J. Phys. Chem. 1993,97,1025-1031
1025
Deterministic Chaos in Serially Coupled Chemical Oscillators Sory I. Doumbouya, Amo F. Miinster, Christopher J. Doona, and Friedemann W. Schneider' Institute of Physical Chemistry, University of Wiirzburg, Marcusstrasse 9- I I, 0-8700 Wiirzburg, Germany Received: May 6, 1992
Two oscillatory Belousov-Zhabotinsky reactions are serially mass coupled to generate additional periodic states, period doubling, phase locking, quasiperiodicity, and chaos. The time series for these states are characterized in terms of their power spectra, attractors (reconstructed by singular value decomposition), PoincarC sections, return maps, Lyapunov exponents, and various dimensionalities (Dq spectra). Adding trace impurities to the system does not alter the observed chaos. Model calculations using a partially reversible Oregonator model qualitatively agree with the experimental findings for the coupled oscillators.
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Introduction There has been extensive interest in developing methods for coupling nonlinear systems in order to generate more exotic nonlinear phenomena such as chemical chaos,14 entrainment,s rhythmogenesis? and phase death.' Marek and StuchllJ generated aperiodicityby parallel coupling the Belousov-Zhabotinsky reaction (BZ) by means of mass exchange across a perforated wall common to two adjacent reactors. Electrical coupling of two independent BZ oscillators, in which electrons flow between systems without exchangingreaction mixtures, has also generated cha0s.3.~In neither of these studies was theobserved aperiodicity fully analyzed819in order to confirm the presence of deterministic chaos. In this report we demonstrate that serially coupling two BZ oscillators is an effective method for generating deterministic chaos and other exoticdynamic behavior. Serial coupling involves connecting two CSTR's via an aperture of variable size, thereby permitting mass exchange between the two reactors. Fresh reagents are fed into only the first reactor, and overflow solution is removed only from the second CSTR. One may also view this setup as a single CSTR comprised of independent inflow and outflow chambers that are separated by a perforated wall. Allowing the reaction mixture to transfer by convection between the chambers produces new dynamic behavior in the outflow compartment. Specifically, the observed scenario consists of a route to chaos via a period doubling sequence (PI, Pz, and P4 oscillatorystates) and some states of the Universal sequenceIJ0-12 (the oscillatory P'4, Pj, and P7 states and two independent chaotic states). The time series displaying these behaviors are characterized in terms of their Fourier spectra, reconstructed attractors, PoincarC sections, and return maps. Lyapunov exponents and generalized dimensions are also determined for the chaotic states to confirm their deterministic nature. Calculations simulating the observed phenomenology have been carried out by using a modified Oregonator model,13J4 and comparisons with the uncoupled system are made.
Experimental Section Apparatus and Materials. The coupled CSTR apparatus used in our experiments consists of two vertically stacked reactors with a common wall bearing an aperture (Figure 1). Reagents solutions are flowed into the lower reactor (VI = 4.5 mL). Reaction mixture from V , then enters the upper reactor (Vz = 15 mL) through the aperture, and outflow is removed from Vz by aspiration. The reactors and the perforated wall between them are made of Plexiglas. Whereas excessive noise prevented stirring the propeller in the upper reactor at rates greater than To whom correspondence should be addressed.
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600 rpm, variations in stirring rate from 300 to 1500 rpm in the lower reactor could be achieved by using a magnetic stirring bar. The entire apparatus was lowered into a water bath unit and thermostated at 28 OC. The lower reactor in this arrangement is completely submerged in the water bath, rendering it inaccessible to a redox electrode, and thereby precluding direct detection of the dynamic behavior in the bottom CSTR. A self-designed high-precision piston pump that is driven by a stepping motor was used to control the flow of reactants into the CSTR. Three gas-tight Hamilton syringes (50 mL each) served as the reagent reservoirs. The first syringe contained 0.75 M malonic acid (MA) and 0.0025 M cerous sulfate, the second stored 0.3 M potassium bromate, and the third contained 0.6 M sulfuric acid. Each step of the motor advanced the pistons 100 nm into the syringes, producing a total pulse into the CSTR of 0.22 pL per step. Virtually constant flow of solutions into the CSTR is ensured by electronic control of the motor steps. The stepping frequencies used in these experiments ranged from 43 to 85 Hz, corresponding to residence times of 15-30 min. Determination of the Coupling Constant (QJ. The coupling constant under batch conditions was determined by carrying out the following procedure. Without stirring, both reactors were filled with a 0.6 M sulfuric acid solution. After 0.1 mL of 0.25 M ceric sulfate (cg) was injected into the lower reactor, stirring was initiated and the temporal increase in [Ce+4]in the upper reactor [&)I was monitored with a platinum redox electrode. This procedure was repeated at various sizes of the aperture and at varied stirring rates in the lower reactor (300-1500 rpm), while the stirring rate in the upper reactor (600 rpm) remained constant (Figure 2). The concentration of Ce+4increased in the upper reactor according to the following kinetics equation for 0 1993 American Chemical Society
Doumbouya et al.
