 1 I , 0-8700Wiirzburg, Germany
and K. Bar-Eti School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel (Received: May 1 1 , 1992; In Final Form: June 26, 1992)
We present a system of two identical chemical oscillators which are coupled by means of their flow rates. The individual flow rate of one oscillator is controlled by the output of the other oscillator and vice versa. A delay time is applied between the output of one oscillator and the input to the other one. This delay has been investigated both theoretically and experimentally for the minimal bromate oscillator. By variation of the delay time both oscillators dramatically change their periods and amplitudes. At lower coupling strengths, the periods of both oscillators increase and decrease in a sawtooth fashion with increasing delay times. At high coupling strengths multimode patterns were obtained at high delay times. No quasiperiodic or aperiodic behavior was observed in this coupled system. The experimental results are in general agreement with those calculated with the Noyes-Field-Thompson mechanism.
1. Introduction Investigations of coupled nonlinear oscillators recently have become a subject of interest not only in chemistry but also in other natural scicnca such as biologyl and physiology? Special attention has been paid to systems which involve a time delay in their coupling modesS3Theoretical models of these systems are widely used to explain complex behavior in physiological control systems. The present paper deals with two identical chemical oscillators where the input to one oscillator is determined by the output of the other with an imposed time delay. In general the following coupling modes between two oscillators are noteworthy: CoapIlng of Two Oscillators. This may be achieved by the periodic forcing of an oscillator, Le., periodic modulation of the continuously stirred tank reactor (CSTR) flow rate or by periodic injection of one or more species. Here the forcing oscillator is always independent of the driven one. The response behavior of the driven oscillator is investigated by variation of the frequency and amplitude of the forcing o s ~ i l l a t o r . ~ - ~ Coupling by MISS Exchange between Two Oscillators or by Vuhtion of the Redox Potential. Changes of the concentrations take place by diffusion through a membrane,'* direct mass exchange,Ig sharing a common species15in a batch reaction, or in the case of strong electrolytes by varying the redox potential.*' N0nliat.r Coupling. The forcing function of the oscillator is created by the oscillator itself. In previous work6 we described the experimental and theoretical results of a delayed feedback coupled oscillator and mentioned the analogy to several biological systems. A special case is the feedback coupling of a bi- or multistable system, which may result in chemical oscillation^.^ To whom correspondence should be addressed.
2. Theoretical Investigations of Coupled Chemical Oscillators Mass-coupled oscillators are among the most commonly investigated chemical oscillators in experiment and in theory, where the coupling is achieved by linear diffusion terms in the differential equation system. In the coupled Brusselator model Schreiber and Mareks found in-phase and out-of-phase solutions as well as quasiperiodicity. Hlavacek and co-workersgfound aperiodic time series. Bar-Eli'O investigated the NlT, Oregonator, FKN, Brusselator, Kumar, and Lotka-Volterra models including mass flow between the two oscillators. If the coupling strength is low, the oscillators are nearly independent, i.e., each oscillator operates at its own free-running frequency and amplitude. If the coupling is high, both oscillators display the same frequency and amplitude by mutual entrainment. Oscillations were suppressed at medium coupling strength, and stable stationary states occurred. Further investigations focused on diffusive coupling of two or more Brusselators in their respective steady states. Wang and Nicolis" investigated a coupled Brusselator model numerically and analytically. A bifurcation from a 2-torus to a 3-torus was observed. Lahiri and Ghosal'* found entrainment and quasiperiodicity in a chain of linearly coupled Brusselators. Recently, Hocker and Epstein13obtained a great variety of dynamic behavior in a coupled two-variable system, in which the individual millators worked under different parameters. Bar-EliI4coupled identical FKN models and found steady-state behavior as well as different kinds of symmetric and asymmetric oscillations. 3. Experimental Investigations of Coupled Chemical Oscillators Chemically coupled oscillators were studied by Alamgir and EpsteinIs using the chlorite-iodide and the bromate-iodide os-
0022-3654/92/2096-89 15$03.00/0 0 1992 American Chemical Society
