J. Phys. Chem. 1993,97, 12239-12243
12239
Control of Dynamic States with Time Delay between Two Mutually Flow Rate Coupled Reactors R. Holz and F. W . Schneider' Institut fur Physikalische Chemie, Marcusstrasse 911 1 , 8700 Wiirzburg, Germany Received: April 8, 1993; In Final Form: August 31, 1993'
We present experiments and simulations of two mutually flow rate coupled chemical oscillators (minimal bromate (MB)oscillators), where the flow rate into one reactor is controlled by a species concentration (Ce4+) of the other reactor. A delay time T is introduced between the output (Ce4+ concentration) of one reactor and the input (flow rate) into the other so that the state of one reactor at a certain time t is determined by the state of the other reactor at a previous time t - T . Chaos does not occur in this MB system. Here we report how to control certain periodic and steady states. We observe two special phenomena, namely, oscillator death and rhythmogenesis. Oscillator death is induced by coupling two MB oscillatory states; as a result, two different steady states are produced. The reverse phenomenon, rhythmogenesis, is observed as the appearance of a common oscillatory state when two identical MB stationary states are coupled. Numerical calculations using two coupled NFT models show good agreement with the experimental results.
1. Introduction
In the last 30 years chemists have described single chemical oscillators and their mechanisms. Coupled chemical oscillators have been the subject of many investigationssince they may serve as models for complex biological systems, for example, for the study of transformation processes in living cells, tissues, and networks of neurons. The coupling between two reactors may be achieved in different ways: by mass exchange, electricalcoupling, or flow rate coupling, where the flow rate of reactants is regulated by a given species concentration in thesecond reactor (andviceversa) withor without a time delay. So far, the most common method of coupling two oscillators has been through mass exchange between two In this case the rate of mass exchange (Le. rate of diffusion through an orifice between two reactors) is proportional to the difference in concentrations of a given species in the two reactors. The motivation of the present work is to study the effect of a delay time in the mutual coupling between the two reactors on the stability of certain dynamic states. This effect may best be studied by flow rate coupling. In earlier work* we studied delayed feedback using the minimal bromate oscillator in a single reactor. The period of the controlled oscillations changed in a sawtooth fashion, with overlapping regions of birhythmicity as a function of delay time. Recently we applied time-delayed mutual flow rate coupling between two minimal bromate (MB) oscillatorsg and two Belousov-Zhabotinsky (BZ) oscillators.10 The output of one continuous flow stirred tank reactor (CSTR) determined the flow rate into a second CSTR and vice versa, including a delay time between output and input. A spectrumof periodic and stationary states were stabilized when the two MB oscillators were coupled with variable delay times. It was shown that both coupled oscillators change their periods and amplitudes with increasing delay and the periods of both oscillators behave in a sawtooth fashion at low coupling strengths. At high coupling strengths multimode patterns are generated at long delay times in the MB oscillators. This experimental behavior is in agreement with simulations of the NFT model. In the BZ system mutual flow rate coupling of two chaotic states without time delay retains chaos in both reactors at low coupling strength.IO For increasing To whom correspondence should be addressed.
*Abstract published in Advance ACS Abstracts, November 1, 1993.