1026 The Journal of Physical Chemistry, Vol. 97, No. 5, 1993 a) EXPERIYLV 7
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in which k, (determined by exponential fitting using the program PLOTFIT) is the exchange rate constant. For zero flow, the product kcV2 yields the value for Qc (mL s-l). However, we expect Q, to vary from these values during flow experiments. Nonetheless,this procedure was useful in determiningthe optimal aperture dimensions and stirring rates that would facilitate our exploration to find chaos in the serial-coupled BZ system under the present conditions. Effects of Varying couplinsStwngth. The influenceof coupling strength on the observed dynamics was determined by carrying out flow experiments with the BZ and varying the parameters governing the coupling strength. While the aperture size was kept constant (5.0mm2)and the stirring rate in the upper reactor was maintained at 600 rpm, the stirring rate in the lower reactor was varied from 300 to 1500 rpm. At a flow rate of 8.6 X 10-4 s-I ( T = 20 min), incremental increases in the lower stirring rate induced successive transitions in dynamic states from C j (300 rpm) to p l 4 (600 rpm), CI(900 rpm), PI (1200 rpm), and P3 (1500 rpm). At the largest aperture setting (7.0 mm2) and constant stirring rates the system does not settle into a stable chaotic or PI state, rather only transient behavior fluctuating between the two states is observed. Aperture sizes of 1.5 mm2 interferred with the regular flow of solution between VIand V2, and noisy time series were measured in the upper reactor. Thus, the aperture size and stirring rates must be of some optimal setting in order to produce effective coupled mass transfer between the solutions in VI and V2. Further, coupling strength acts as a bifurcation parameter, which determines in part the dynamic state of the system observed in the upper reactor.
Results md Discussion Complex Behavior. Decreasing the flow rate through the total CSTR from kr = 1 X S-I (residence time T = 17 min) to 6.0 X lo4 s-I led to transitions between an array of periodic and chaotic states in the upper reactor of the serially coupled BZ system (Figure 2). These observations are described below. The aperture size (5.0 mm2) and stirring rates in the upper reactor (600 rpm) and lower (900 rpm) CSTR were held constant in order to best approximate constant coupling strength, and variations in Q, arising from changes in the flow rate were neglected.
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As kr decreases to 8.5 X lo4 s-I ( T 20 min), the system undergoestransitionsvia period doubling bifurcations from simple periodicity (PI oscillations) to P2 oscillations, in which each cycle is comprised of one large- and one small-amplitude oscillation, followed by the first chaotic state CI. With continuing decreases in flow rate the system enters a p14 state, appearing through a tangent bifurcation, followed by a P3 state. Although the Y4 state should lose stability to a C2 state, CZwas not observable within our experimental resolution. Further flow rate decreases revealed a C3 chaotic state succeeded by a Py periodic window and a P4 and a P2 state. Although we were unable to resolve a C4chaotic state following the P7 window, its existenceis suggested by the transition from P4 to P2, a reverse period-doubling bifurcation. The dynamic behavior is very much different for the serially coupled BZ oscillators than for the uncoupled system. At kr greater than 6.0 X 10-4 s-I ( T = 30 min) the free-running BZ shows only PI oscillation^^^-'^ and not a sequence of complex behavior as described above. The uncoupled BZ at the lower flow rates designated by an asterisk in Figure 3a passes through a sequence of periodic and chaotic ~tates.15-I~Therefore, serial coupling has the effect on an oscillatory BZ system of inducing complex behavior similar to that seen at lower flow rates. Dynamical System Analysis. Although aperiodic time series and the known period doubling route to chaos have been observed, they alone do not sufficientlydistinguishthe deterministic nature of the observed aperiodi~ity.