8916 The Journal of Physical Chemistry, Vol. 96, No. 22, 195'2 cillators combined in one CSTR. They found birhythmicity and compound oscillations which were not present in the separated oscillators. Citri and Epstein16 proposed a 20-step mechanism with 12 species for this coupled system and found good agreement with experiments. Another chemically coupled system was devised by Maselko and Epstein,17 in which they mixed the chloritethiosulfate and the thiosulfate-iodide system. They found birhythmicity and three coexistent stable steady states. Marek and StuchlIs investigated a BZ system in two reactors coupled by mass exchange. They found entrainment and irregular behavior. An extended system with seven coupled reactors displayed stable steady states.Is Sawada and co-w~rkers'~ studied the phase relationship between two mass coupled BZ reactions. Bar-Eli and Reuveni" coupled two BZ reactions by pumping the solutions back and forth between two reactors. Crowley and Field2I coupled two BZ reactions by means of their redox potentials. In one CSTR, the redox potential, Le., the Ce(III)/Ce(IV) ratio, was measured and applied to the other reactor by a large working electrode and vice versa. These authors found in-phase and out-of-phase entrainment. Another experiment conducted by Marek and coworkersZ2used mass coupling of two BZ reactions. The pulse perturbation of one oscillator by Br- solution led to complex periodic behavior, which could be described by the Farey arithmetic. This behavior could be simulated with the Oregonator. DeKepper and co-workersZ3investigated the chlorite-iodide reaction in a CSTR. When two systems were coupled by mass exchange, oscillations were generated. Crowley and Epstein observed the supression of oscillations, multistability, and in-phase and out-of-phase entrainment in a coupled BZ system with mass exchange.24 Flow rate coupling with delay in a single oscillator has been investigated experimentally and theoretically in the minimal bromate systema6 In this work we use two identical minimal bromate oscillators which are delay-coupled by means of their flow rate, i.e., the output of one oscillator determines the input to the second oscillator in a cyclic interaction. When the coupling strength is sufficiently high, the two coupled chemical oscillators are expected to be entrained. This work answers the question of how the period and amplitude of the entrained oscillators behave as a function of delay time. 4. The Coupling System 4.1. The Minimal Bromate Oscillator. The chemical system used in this investigation as well as in our previous one6 is the minimal bromate oscillator (MBO). The single MBO was predicted by Bar-Eli to o ~ c i l l a t e .The ~ ~ experimental verification of this oscillator occured independently by Geiselerz6and by Orban et a1.2~in a very narrow parameter region.z8 We use the following NFT mechanismz9with a set of "newn rate constants:jO Br0,- Br- + 2H+ HBr02 + HOBr (1)
-
+
k , = 2.0 Me3 s-l HBr02 + Br-
+ H+
k2 = 3 X lo6 M-2 s-I k, = 3
-
-
lo9 M-2 s-I
X
Br03- + HBr02 + H+ k4 = 42 M-' X
k6 = 3
-
lo4 M-2 s-I
2HBr02 X
(2)
M-l
s-I
Br2 + H 2 0
(3)
k-, = 2 s-l
2Br0,'
+ H20
(4)
k-, = 4.2 X lo7 M-l s-l
S-I
Ce3++ Br02' + H+ k5 = 8
s-I
2HOBr
k..2 = 2 X
+ Br- + H+
HOBr
-
k-l = 3.2 M-I
Ce4++ HBr02
k-5 = 8.9
Br03- + HOBr
lo3 M-I s-'
k-6
X
(5)
lo3 M-I s-l
+ H+
(6)
= 1 X 10 M-2 s-'
The concentrations were 0.1 M BrO,-, 0.3 mM Br-, 0.3 mM Ce3+,and 0.75 M H+ in the simulations. The model system shows
Weiner et al. stable oscillationsin the kf range 4.124 X 10-3s-l to 5.816 X lo-, s-I. The lower kfvalue marks a supercritical and the upper a subcritical Hopf bifurcationa6 The experimental concentrations were chosen to minimize the period and to maximize the amplitude of the oscillations. With the reactor concentrations 0.1 M KBr03, 0.4 mM KBr, 0.3 mM Ce(N0,)3, in 1 M H2S04,the experimental MBO oscillates in the flow rate range from 2.5 X lo-, s-I to 3.5 X 10-3 s-l. The period rises from about =340 s to about 4 8 0 s. 4.2. Mutual Coupling. Mutual coupling of the two reactors has been realized by coupling the output of one oscillator, i.e., the measured ceric ion concentration to the other. The flow rate of one oscillator is modulated by the measured [Ce4+]of the other oscillator, respectively, according to
)
[Ce4+I2(t- D2)- [Ce4+laV [Ce4+1av
(7)
[Ce4+],(t- D l ) - [Ce4+Iav D 4 + 1 8 V