0022-3654/93/2097-12239304.00/0
coupling strength the Hausdorff dimension passes through a maximum (=2.5). At high coupling strengths a transition from chaos to periodic states takes place. Our mode of coupling9JO uses the difference between the delayed output signal and an average output signal because the latter is convenient to realize experimentally. This method is similar to the delayed feedback control by Pyragas," who achieved a continuous control of chaos by using the difference between the delayed output signal and the output signal itself. Marek and co-workerP have carried out detailed experimental and theoretical investigationsof coupled reaction-diffusion cells. Mankin and HudsonIzinvestigated the effectsof thermal coupling via heat transfer across an interface between two oscillatory reactors. Crowley and Field used electrical coupling between two Belousov-Zhabotinsky reactions.*3J4 In this work we demonstrate experimentally and numerically two phenomena in the minimal bromate oscillator, which represents the BelousovZhabotinskyreaction without malonicacid the loss of oscillations (oscillator death) when two limit cycles are continuously coupled by the flow rate without time delay and rhythmogenesis? which stands for the stabilization of oscillations from the coupling of two identical steady states in two reactors which are continuously coupled via the flow rate including a time delay. For linear flow rate coupling, quasiperiodicity or chaos has not been observed in the MB system. 2. Coupled System
For the following study we chose the flow rate coupling of two minimal bromate oscillator^,^^ because the MB system is particularly sensitive to changes in the flow rate. This reaction is known as the inorganic part of the BZ reaction, and it displays oscillations only under flow conditions in a CSTR and by adding B r as one of the inflow components. The free-running MB oscillator displays relatively simple periodic behavior (PI) in the oscillatory region with a supercritical Hopf bifurcation at low flow rates and a subcritical Hopf bifurcation at high flow rates. The amplitudes and periods of the oscillations increase with increasing flow rate. Although there are still discussions about the correct values of some rate constants,16 the NFT mechanism" of the minimal bromate system is commonly accepted. Bar-ElilE predicted oscillations in the uncoupled system by numerical simulations, which could be verified experimentally by Geiselerl9 and by O r b h and co-workers.20 The NFT mechanism is 0 1993 American Chemical Society
Holz and Schneider
12240 The Journal of Physical Chemistry, Vol. 97,No. 47, 1993
Br0,-
+ Br- + 2H+ F! HBrO, + HOBr
k , = 2.0 M" s-'
HOBr + Br-
(3)
t
k-3 = 2 s-'
+ HBrO, + H+ F! 2Br0,' + H,O
k, = 42 M-, s-l Ce3+
(2)
+ H+ F! Br, + H,O
k, = 3 X lo9 M-' s-' BrO;
A
k-, = 3.2 M-' s-I
+ Br- + H+ F! 2HOBr
HBrO,
(1)
t ,
,Lg_l
(4)
k , = 4.2 X lo7 M-'s-l
+ BrO,' + H + e Ce4++ HBrO,
(5)
B 3 Inlet tube# outlet tube
k-, = 8.9 X 1O3 M-' s-'
k , = 8 X lo4 M-' s-'
themlitor
2HBr0, k6 = 3 x 10,
F!
S-'
BrO;
+ HOBr + H+
k , = 1 X lo4 M-'
(6) s-l
where we use the "new" rate constants given by Bar-Eli and Field16 in our numerical simulations. Calculations of the delay-coupled oscillatorshave been carried out by numerical evaluation of the differential equations arising from the NFT mechanism (eqs 1-6). They are of the form
+
(7) dC/dt = f l C ) kf3JC0- C) whereflC) represents the kinetic mechanism, C represents the concentrations, COstands for the concentrations of the inflow components, and kti is the flow rate into reactor i. 3. Flow Rate Coupling The flow rate into one CSTR is controlled experimentally by the Ce4+-ionconcentration in the other reactor. In addition a time delay Ti is imposed between the output of one reactor and the input to the other. We use the following equations for "flow rate coup1ing":g
where
kf,i is the flow rate into the coupled ith reactor, ko,i is the flow rate into the uncoupled reactor i, and 81is the coupling strength. In the following we will simply use 8, which refers to the case where 81 = 8 2 . If 8 = 0, the two CSTRs are uncoupled. The values of [Ce4+lsV,i are the average values between the maxima and minima of the oscillations of the uncoupled oscillators in each reactor i, or with respect to the steady states, they are the averagevalues between the low and the high steady states in each uncoupled reactor. The high (low) steady state is the maximal (minimal) Ce4+concentration which is reached in a given reactor due to the choice of ko. The indexes stand for reactors 1 and 2,
-1
cm
-.
Figure 1. (A) Diagram of the experimental setup. (B) Two CSTRs containing the MB system fed by three syrings, respectively. The data from the optical density of ceric ions measured by the two photometers are transmitted to two PCs (contained in delay 1 and delay 2). The computers calculate the mutual inflow rate of reactants into the respective CSTRs with or without time delay.
respectively. [Ce4+],,irepresents a requiredreferencestate, whose chosen value (see above) is somewhat arbitrary. 4. Experimental Section A schematic diagrasm of the experimental setup is shown in Figure 1. The two CSTRs are specially designed spectrophotometric cells (magnetically stirred) of 1-cm path length. They are fed by three identical syringes which are driven by precise syringe pumps. The pumps contain a stepping motor whose frequency is controlled by the light absorption of ceric ions at 350 nm in the other CSTR. The output from the photomultipliers of the spectrophotometers is connected to computers, which store the data and calculate the flow into the respective other reactor withorwithout adelay time,accordingtoeqs8. The threesyringes contain (a) aqueous KBr03, (b) aqueous KBr and Ce(NO3)3, and (c) dilute HzSO4. All solutions, tubeconnections,and CSTRs were thermostated at 25 OC. 5. Results and Calculations
Coupling of Two Oscillatory States. Without Time Delay: Oscillator Death. When two MB oscillators are coupled via the flow rate with increasing coupling intensity, oscillator death is observed for the followingconditions. The bromateconcentrations and the flow rates ko are different in the two reactors (CSTRl, 0.10 M KBrO3, ko,, = 3.1 X l e 3 s-I; CSTRz, 0.13 M KBrO3, ko,z = 3.7 X s-l), while the concentrations of B r (0.003 M) and Ce3+(1.2 mM) are identical in the two reactors. With these
Control of Dynamic States
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12241 500
600
1
4
e
‘ 1 400
I
I
/
I
t
/
i
*
200
200 0
0.2
0.4
06
08
1 coupling strength [beta]
12
0
14
Figure 2. Calculated (+) and experimental ( 0 )oscillation periods vs j3 resulting from flow rate coupling of two MB systems.
