4407
J . Phys. Chem. 1987, 91, 4407-4410
Periodic and Aperiodic Regimes in Coupled Reaction Cells with Pulse Forcing M. Dolnik, E. Paduiikovi, and M. Marek* Department of Chemical Engineering, Prague Institute of Chemical Technology, 166 28 Prague 6, Czechoslovakia (Received: November 10, 1986; In Final Form: February 20, 1987)
Experimental observations of periodic and aperiodic regimes in two coupled cells, where one cell is forced, are discussed. The parametric plane "intensity of interaction between the cells - frequency ratio of perturbations" documents the presence of synchronized periodic regimes corresponding to the first two rows of the Farey sequence together with an intermediate aperiodic behavior. Transient extinction of oscillations was observed at higher levels of intensity of the interaction between the cells.
Introduction Nonlinear chemical systems exhibiting both periodic and aperiodic oscillations have become prototypes for studies of nonlinear dynamics. Aperiodic oscillations are observed, for example, in continuous stirred tank reactors with constant inlet concentration^,^^ in single flow-through reaction cells with periodic variation of inlet concentration^,^-' and in a system of two coupled reaction cells with mutual mass8-10,11 and electric chargeg exchange. In the system of coupled cells, there are observed (depending on the intensity of an interaction on the initial ratio of the frequencies) periodic, quasiperiodic, and chaotic oscillations, and both transient and permanent stopping of oscillations in one or both oscilIator~.~.'~ Single, coupled, and forced oscillating reaction cells also serve as experimental models for various excitatory biological systems.12 However, theoretical and modeling studies are much more numerous than experimental studies. Here we present several experimental observations of dynamic regimes in a new experimental arrangement-two coupled flowthrough reaction cells with a concentration pulse forcing. It is actually a system of three oscillators with uni- and bidirectional coupling. It models both simple types of excitatory tissues and forms at the same time the simplest approximation of distributed systems with pulse perturbations at the boundary.
*
I
-
1
1
.
C
1
I
Experiments The standard Belousov-Zhabotinsky oscillatory reaction,13 oxidation of malonic acid by bromate in the presence of H2S04, with Ce3+/Ce4+catalyst, was used in the experiments. Bromide ions form an important reaction intermediate and the actual concentration determines the length of the oscillation period. The experimental arrangement is schematically shown in Figure 1. Pairs of solutions (A, 0.05 M KBrO, and 0.05 M malonic acid, and B, 0.001 M Ce(S04)2and 1.5 M H2S04)were fed into (1) Swinney, H. L.; Roux, J. C. In Nonequilibrium Dynamics in Chemical Systems, Vidal, C., Pacault, A,, Eds.; Springer-Verlag: West Berlin, 1984; p 124. (2) Epstein, I. R. J . Phys. Chem. 1984, 88, 187. (3) Hudson, J. L.; Mankin, J. C. J . Chem. Phys. 1981, 74, 6171. (4) Marek, M.; Schreiber, I. Stochastic Behauiour of Deterministic System (in Czech);-Academia: Praque, 1984. ( 5 ) Dolnjk, M.; Schreiber, I.; Marek, M. Phys. Lett. 1984, 100A, 316. (6) Dolnik, M.; Schreiber, I.; Marek, M. Physica 1986, 21D, 78. (7) Schneider, F. W. Annu. Rev. Phys. Chem. 1985, 36, 347. (8) Mankin, J. C . ; Hudson, J. L. Chem. Eng. Sci. 1986, 41, 2651. (9) Crowley, M. F.; Field, R. J. J . Phys. Chem. 1986, 90, 1907. (10) Marek, M. In Lecture Notes in Biomathematics, "Modelling of Patterns in Space and T i p " ; Springer-Verlag: West Berlin, 1984; p 214. (1 1) Marek, M.; Dolnik, M.; Schreiber, I. In Selforganization by Nonlinear Irreuersible Process, Ebeling, W., Ulbricht, H., Eds.; Springer-Verlag: West Berlin, 1986; p 133. ( 1 2) Holden, A. V. Bull. Math. Biol. 1983, I S , 443. (13) Field, R. J.; Burger, M., Eds. Oscillations and Trauelling Waves in Chemical Systems; Wiley: New York, 1985.