The parameters D, and D2 denote the delay times and and f12 determine the intensity of coupling. The average flow rate and the average ceric ion concentration [Ce4+Iavare identical for both reactors. The indexes belong to reactor 1 and reactor 2, respectively. 5. Experimental Section 5.1. Mutual coupling with Delay. Each CSTR consists of a shortened quartz-cuvette with 1-cm path length. The bottom is rounded for better stirring with a specially designed Teflon stirrer. The Teflon lid is dome-shaped for better removal of gas bubbles. The stirring is performed magnetically at 500 rpm. The top of each CSTR has three inlet tubes, one outlet tube, and a glass tube covering a thermistor for temperature control. The volume was determined by measuring the fust-order kinetics of a dye diluted with buffer, i.e., by monitoring the rate of change of absorbance for a known flow rate. The ceric ion concentration was measured continuously (Beckman ACTA 5260) at 350 nm. At this wavelength only the ceric ion absorbs. The optical density ranges from 0 to 1 with a resolution of 0.001 optical density (OD) units. The spectral resolution was 1 nm. A second nearly identical setup was constructed (Perkin-Elmer Lambda 7). The average amplitude of an oscillation is typically between 0.6 and 0.9 OD units, where the extinction coefficient of Ce4+is 5.0 X lo3 at 350 nm. Each reactor was fed simultaneously through three inlet tubes by three gas-tight 50-mL syringes which are moved by a precise infusion pump. The entire equipment, i.e., syringes, tubes, cuvette holder, etc., was thennostated at 25.0 f 0.1 OC. The three syringe6 contained aqueous KBrO,, dilute H2S04, aqueous KBr, and Ce(NO,),, respectively. All chemicals were purification grade p.a. and were used without further treatment. All solutions were prepared from water (conductance SlO-' Q-I) purified by ion exchange. 5.2. Data Sampling and Pump Control. The analog output of the ACTA 5260 was fed to a two-channel x / t recorder. The data from the BCD interface were read into a personal computer (MS-DOSstandard) via a parallel 1/0board,decoded, and stored. The Lambda 7 spectrometer also features an analog output and a serial RS232 interface. The analog output was recorded by a two-channel x / t recorder. The data for pump control were processed in the same way as described above. The PC controlled the pump by means of a 12-bit analog/digital converter board. The pump speed was directly proportional to the input voltage in an interval of -2 to 0 V. Figure 1 shows a schematic diagram of the experimental setup. 6. Results
In the experiments as well as in the calculations the minimal-bromate oscillations are started simultaneously and independently in both reactors under identical conditions of concentrations and flow rate. A phase difference is usually observed
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8917
Mutually Coupled Oscillators with Time Delay
.
n
.
.
.
,
.
.
.
.
,
.
,
t
-. I B=
t -0
i Ob
1
beta Figure 2. R e d u d periods (coupling without delay) vs mental, ( 0 )calculated.
I
L--> PUMP
1
.
CSTR
8: (A)experi-
1
delay (reduced) Figure 3. Asymmetrical coupling (D2= 0): reduced response period vs reduced delay: 8 = 1.0, (A) experimental, ( 0 )calculated.
At larger delays overlapping regions occur between D l / P o = 3.4 and 4.2. This is not a hysteresis phenomenon. In some cases
it is difficult to determine the main frequency in the power spectrum. For instance, at D,/Po = 3.8 the unambiguous determination of the frequency seems to be impossible; one finds two different frequencies simultaneously which belong to the upper and the lower branch. From D l / P o= 4.0, a third branch starts in the overlap region and the measured periods are found in either of the three states. At large delays, the transient time may be longer than the measurement. Therefore, a stable limit cycle may not be reached. This is true especially near the transition points between the branches. Perturbations may induce transition between the different branches. At small delays, i.e., in the region of the fmt branch, experiment and simulation are in good agreement. At larger delays the experimental periods stay longer on a given branch than in the simulations. The normalized starting points depend on the values of the free-running period, which varies somewhat from one experiment to the next. Thus the absolute position of the branches is not known with great precision. The slopes of the experimental decrease in the same manner branches (0.82,0.533,0.287,0.177) as the computational ones with increasing delays. Computations show that the time required to reach a stable limit cycle as well as the period of the limit cycle itself depends on the initial phase between the two oscillators. The transient time for the initial out-of-phase coupling is considerably shorter than for the in-phase coupling. The power spectra show different periods; the phase coherent coupling has the lower frequency, Le., higher period. With these computational results, the experimental observation can be readily explained. Simulations. The results of the simulations show general similarities with those for the feedback coupled oscillator? The reduced period rises from 0.73 at zero delay to about 1.3 at a
Weiner et al.