values the free-running periods in CSTRl and CSTR2 are 300 and 170 s, respectively. (In previous work9we used two identical reactors and obtained identical free-running periods.) When the two reactors are coupled, the two subsystems become entrained and they oscillate with the same frequency but 180’ out of phase. With increasing @ the periods (Figure 2) and amplitudes in both subsystems increase until a threshold value for @ = 1.4 is reached, where two steady states suddenly appear, one of high and the other at low Ce4+ concentration. If the Ce4+ concentration in one of the CSTRs is lower than [Ce4+Iav,it causes a decrease of the flow rate in the other CSTR and hence an increase in the Ce4+ concentration. If the ceric ion concentration in one subsystem is higher than thevalue for [Ce4+Iav, it causes a decrease of Ce4+ions in the other subsystem. When the coupling strength fl is sufficiently high, both subsystems push each other out of the regions of oscillations. They reach stable steady states, one at high and the other at low ceric ion concentration. It is not predictable which of the two steady states is reached in a particular reactor. Coupling of Steady States. (a) Without Time Delay: Rhythmogenesis. There are two possibilities for the flow rate coupling of steady states of two MB systems, namely, either symmetrical or asymmetrical coupling. Symmetrical coupling indicates that the flow rates of the coupled MB systems correspond to steady states either above the subcritical or below the supercritical Hopf bifurcation. Asymmetrical coupling is obtained when one MB system is below (above) and the other MB system is above (below) the corresponding Hopf bifurcation. Most interesting is the caseof coupling two symmetrical steady states, both of low or high ceric ion concentrations (high or low flow rates). The concentrations are identical with those in the experiments on oscillator death without delay. When low flow rates are used (experiment, k ~= ,2.4~ X l e 3s-l, k0,2 = 3.3 X s-I; simulation, ko.1 = 3.4 X 10” s-I, k0,2 = 3.7 X le3 s-I (Figure 3a)), symmetrical coupling of two steady states leads to chemical oscillations which are identical in both reactors but about 180’ out of phase (Figure 4a). At higher @ (Figure 4b) the oscillations become irregular due to the close proximity of the subcritical Hopf bifurcation. At sufficiently high /3 the oscillations cease and two different steady states are obtained in the two reactors, one below and the other above the respective Hopf bifurcation. Symmetrical coupling a t high flow rates (experiment, ko,l = 6.5 X 10-3 s-1, k0.2 = 9.0 X 10-3 s-1; simulation, ko,,= 4.2 X 10-3 s-I, ko.2 = 4.7 X 1O-’s-I) leads to oscillations of identical amplitudes in both reactors which are 180’ out of phase. With increasing /3 both periods (Figure 3b) and amplitudes decrease until they disappear to form two steady states, one being below and the other above the respective Hopf bifurcations. Due to the symmetry of the setup, it is impossible to predict which of the two steady states is established in a given reactor.
400
325 300
’ 02
04
06
08 1 12 1 4 coupling strength [beta]
16
18
2
h
I
‘
0
01
02
03
04
05
06
07
08
I 09
coupling strength [beta]
Figure 3. Calculated periods vs j3 for coupled steady states of (a, top) high and (b, bottom) low ceric ion concentration (parameters, see text). SS = steady state.