**~ Therefore, we have used dynamical system theory to analyze the observed Pz,Pa, P7, P3, and C3 states in terms of their time series, Fourier spectra, reconstructed attractors, Poincar6 sections, and return maps. The time series and Fourier spectra shown in Figure 4 confirm the identity of these states. The Fourier spectra for periodic states P2, P4, P3, and P7 show the fundamental frequency as the second, fourth, third, and seventh harmonics, respectively. The Fourier spectrum for the chaotic state C3 is broad banded. The return maps for P2 and P4 (data not shown) reveal two and four point clouds, respectively,scattered by experimentalnoise, a further indication of a period-doubling bifurcation. I n addition to the attractor, PoincarC section, and return map, further analyses were carried out on the Cjstate. Figure 5 shows the Lyapunov exponents, the log-log plot used to determine the correlation dimension (Om), and a spectrum of generalized Rmyi dimensions (D, as calculated by the nearest-neighbor method2+22) fork = 100inlinearscaling. The"strange"attractor, therelatively continuous curve representing the PoincarC section, and the single maximum seen in the return map are all strong indications that C3 truly is deterministic chaos and not the result of amplified noise. The Lyapunov exponents shows signs of +, 0, and - (Am,, = 0.1 bits/s), which are also characteristic of chaos. The value of D,,, calculated by the sphere-countingalgorithm23 is greater than two(-2.3),anditisingoodagreementwiththeDqspet", which yields values of the Hausdorff dimension (Do = 2.52), the
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Trace Impurities. The chaos observed in the current system is insensitive to trace impurities. Whereas in all of the experiments described thus far only purified malonicacid was used,chaos was
Doumbouya et al.
1028 The Journal of Physical Chemistry, Vol. 97, No. 5. 1993
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has been taken into consideration. The parameter values, reagent concentrations, reactor volumes, and Qc are given in Table 11. The temporal variations of concentrations (represented by x) in the individual reactors (designated by the subscripts 1 or 2) are given by the following equations, combining coupling terms, flow terms, and the set of ordinary differential equations representing the mass-action kinetics of the chemical mechanism (abbreviated as
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Figure 6. (a) Experimental time series and (b) Fourier spectrum of the quasiperiodicstate with V I= V2 = 4.5 mL. The Fourier spectrum shows two fundamental frequencies (WI = 1.5 X Hz and w2 = 2.09 X lo-) Hz) and their linear combinations (w1 - w2 = 1.33 X Hz) and (WI 02 1.77 X Hz).
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the dynamic behavior."+I9 The source of chaos observed in the serial-coupled BZ appears to be of a different origin than that in the uncoupled system. Influence of Reactor Volumes. Increasing the volume ratio of the reactors from Vl/V2 = I / 3 to Vl/V2 = 1 produces changes in theobserveddynamics. Using two 4.5-mL reactors, an aperture setting of 5.0 mm2, a lower stirring rate of 900 rpm, and the same reagent concentration as in the previous experimentsphase locking and a quasiperiodic route to chaos were observed.28 Quasiperiodicity is indicated29 by the time series and Fourier spectrum (Figure a), in which two fundamental frequencies are seen. Simulations. Calculations using a partially reversible fourvariable Oregonator model were carried out to simulate the phenomenology occurring in both reactors in the serially coupled system. The Oregonator13J4was chosen because it gives good agreement with the BZ, it is relatively insensitive to parameter values,5 and it has been used in modified forms to simulate the behavior in other coupled BZ systems.3.5JO-32 It should also be noted that the Oregonator model does not predict the Occurrence of any chaotic states in the "free running" mode. The four variables reflect the concentrations of B r , HBr02, BrOp, and Cd+in each reactor. Listed in Table I are the component reactions of the model, in which the reversibility of HBr02 autocatalysis33
The factor 3 indicates that three feed solutions were used. Numerical simulations were made by using a Gear integrator routine34on a Siemens 7860L computer. The simulated sequences of periodic and chaotic behavior occurring in the upper reactor are in good qualitative agreement with those observed experimentally (Figure 3b). At high flow rates (kf= 1.5 X 10-3s-I) the systemundergoesa Hopf bifurcation, signifying the transition from a steady state to PIoscillations. As the flow rate is decreased further, the simulations show period doubling and a sequence of transitions between several periodic and chaotic states (Figure 3b). Further analysis has been carried out on the simulated CItime series (Figure 7). The Fourier spectrum, reconstructed attractor, PoincarB section, and return map all suggest chaotic motion. Thus, despite simplifications and quantitative differences between the observed and simulated behaviors, the model provides some indication of how chaos arises in the coupled system. The simulated behavior for the lower reactor shows much different character from that of