0022-3654/87/2091-4407$01.50/0
1
3
4
2
Figure 1. (a) Experimental arrangement: A, Ce(SO& and H,SO,; B, malonic acid and KBrO,; C, KBr; P, pump; T, thermostat: SV, solenoid valve; CA, capillary; PE, platinum electrode; CE, calomel electrode; SB, salt bridge; Th, thermometer;S, stirrer; ER, electromagnetic relay. (b) A cross section showing the coupling of the reaction cells: 1, reactor R,; 2, reactor R,; 3, movable barrier; 4, permeable glass cloth.
two reaction cells (volume 70 mL). Reaction temperature (35 "C) and the intensity of stirring (400 rpm) were chosen in such a way that the level of noise in both cells was below 2% relative to the oscillation period of the uncoupled cells. The course of the redox potential in the cells was followed by a pair of Pt and calomel electrodes. Pulse addition of bromide ions was controlled by a microcomputer. Here results are discussed where only one of 0 1987 American Chemical Society
4408
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987
Dolnik et al.
R2
R1
log
I I1
105s
950
1050
1050
95G
95c
Hl
log P
0
1050
‘050
9 5c
950
1056
1055
950
95:
e)
0 T :s:
c
5 .
f [Hz]
’” ’
f [Hz] O * ’
Figure 2. Characteristic course of the redox potential in reaction cells. The vertical lines denote the time of the pulse perturbation: R,, “perturbed”cell: R2, “unperturbed”cell; TF= 90 s; a = 5 X IOd M Br-; intensity of interaction. (a) z = 0. (b) z = 6. (c) z = 9. (d) z = 15.
Figure 3. Power spectra evaluated from the recordings of redox potential: (a) power spectra for R , and z = 6 up t o f = 0.5 Hz; (b) z = 0; (c) z = 6, (d) z = 9, (e) z = 15; ---, forcing frequency; -.-., frequency of autonomous oscillations; TF = 90 s; a = 5 X M Br-.
reaction cells, denoted R,, was perturbed (R2 is the unperturbed cell). The coupling arrangement of the reaction cells is depicted in Figure 1b. The reaction cells were machined from plexiglass and were connected by an aperture with a movable barrier made from Teflon. The aperture was covered by a permeable glass cloth (made from bunches of fibers of 0.5 mm with an average mesh size 0.62 mm), which limits convection currents between the cells. The intensity of the mass exchange between the cells was controlled by changing the dimension of the aperture in the common wall of the cells and was set at a sequence of values z = 0, 3, 6, 9, 12, and 15, corresponding to the values of mass exchange coefficients 103K= 0,0.87, 4.05, 7.21, 10.37, and 13.61 s-’. The mass exchange coefficients were determined experimentally by using a standard tracer response analysis.I6 The values of the coefficients thus include both convective and diffusive fluxes between the cells; the fluxes were approximately (on a time average) the same in both directions. Other ways of coupling are also possible. Electrical connection of the cells can be also used;9 however, it represents a qualitatively different type of coupling. Results for two concentration levels (“forcing amplitude, a”) of bromide ions used for the perturbation, 5 X and 5 X M Br- (concentration after dilution in one reaction cell), are reported. When the effects of single pulse additions on the phase shift of the oscillations (expressed as the phase transition curve, PTCI4) were studied in the single reaction cell, then the use of these two Br- ion concentrations resulted in two different types of PTC.5.6 The regime in the reaction cell will be in the case of periodic perturbations called periodic if in the course of q-pulse perturbations, p oscillations always occur; hence this periodic regime
is characterized by the phase-locking ratio p / q . Characteristic time courses of the redox potential for the pulse M and the period of perturbations T F = concentration 5 X 90 s in two reaction cells are shown in Figure 2 and the corresponding power spectra in Figure 3. The initial periods of oscillations (without coupling) were equal to 90 s in R1 and 59 s in R2. The addition of the bromide pulse is reflected by a sharp decrease of the potential in Figure 2. At zero coupling intensity between R l and R2 ( z = 0), the regime in R l was fully synchronized, Le., p / q = 1/ 1. At high coupling intensities ( z = 12 and z = 15) the regime was again periodic but now with the phase locking ratio p / q = 3/2. The course of oscillations in R, is affected both by periodic pulse perturbations and by the course of oscillations in cell R2 (particularly for strong coupling intensities). In the case of weak interaction between cells ( z = 3) the frequency in the perturbed reactor R, was equal to the frequency of the perturbation ( p / q = l / l ) , but in the unperturbed reactor R2 a periodic regime with a frequency which was close to the original autonomous frequency was recorded. The power spectra were computed from 4096 values of the redox potential (sampling frequency 1 Hz) by a FFT algorithm. These are presented in the figures only to a frequency of 0.1 Hz, as most important characteristic frequencies occur in this range. The power decreases whenf = 0.5 Hz, which indicates the insignificant role of noise at this frequency, as is illustrated in one case (cf. Figure 3a). We can identify periodic regimes in R l under the conditions depicted in Figure 3, b and e ( z = 0; 15), and aperiodic regimes in Figure 3, e and d ( z = 6; 9). The same form of the power spectra in Figure 3e for both cells indicates synchronization of oscillations. Hence coupling to other oscillating cell can cause a change to other phase locking ratio when a certain minimum level of the intensity of the interaction is reached. The values of the powers for both reactors at characteristic frequencies fA andfF cf, is the frequency of autonomous oscil-
(14) Winfree, A . T. The Geometry of Biological Time: Springer-Verlag: New York. 1980.