On the other hand, unsymmetric coupling of two steady states preserves the steady states. The differences in Ce4+concentrations in the two reactors increase with coupling strength until limiting values are reached. The limiting value is determined by the high ceric ion concentration in one reactor extrapolated to kf= 0. This determines both the corresponding flow rate into theother reactor according to eqs 8 and the limiting value of the steady-state concentration. (b) With Time Delay. If the concentrations are adjusted so that the isolated subsystems are unable to oscillate over the whole range of flow rates, coupling according to eqs 8 with time delay causes chemical oscillations for t h e following reactor concentrations: 1 M H+, 0.8 mM B r , 0.3 M Br03-, and 0.3 mM Ce3+, which are identical in both reactors. For ko = 5.0 X l t 3 s-I, steady states of low ceric ion concentration in the uncoupled reactors are observed. If one starts the coupling procedure with a delay of T I = 7 2 = 150 s, one obtains relaxation oscillations in both reactors with a period of 400 s. The oscillations in the two reactors are in phase. A further increase of T in steps of 150 s shows a linear dependence of the period of oscillations Tin each subsystem. If T I= r2/2, one finds that T depends linearly on the sum of T I and 72 (Figure 5 ) . The same behavior is observed when two steady states of high Ce4+ concentration are coupled (ko = 1.2 X 10-3 s-1, @ = 8.0), where T I = 7 2 (or T I = ~ 2 / 2 ) .The oscillations are also in phase (Figure 6) and follow the relation T = T I + 7 2 + A, with A 100 s. However, rhythmogenesis is not found if the sum of the time delays is below a threshold value of -200 s. The computer simulations give similar results. Here, the imposed time delay causes the two subsystems to switch back and forth between the two steady states, one of high and one of low ceric ion concentration. The two systems require some time to adjust to the change in the flow rates caused by the relatively high coupling strength 8. These particular oscillations
-
12242 The Journal of Physical Chemistry, Vol. 97, No. 47, 1993
Holz and Schneider
l 1400
600
i
400
l' o
I 2000
2200
2600
2400
2800
200 200
3 m
time [E] loo0
400
600
800 1000 tau1 + tau2 [SI
1200
1400
I 1600
Figure 5. Linear dependence of the oscillation period on the imposed time delay, when two nodal MB systems (concentrations, see text) are coupled by flow rate (0,low steady states, 71 = 1 2 ; +, low steady states, T I = 272; 0 , high steady states, T I = 72; X, high steady states, T ] = 272). CSTR 1
1000
0 f
L
0
.3
200 01000
1500
2000
2500
3000
3500
4000
8
time [SI
loo0
800
-.,
1
200
i
1
1000
2000
3000
4000 5000 time [s]
6000
7000
8000
1000
2000
3000
4000 5000 time [SI
6000
7000
8000
600
400
7 nl
1
Fipre4. Experimental timeseries (both CSTRs) obtained from coupling of two steady states of high ceric ion concentration; OD = optical density of Ce4+ at 350 nm. (a, top) fl = 0.5. (b, middle) fl = 1.0; here the subcritical region is approached. The oscillations are irregular due to the fact that this region is extremely sensitive to experimental fluctuations. (c, bottom) fl 1.5. The coupling system approaches two asymmetric steady states.
are not due to the chemical dynamics of the coupled subsystems but to the mode of coupling of eqs 8. 6. Discussion
Figure 6. Time series in MBI and MB2 obtained by mutual flow rate coupling of two nodal steady states ( T I = = 400 s, j3 = 5.0). OD = optical density of Ce4+ at 350 nm.