the upper reactor over the flow rate range investigated. At the highest flow rates, the lower reactor shows only stationary behavior. At kr = 1.5 X s-I, simple PI relaxation oscillations evolve, which persist throughout the flow rate range we examined. Quantitative differences arising between experiment and simulation result from some of the approximations and simplifications used in the model. Mass coupling involves the exchange of all species between reactors, and so the concentrations of the reagents in excess are assumed to be equally distributed in both reactors. More important than the volume of the individual reactors used to simulate the behavior is the ratio of VIto V2. Thus, the discrepancy between VI and V2 for the simulations from those of the experimental was one purely for convenience, and it helped us predict the optimal reactor sizes for these experiments. Nevertheless, setting V2 = 15 mL yields similar behavior in the simulations. Although we were able to simulate qualitativelythe observed behavior over a wide range of Qcvalues, Qc was estimated as 5 X mL/s, a value similar to that used by GyBrgyi and Field" in their modeling studies of chaos in the BZ as a coupling problem. We do not use the value of Qc calculated earlier from the batch experiments because Qc is expected to change during flow experiments.
Doumbouya et al.
1030 The Journal of Physical Chemistry, Vol. 97, No. 5, 1993
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Conclusions The primary goal of this study was to generate chemical chaos in the BZ by means of serial mass coupling. The deterministic nature of the chaos observed in the upper reactor has been confirmed by establishing the period-doubling route to chaos, as well as by rigorous analysis of the aperiodic time series using the tools of dynamical systems theory (the reconstructed attractor, PoincarC section, return map, Fourier spectra, Lyapunov exponents, and Renyi dimensions). The chaos arises from coupling, and it appears to be different from that seen in the uncoupled system, which is sensitive to trace impurities. Numerical simulations based on a four-variable Oregonator support the experimental findings. Chemical chaos arises from interactions between independent oscillatory cycles. We surmise that the method of serial coupling CSTR's promotes the evolution of independent oscillatory cycles that interact to produce chaos. Restricting the flow of fresh reagents into VIonly, allowing reaction mixture to fill V,,and
removing outflow from only V,may create disparate compositions within the reactors. Correspondingly, independent oscillators may arise reflecting these differencies in composition. The concerted action of flow and coupling would produce interactions between the cycles. Whereas flow (QJ contributes unidirectional mass transfer from VIinto VZ,coupling (Qc) exchanges mass bidirectionallybetween reactors. Therefore, chaos in the present system may arise as a result of the mass exchange between independent oscillatory cycles that were induced by the reactor flow setup. The volumes of the reactors comprising the coupled system also influence the observed behavior. The residence times of the reactors determine the extents to which reactions can occur, which influences the compositions of the reaction mixture contained within each CSTR. Therefore, reactor volumes may also promote the formation of distinct oscillatory cycles, and thereby influence the development of complex behavior. The coupling strength Qc may be considered as a bifurcation
Chaos in Chemical Oscillators parameter that, when varied, induces transitionsbetween dynamic states. Aperture dimensions, stirring rates, and reactor volumes combine to determine the coupling strength in our experimental setup. Whereas in the absenceof adequatecoupling only periodic relaxation oscillations (PI)or steady states were observed, increasing Qcthrough larger aperture sizes or elevating stirring rates leads to entrainment and nonperiodic behavior in the upper reactor. It would be of interest to find an experimental setup amenable to detection in the lower reactor in order to support the results of our modeling and to determine the effects of serial mass coupling on reactions other than the BZ. These studies would yield insight into the behavior occurring in biological processes on the basis of the interactions between chemical oscillator^.^^
AcLwwledgment. This work was supported by the Stiftung Volbwagenwerk and the Fonds der Chemischen Industrie. We thank T.-M. Kruel for the implementation of the computer programs for thedimensionalanalysis. C.J.D. would like to thank the National Science Foundation for providing a postgraduate fellowship.