8918 The Journal of Physical Chemistry, Vol. 96, No. 22, 199'2
t
n
c
delay (reduced) Figure 4. Asymmetrical coupling (D2= 0): reduced response period vs reduced delay: 6 = 2.0, (A) experimental, ( 0 )calculated. 1 0 Y c
delay (reduced) 6 = 2.0, calculated firing
Figure 6. Asymmetrical coupling (D2= 0): numbers vs reduced delay.
1
I-\
I
delay (reduced) Figure 7. Symmetrical coupling (Dl = 4):reduced rsponse period vs reduced delay: 6 = 1.0 (A) experimental, ( 0 )calculated.
time/s Figure 5. Asymmetrical coupling (D2= 0): fi = 2.0, calculated time series with the reduced delays (a, top) 4.91 and (b, bottom) 1.23.
n
P P)
0
reduced delay of one; then it drops sharply and rises again in a sawtooth fashion (Figure 3). The reduced periods start with values less than unity. The starting points of the branches shift toward higher reduced periods with increasing j3. The end points are slightly above unity with increasing delay, the reduced period of one is approached. One interesting observation is the decrease of the slopes of the branches with increasing delays. The values of the slopes are 0.595,0.421, 0.327, 0.291, and 0.141 (Figure 3). For j3 = 2.0, the results are similar to those for j3 = 1.O (Figure 4). The reduced periods increase from 0.69 at D l / P o= 0.25 to 1.39 at 1.03, respectively. With larger delays the time series become more complex. The exact determination of the periods of these complex time series is difficult. For the sake of simplicity, all complex oscillations are represented with only one period in the period versus delay plots. It is interesting to note that the complex time series (Figure 5) can be analyzed in terms of Farey patterns. If the largest amplitude of an oscillation pattern is denoted by L (large) and all other amplitudes by S (small), then a given Farey pattern may be described by Laor by the firing number,f = ( z S ) / ( z L+ XS)(Figure 6). The sum of two firing numbersf, +fiis q u a l to (SI Sz)/(LI SI+ .C, + S2)in analogy to the Farey sum. 63. SymmetricplCoupliao (Dl I Dz) ni&DAY.E X&" and Simulation. Fwre 7 shows the reduced period versus delay time for 0 = 1.O. Evidently, a similar sawtooth like behavior occurs as in section 6.2. The slopes of the branches decreased with
+
+
2
-0
2
- ( I (
P 0
'50
delay (reduced) Figure 8. Symmetrical coupling (D,= D2):reduced response period vs reduced delay: @ = 2.0, (A) experimental, (0)calculated.
increasing delay, i.e., 1.571,0.949,0.700,0.527,0.299, and 0.165. The average value of the reduced period approaches -1.5 at large delay times. Overlapping regions are also observed. For 0 = 2.0, the results are qualitatively similar to those for j3 = 1.0. The slopes of the branches shown are 2.073, 1.140,0.856, 0.548, and 0.441 (Figure 8). At reduced delay times larger than -0.5, the reduced periods increase and decrease in a sawtooth fashion as a function of DIP0 converging to unity. For = 2.0 the periods increase and decrease in the familiar sawtooth behavior, but with larger amplitudes and s l o p (Figure 8). At D/Po = 1.0 more complex oscillations occur, which consist of a series of small and large amplitudes. Lnpatterns are followed
Mutually Coupled Oscillators with Time Delay by lo patterns which give way to LmLkpatterns where n L 3, n = m k, and k = m - 1. Thus, this series of complex periodic oscillation patterns is interrupted by simple periodic motions. The firing numbers do not increase continuously with the delay times. The 1” and Pikpatterns do not directly follow each other, instead they are separated by a periodic region of oscillations.