lag;I;:i
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987 4409
Periodic and Aperiodic Regimes in Two Coupled Cells
15
‘
6
O
g
P
r
:
Z
x
X
4
12
x
9 Z
X
0
6
12
0
z
6
l 2z
Figure 4. Amplitudes of the power spectra obtained from recordings of redox potential TF= 90 s, a = 5 X M Br- at (a) forcing frequency of 0.01 11 1 Hz and (b) frequency of autonomous oscillations of 0.01724 Hz: 0 , values computed for R,; X, values computed fro R,.
“I
1.0 ;::
4, L
,
.
,
..I._
I
2oJ
40
60 ti!Sl t:
I. rlSl
..
is1
6o
1 ’*
401
i’l 0
, 20
,a 40
,
1 2ol
60t,lsl
0
c
io
1.5
’
2.0
TF 0 ‘
..
20
3
R2
R1
0
6
io
6cI t,!s 1
Figure 5. Stroboscopic map ti+, VS. ti for TF= 90 s, a = 5 X lo4 M BrC (a) z = 0, (b) z = 6, (c) z = 9, (d) z = 15. Numbers denote subsequent points of the map.
lations,jp is tne rrequency or perioaic perruroations) are piotrea vs. z in Figure 4. The powers remain constant for values of z > 9 (where oscillations in both cells are almost always synchronized). Figure 4b also presents a clear example of the stabilization of the frequency of a chemical oscillator with respect to external perturbations by coupling with another nonlinear oscillator. A stroboscopic map was constructed to obtain additional insight into the character of the observed regimes. The construction is schematically shown in Figure 2a. The state of the system at the time of the perturbation is characterized by ti ( t i was used as the
Figure 6. Parametric plane (frequency of perturbation-intensity of interaction), a = 5 X M Br-; periodic regimes in R, with the phase locking ratios 1:l (0),4:3 (*), 3:2 (+), 5:3 (A), 2:1 ( O ) , 7:4 (X); A, aperiodic regime.
accuracy of its measurement was better than that of the redox potential). The value of ti thus denotes the time elapsed between the last jump of the redox potential and the pulse perturbation. The map in Figure Sa, constructed for R,, corresponds to a one-periodic regime. The periodic regime in R, in the stroboscopic map is here characterized by points which are distributed along two straight lines. The distributions of points in both stroboscopic maps in Figure 5b (intermediate level of mass exchange) do not allow an interpretation based on 1D mapping. The dynamics is aperiodic (chaotic) as it follows from the random order of visiting of different parts of the stroboscopic maps (cf. sequence of numbers). When the intensity of the interaction is further increased (z = 9), the stroboscopic maps agains become approximately onedimensional, cf. Figure 5c. In this case the points are located along curves which can be described by a nondecreasing function ti+l =Ati).This suggests the presence of a quasiperiodic or periodic behavior. The regimes in both reactors are fully synchronized. For a still higher intensity of interaction ( z = 15) a two-periodic regime was observed, as is illustrated in Figure 5d. The results of experiments observed in R1 are summarized in the parametric plane “intensity of interaction-period of perturbation” for the “forcing amplitude” 5, X lo4 M Br- in Figure 6. The period of perturbations is related to the average “free” length of the period of oscillations ( T o )in R,. (Practically only one oscillation was affected by the pulse addition; hence the oscillations without a perturbation are called “free”.) The dashed lines denote the estimated locations of boundaries between periodic and aperiodic regimes. These rough estimates depend on the level of experimental noise; weaker resonances (with higher values of p l q ) are imbedded in aperiodic regions. Relatively large “Arnold tongues” exist for the first two complete rows of the Farey sequenceI5 111, 413, 312, 513, and 211. In one experiment the regime 7/4 was also found (belonging to the next row of the Farey sequence, cf. Figure 6). “Arnold tongues” of periodic regimes with higher numbers in the ratio p / q are narrow and they were not observed due to the presence of noise. Dynamic regimes in the unperturbed reactor R, are for high and intermediate values of z quite similar to regimes in R1 (cf. Figures 2, 3, and 5 ) . In reactor R2,oscillations were observed (15) Hardy, G. H.; Wright, E. M., Eds. An Introduction to the Theory of Numbers, 4th ed.; Claredon Press: Oxford, 1954. (16) Marek, M.; HavliEek, J.; VIEek, J. “Network of Coupled CSTR’sModel Experimental System“, Proceeding of the Fourth European Conference of Mixing, April 1982; Organized by BHRA Fluid Engineering, Cranfield, Bedford, England; Paper 54, p 339.