R e f e m Uni Notes ( I ) Marek, M.; Stuchl, I. Biophys. Chem. 1975, 3, 241. (2) Marek. M. In SynergericsFar From Equilibrium; Pacault, A., Vidal, C., Edr.; Springer Verlag: Berlin, 1979; p 12. (3) Crowley, M. F.; Field, R. J. J. Phys. Chem. 1986, 90,1903. (4) Crowley, M. F.; Field, R. J. In Nonlinear Oscillations in Biology and Chemistry, Othmer. H.G., Ed.; Springer Verlag: Berlin, 1985; Lect. Notes in Biomath. Vol. 66, p 68. (5) Crowley, M. F.; Eptein, 1. R. J . Phys. Chem. 1989, 93, 2496. (6) Boukalouch, M.; Elzegeray. J.; Arneodo, A.; Boissonade, J.; De Kepper, P. J. Phys. Chem. 1987.91, 5843. (7) Eptein, I. R. React. Kinel. Catal. Len. 1990, 42 (2), 241. (8) Swinney, H. L.; Roux, J. C . In Non-Equilibrium Dynamics in Chemical Systems; Vidal, C., Pacault, A., Eds.; Springer: Berlin, 1984, p 124.
The Journal of Physical Chemistry, Vol. 97, No. 5, 1993 1031 (9) Argoul, F.; Arneodo, A.; Richetti, P.; Roux, J. C. Acc. Chem. Res. 1987, 20, 436. (10) Schuster, H. G. Deterministic Chaos, An Introduction; VCH: Weinheim, 1989. ( 1 1) Marek, M.; Schreiber, 1. Behavior of Deterministic Dissipative System; Academia: Praha. 1991. (1 2) Collet, P.; Eckmann, J. P. Irerated Maps ofrhe Interval as Dynamical Systems; Birkhauser: Boston, 1980. (13) Field, R. J.; Noyes. R. M. J. Chem. Phys. 1974, 60, 1877. (14) Field, R. J.; Kbrb, E.; Noyes, R. M.J . Am. Chem. Soc. 1972, 94, 8649. (15) Noszticzius, Z.; McCormick, W. D.; Swinney, H. L.J . Phys. Chem. 1987, 91, 5129. (16) Coffman, K. G.; McCormick, W. D.; Swinney, H. L.; Roux, J. G.
In Non Equilibrium Dynamics in Chemical Systems; Vidal, C., Pacault, A., Eds.; Springer Verlag: Berlin, 1984. (17) Coffman, K.G.; McCormick, W. D.; Noszticzius, Z.; Simoyi, R. H.; Swinney, H.L.J. Chem. Phys. 1987,86, 119. (18) Schneider, F. W.; Miinster, A. F. J . Phys. Chem. 1991, 91. (19) Schneider. F. W.; Miinster,A. F. In DissipatiueStrucrurein Transport Processesand Combustions;Meinkbhn, D., Ed.;Springer: Berlin, Heidelberg, 1990. (20) (21) (22) (23) (24)
Renyi, A. Probability Theory; North-Holland: Amsterdam, 1970. Radii, R.; Politi, A. Phys. Rev. Lett. 1984, 52, 1661. Van de Water, W . ;Schram, P. Phys. Rev. Lett. 1988, 37, 31 18. Grasberger, P.; Procaccia, I. Phys. Rev. Lett. 1983, 58, 346. Mandelbrot, B. B. The Fractal Geometry ofNature; Freeman: San Francisco, 1982; pp 193-198. (25) Farmer, D.; Ott, E.; Yorke, J. Physica 7 0 1983, 153. (26) Wolf.A.;Swift.J. InStarisricalPhysicsandChaosin Fusion Plasma; Horton, W., Reichl, L.,Ma.; Wiley: New York, 1984; p 11 1. (27) Ruelle, D. Ann. N.Y. Acad. Sci. 1978, 317, 408. (28) Doumbouya, 1. S.Thesis, University of Wiirzburg, 1992. (29) Scott,S. K.ChemicalChaos; OxfordUniversity Press: Oxford, 1991, p 158. (30) Gybrgyi, L.;Field, R. J. J. Phys. Chem. 1988, 92, 7079. (31) Gybrgyi, L.;Field, R. J. J . Phys. Chem. 1989, 93, 2865. (32) Gybrgyi, L.;Field, R. J. J. Phys. Chem. 1989, 91, 6131. (33) Field, R. J.; Fbrsterling, H.-D. J . Phys. Chem. 1989, 91, 5400. (34) Gear, C. W. Numerical Initial Value Problems in Ordinary Differential. (35) Epstein, I. R. Physica D 1991, 51, 152 and references therein.