+
7. Discussion When two chemical oscillators are coupled by the flow rate without delays, identical periods and amplitudes are produced in both reactors. At low coupling strengths, the period of the coupled system is close to the free running period. When the coupling strength is increased, the period of the coupled system increases in the experiments but it was found to decrease in the calculations (Figure 2). The reason for this discrepancy is not known. If the coupling strength exceeds a critical value (B > 1.0 in the experiments, @ > 1.1 in the calculations), the oscillations in both reactors cease and each reactor attains its own steady state where the steady-state concentrations of Ce(1V) are different. Due to the symmetry of the setup, it is impossible to predict which of the two steady states is established in a given reactor. The use of delay times introduces a considerable complexity in the experiments. Yet, mutually coupled oscillators with asymmetric delay times (D2= 0) show a behavior which is similar to the simple feedback coupled system at small coupling strengths. The periods increase and decrease in a sawtooth fashion. At larger coupling strengths multimode oscillations are observed. A similar situation is encountered for symmetric delays (Dl = DJ where the familiar sawtooth pattern is also observed. At higher delays, more complex oscillations occur. The Hoquet multipliers have been calculated from the monodromy matrixes of all complex time series, and they indicate stable periodic oscillations for each delay time, i.e., no evidence for deterministicchaos has been found. In all cases the calculated values for PIPo agree only qualitatively with experiment. Due to the limited measurement time the experiments may still contain some transients. The experimental data show some scatter close to the transition points in the “sawtooth” curves. Therefore, one obtains large overlapping regions. This phenomenon, however, is not a hysteresis. Our calculations according to method B did not indicate any hysteresis in contrast to the results of the single feedback coupled oscillator, which exhibits hysteresis behavior and biryhthmicity.6 Biryhthmicity is well-known in coupled chemical oscillators. Decroly and G ~ l d b e t e found r ~ ~ this behavior in their theoretical investigations of a coupled allosteric enzyme model. Epstein and co-w~rkers’~J’ and Gybrgyi and Field” obtained birhythmicity in their investigations of chemically coupled oscillators. Multimode oscillations have already been found in many chemical systems, for instance, in the free running BZ and in chemically coupled systems.15 In many cases the patterns of these oscillations can be described by means of the Farey arithmeti~.~~J~J’ It is remarkable that quasiperiodic or chaotic behavior has not been found in the present coupling experiments and simulations that include delay times. On the other hand, chaos has been found to occur in relatively simple differential delay equation^.^^ However, the absence of chaos is not surprising. In the oscillating
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8919