4410
The Journal of Physical Chemistry, Vol. 91, No. 16, 1987
I1
Dolnik et al.
1000 s I
4
c
Figure 7. Transient extinction of oscillations as a function of the intensity of interaction, "forcing amplitude" 5 of the redox potential in R,; lower curve, recording in R,.
with a frequency which is close to autonomous for low values of the intensity of interaction. The courses of the redox potential for a lower level of the forcing amplitude (corresponding to a concentration of Br- ions equal to 5X M Br- and a perturbation period TF= 100 s) are depicted in Figure 7. When the cells were decoupled ( z = 0), the periodic regime with the phase locking ratio 3/2 was found in the perturbed reactor R1 and it was stable for a low intensity of interaction (z = 3). When the intensity of interaction was increased to z = 6, an apparently intermittent aperiodic regime was observed. Long periodic intervals with the phase locking ratio 1/ 1 occurring in R, alternated with short aperiodic bursts. The oscillatory regime in R, reflected the behavior of R,; oscillations with the same frequency but in an opposite phase ("antiphase synchronization") were observed in the reactor. At still higher intensities of interaction ( z = 9, 15) a transient extinction of oscillations was observed, cf. Figure 7 . It occurred simultaneously in both reactors, but only in situations where the pulse perturbation took place immediately after the fast increase in the redox potential (corresponding to high concentration of Ce4+ ions). In this phase of the reaction sequences pulse addition of bromide accelerated the formation of bromides and this is connected with the fast consumption of Ce4+ions (they are reduced to Ce3+ ions). Hence an unstable structure arises, where Ce3+ ions dominate in R, and Ce4+ ions are prevalent in R2. This behavior corresponds to such a type of a phase space where a limit cycle and a stable stationary state coexists. If the pulse addition of Br- ions shifts the trajectory in the phase space into the neighborhood of a stable stationary point, then extinction of oscillations is observed. Further additions cause a shift of the
X
1
M Br-: upper curve, recording
trajectory into the region of attraction of the limit cycle and thus only a transient extinction of oscillations is observed when periodic perturbations are applied. The time interval of the transient extinction increased with increasing intensity of interaction between the cells. Permanent extinction of oscillations was not observed when periodic perturbation was used. On the other hand, if only a single pulse perturbation was applied than a permanent extinction of oscillations was observed for higher intensities of interaction. More complete results on single-pulse perturbations will be presented elsewhere.
Conclusions Periodic and aperiodic regimes observed in the system of two coupled oscillations form a typical example of dynamic dissipative structures. The regimes in the pulse-perturbed coupled oscillators can be controlled by the choice of three parameters-amplitude and frequency of perturbations and the intensity of interaction. Transient extinction of oscillations in pulse perturbed coupled cells was observed at intermediate values of the forcing amplitude, similarly as in the case of two autonomous coupled chemical
oscillator^.^^^^ The observed transitions between different phase synchronized regimes via aperiodic regimes depended on the intensity of interaction for high amplitudes of perturbations; this offers a number of possibilities for interpretations of similar phenomena observed in cardiac arrhytmias, neurobiology, and excitatory tissues in general. Acknowledgment. The authors thank Dr. I. Schreiber for stimulating discussions and helpful comments.