One of the most interesting effects that can result from the coupling of two nonlinear oscillators is the transition from a common oscillatory state to two steady states. This phenomenon has been referred to as oscillator death or phase deathn5 BarEli2' found this behavior in calculations of a coupled Oregonator model, and it was found experimentally in mass-coupled BZ reactions by Crowley and Epstein.5 In this work the experimentally observed oscillatory death is caused by the linear coupling of two oscillatory states without time delay, and it is the consequenceof a sufficiently high coupling strength which leads to a 180' out-of-phase shift of the two oscillators. When the two oscillatory MB states are coupled, the
two subsystemscross their respective Hopf bifurcations into their steady-state regions, where one of the two CSTRs shows a steady state of high and the other of low ceric ion concentration. Rhythmogenesishas been found in numerical studies of simple chemical model systems by Kumar and co-workers22 and by Boukalouch et al.23 Experimentalconfirmationhas been achieved by mass coupling in the chlorite-iodide reacti0n.~3 Rhythmogenesis is also observed in the present experiments when two yet uncoupled MB systems show either nodal or focal steady states. For the case of nodal steady states (a perturbation decays exponentially to its steady state) the periods of oscillations in each reactor are identical and equal to the sum 71+ 7 2 + A, where
Control of Dynamic States A = 100 s for the present choice of concentrations. Here the oscillations resulting from coupling arise not from the dynamic behavior of the coupled chemical systems but from the method of coupling with time delay ( q s 8). When the two noncoupled MB systems showfocal steady states (a perturbation decays with a damped oscillation to its steady state), the resulting coupled limit cycles arise from the chemical nonlinearities of the experimental subsystems where the two focal steady states are coupled above or below their respective Hopf bifurcations without or with delay (the latter is not shown here). All experiments have been simulated on the computer by the flow rate coupling of two NFT models using the ‘‘new- rate constants with good agreement. The reported control of oscillatory death and rhythmogenesis from focal steady states may be of consequence for biological systems. Physiologically, the signal transmission between two cells requires a finite time T even though the cells may fire synchronously. Acknowledgment. This work was supported in part by a grant from the Stiftung Volkswagenwerk and from the Fond der Chemischen Industrie. References and Notes (1) Alamgir, M.; Eptein, I. R. J. Am. Chem. Soc. 1983, 105, 2500. (2) (a) Marek, M.; Stuchl, I. Biophys. Chem. 1975,3,241. (b) Stuchl, I.; Marek, M. J . Phys. Chem. 1982, 77, 2956. (c) Dolnik, M.; Marek, M. J . Phys. Chem. 1988,92.2452. (3) Marek, M. In Temporal Order; Rensing, R., Jaeger, N. I., Eds.; Springer Verlag: Berlin, 1984; p 105.
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12243 (4) Marek, M. In SeljOrganization by Nonlinear Irrewrsible Processes; Ebcling, W., Ulbricht, H., Eds.; Springer Verlag: Berlin, 1986; p 133. (5) Crowley, M. F.; Epstein, I. R. 1. Phys. Chem. 1989, 93, 2496. (6) Bar-Eli, K.; Reuveni, S. J. Phys. Chem. 1985,89. 1329. (7) (a) Doumbouya, I. S.;Monster, A. F.; Doona, C.; Schneider, F. W. J . Phys. Chem. 1993, 97, 1025. (b) Hauser, M.; Schneider, F. W.To be published. (8) Weiner, J.; Schenider, F. W.; Bar-Eli, K. J. Phys. Chem. 1989,93, 2704. (9) Weiner, J.; Holz, R.; Schneider, F. W.; Bar-Eli, K.J. Phys. Chem. 1992, 96. 8915. (10) Zeyer, K. P.; Holz, R.; Schneider, F. W. Ber. Bunsen-Ges. Phys. Chem. 1993, 97, 1112. (11) Pyragas, K. Phys. Le??.A 1992, 170,421. (12) Mankin, J. C.; Hudson, J. L. Chem. Eng. Sci. 1986, If, 2651. (13) Crowley, M.F.; Field, R. J. J . Phys. Chem. 1986, 90. 1907. (14) (a) Crowley, M. F.; Field, R. J. In Nonlinear Phenomena in Chemical Dynamics; Vidal, C., Pacault, A., Eds.; Springer Verlag: Berlin, 1981; p 147; (b) k c ? . Notes Biomath. 1986, 66, 86. (15) Bar-Eli, K.; Geiseler, W. J . Phys. Chem. 1983, 87, 3769. (16) Bar-Eli, K.; Field, R. J. J. Phys. Chem. 1990, 94, 3660. (17) Noyw,R. M.;Field.R. J.;Thompson,R. C. J. Am. Chem.Soc. 1971, 93,-7315. (18) Bar-Eli, K. In Nonlinear Dynamics in ChemicalSysrems; Vidal, C . , Pacault, A., Eds.; Springer Verlag: Berlin, 1981; pp 228-239. (19) (a) Geiseler, W. Ber. Bunsen-Ges. Phys. Chem. 1982,86, 721. (b) Geiseler, W. J. Phys. Chem. 1982, 86, 4394. (c) Geiseler, W.; FOllner, H. Biophys. Chem. 1977, 6, 107. (20) Orbin, M.; DeKepper, P.;Epstein, I. R. J. Am. Chem. Soc. 1982, 104, 2657. (21) Bar-Eli, K. J. Phys. Chem. 1984, 88, 3616. (22) Kumar, V. R.;Jayaraman, V. K.;Kulkarni, B. D.; Doraiswamy, L. K. Chem. Eng. Sci. 1983,38, 673. (23) Boukalouch, M.;Elezgaray, J.; Arneodo, A.; Boissonade, J.; De Kepper, P. J. Phys. Chem. 1987, 91, 5843.