region the minimal bromate oscillator displays only periodic behavior over the whole range of flow rates. Apparently coupling through the flow rate does not produce any additional bifurcations which would be necessary for chaos to occur. Registry No. Br03-, 15541-45-4; Br-, 24959-67-9; Ce, 7440-45-1. References and Notes (1) Winfree, A. T. The Geometry of Biological Time; Springer Verlag: New York, 1980. (2) Glass, L.; Mackey, M. The Rhyrhms of Lve; Princeton University Press: Princeton, NJ, 1988. (3) Glass, L.; Larocque, D.; Beuter, A. Marh. Biosci. 1988, 90, 111. (4) (a) Schneider, F. W. Annu. Rev. Phys. Chem. 1985, 36, 347; (b) Weiner, J. Diplomarbeit, Universitiit Worzburg, 1986. (5) (a) Rhemus, P.; Ross, J. J . Chem. Phys. 1983, 78, 3755. (b) In Oscillations and Travelling Waves in Chemical Systems; Field, R.J., Burger, M., Eds.; Wiley-Interscience: New York, 1985. (6) Weiner, J.; Schneider, F. W.; Bar-Eli, K. J . Phys. Chem. 1989, 93, 2704. (7) Shanks, B. H.; Bailey, J. E. Chem. Eng. Commum. 1987, 61, 127. (8) Schreiber, I.; Marek, M. Physica 1982, 5D, 258. (9) Nandarpurkar, P.; Hlavacek, V.; Degreve, J.; Jansen, R.; Van Rompay, P. Z . Naturforsch. 1984, 39A. 899. (10) (a) Bar-Eli, K. J . Phys. Chem. 1984,88, 3616; (b) J. Phys. Chem. 1984,88, 6174; (c) Physica 1985, 140. 242. (1 1) Wang, X.-J.; Nicolis, G. Physica 1987, 260, 140. (12) Lahiri, A.; Ghosal, S.S.J . Chem. Phys. 1988,88, 7459. (13) Hccker, C. G.; Epstein, I. R. J. Chem. Phys. 1989, 90, 3071. (14) Bar-Eli, K . J . Phys. Chem. 1990, 94, 2368. (15) (a) Alamgir, M.; Epstein, I. R. J . Am. Chem. SOC.1983,105,2500, (b) J. Phys. Chem. 1984.88, 2848. (16) Citri, 0.; Epstein, I. R. J . Phys. Chem. 1988, 92, 1865. (17) Maselko, J.; Epstein, I. R. J. Phys. Chem. 1984, 88, 5305. (18) (a) Marek, M.; Stuchl, I. Biophys. Chem. 1975,3,241. (b) Stuchl, I.; Marek, M. J . Chem. Phys. 1982, 77, 2956. (19) (a) Fujii, H.; Sawada, Y. J . Chem. Phys. 1978,69,241. (b) Nakajima, K.; Sawada, Y. J . Chem. Phys. 1988, 72,2231; (c) J . Phys. SOC.Jpn. 1981, 50, 687. (20) Bar-Eli, K.; Reuveni, S.J. Phys. Chem. 1985,89, 1329. (21) (a) Crowley, M. F.; Field, R. J. In Nonlinear Phenomena in Chemical Dynamics; Vidal, C., Pacault, A., Eds.;Springer-Verlag: West Berlin, 1981; p 147; (b) Lecture Nores Biomath. 1986, 92,2452; (c) J. Phys. Chem. 1988, 90, 1907.. (22) (a) Dolnik, M.; Padusakova, E.; Marek, M. J . Phys. Chem. 1987,91, 4407. (b) Dolnik, M.; Marek, M. J . Phys. Chem. 1988, 92, 2452. (23) Boukalouch, M.; Elezgaray, J.; Arneodo, A.; Boissonade, J.; DeKep per, P. J. Phys. Chem. 1987, 91, 5843. (24) Crowley, M. F.; Epstein, I. R. J. Phys. Chem. 1989, 93, 2496. (25) Bar-Eli, K. In Nonlinear Phenomena in Chemical Dynamics; Vidal, C . , Pacault, A., Eds.; Springer-Verlag: West Berlin, 1981; p 228. (26) Geiseler, W. Ber. Bunsen-Ges. Phys. Chem. 1982, 93, 2496. (27) Orbin. M.; DeKepper, P.; Epstein, I. R.J. Am. Chem. Soc. 1982,104, 2657. (28) (a) Geiseler, W. J . Phys. Chem. 1982, 86, 4394. (b) Bar-Eli, K.; Geiseler, W. J. Phys. Chem. 1983, 87, 3196. (29) Noyes, R. M.; Field, R. J.; Thompson, R. C. J. Am. Chem. Soc. 1971, 93, 7315. (30) Bar-Eli, K.; Field, R. J. J . Phys. Chem. 1990, 94, 3660. (31) Field, R. J.; FGrsterling, H. D. J . Phys. Chem. 1986, 90,5400. (32) Decroly, 0.; Goldbeter, A. Proc. Narl. Acad. Sci. U.S.A. 1982, 79, 6917. (33) (a) GyGrgyi, L.; Field, R. J. J . Phys. Chem. 1988, 92, 7079; (b) J . Phys. Chem. 1989, 93, 2865; (c) J . Chem. Phys. 1989, 93, 6131. (34) Schneider, F. W.; Miinster, A. J. J. Phys. Chem. 3991, 95, 2130. (35) Dolnik, M.; Finkeova, J.; Schreiber, I.; Marek, M. J . Phys. Chem. 1989, 93, 2764. (36) Maselko, J.; Swinney, H. L. J. Phys. Chem. 1986, 90,6430. (37) Monster, A. Diplomarbeit, Universitiit WDrzburg, 1988. (38) (a) Ikeda, K.; Matsumoto, U. Physica 1987, 290, 223. (b) Hale, J. K.; Sternberg, N. J. Compur. Phys. 1988, 77, 221.
