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868 Journal of Chemical Education Vol. 73 No. 9 September 1996

In the Laboratory

Nonlinear Dynamics of the BZ Reaction:A Simple Experiment That Illustrates Limit Cycles, Chaos,Bifurcations, and Noise

Peter Strizhak1 and Michael Menzinger

Department of Chemistry, University of Toronto, Toronto, ON M5S.1A1

1Permanent address: L. V. Pisarzhevskii Institute of Physical

Chemistry, National Ukrainian Academy of Sciences, prosp. Nauki,

31, Kiev, Ukraine 252039.

Chemical reactions not only proceed monotonicallyas described by classical kinetics, but can also show pe-riodic and chaotic oscillations, multistability, excitabil-ity, and the formation of chemical waves and spatial pat-terns (13). These provide some of the most striking ex-amples of the nonlinear phenomena that pervade thephysical and biological world. Nonlinear chemical sys-tems are capable of the kind of timing, switching, and

signal propagation functions that are the elements of bio-logical and neuronal processing. For instance, Ross andco-workers showed recently that networks of bistablechemical reactions represent chemical realizations ofneural networks that are capable of learning and memory(4). Rovinsky and Menzinger showed that spatially ex-tended, excitable systems can act as the kind of analog-to-frequency converters that are found in sensory recep-tors (5). Thus, nonlinear chemical phenomena, apart fromtheir intrinsic interest, are also vehicles for understand-ing nonlinear phenomena in a larger context.

Oscillating chemical reactions have been the sub-jects of several articles in this Journal (620). Generaldynamic principles were discussed (6, 7) and experimentsin flow reactors received particular attention (8, 9). Ape-

riodic (chaotic) oscillations were discussed in this con-text. The history, mechanism, and recipes of theBelousovZhabotinsky (BZ) reaction were reviewed (911). Chemical waves in reaction-diffusion systems weredescribed (1215). Recently, a report appeared of tran-sient chaotic oscillations in the BelousovZhabotinskysystem in a stirred batch reactor(21).

This paper revisits the well-known oscillations of theBZ reaction (911) in a stirred batch reactor. As this closedsystem drifts slowly toward thermodynamic equilibrium,it passes through different regimes of steady state, os-cillatory, and chaotic dynamics and undergoes the asso-ciated bifurcations. While the nonstationarity of batchexperiments is often considered a drawback, it providesthe careful observer with a rich sequence of dynamic phe-

nomena that illustrate some of the key concepts of non-linear dynamics. The experiment may be viewed as aself-organized walk through different dynamic domainsof parameter space and the unfolding of the associateddynamics. Depending on the initial conditions, the reac-tion eventually ceases to oscillate either suddenly orgraduallyscenarios known as subcritical andsupercritical Hopf bifurcations. The experiments areviewed as illustrations of key concepts of the qualitativedescription of nonlinear dynamic systems, including thenotions of separation of time scales, phase space, phase

portraits, periodic and aperiodic (chaotic) oscillations, andthe corresponding bifurcations. These concepts are brieflyreviewed in the next section. The experimental setup isdescribed in the following section, after which the resultsare presented and discussed. To conclude, suggestionsare made for modifying the experiment for the use ofteaching laboratories. Given the rich dynamics, there areseveral aspects that remain unexplored and that may be

profitably addressed by undergraduate students, eithersingly or in teams.

Some Concepts of Nonlinear Dynamics

The coupled rate equations that govern the dynam-ics of a homogeneous and isothermal chemical reactionfollow from its reaction mechanism. In compact form theyare written as

dx/dt = F(x;p) (1)

where F is the rate function, x(t) = {x1(t),x2(t),, xn(t)} isthe vector of dynamic variables in n-dimensional phasespace, and p = {p1,p2, , pm} is the vector of control pa-rameters. In a chemical context, x(t) represents concen-

trations and p the constraints (i.e., rate constants, tem-perature, reactant composition, flow rate, etc.).

The concepts of phase space and phase portrait areimportant tools for visualizing the evolution of system1.Phase space is the n-dimensional space with coordinates{x1,x2, ,xn}. Aphase portrait is a trajectory of the rep-resentative point x(t). A dissipative dynamicsystem (e.g.,chemical reaction in which the free energy decreases) ischaracterized by the contraction of flow in phase space(1) and by the convergence of all initial conditions to so-called attractors. These are low-dimensional subsets, ormanifolds, of phase-space to which the system eventu-ally converges after transients have died out. The dy-namics is greatly simplified by neglecting the transientsand focusing attention on the asymptotic dynamics on

these low-dimensional attractors, which organize the flowin phase space.

Examples of attractors are the stable steady statesor stable fixed-point xo

i (zero-dimensional manifold), thestable limit cycle (one-dimensional manifold), and thestrange attractor (higher dimensional manifold with non-integer dimension)(2225). The fixed points xo

i are thesolutions of the steady-state equation dxo/dt = F(xo;p) =0. Since the rate function is nonlinear, the steady-stateequations may have more than one solution xo

i (i=1,2,3), and as a result multistability between coexistingattractors is possible. A linear stability analysis(1, 3) ofa given fixed point reveals whether it is stable or un-stable: this means whether perturbations from the steadystate decay or grow. A stable manifold is called attractor;
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Vol. 73 No. 9 September 1996 Journal of Chemical Education 869

In the Laboratory

Figure 1. Response diagram of (a) supercritical and (b) subcritical

Hopf bifurcation.

Figure 2. Schematic bifurcation diagram in the parameter space

of the batch BZ system, where p1o = [BrO3

]o and p2o= [MA]o. In the

course of its drift through parameter space along the trajectories

(1), (2) the system encounters Hopf bifurcations and bifurcations

to chaos. Subcritical Hopf bifurcations are indicated by the solid

line and supercritical bifurcation by the dotted line.

an unstable manifold, repellor.The type of attractor depends on the chemical ki-

netic term F(x;p) and on the dimension n of phase space.If the systems behavior is determined by the rate of con-sumption and accumulation of a single species, thenstable fixed points or steady states are the only kind ofattractor. In two-variable systems, two types of attractorare possible: fixed points and limit cycles. A limit cycle is

a closed, 1-D curve embedded in phase space, which rep-resents periodic oscillations. For a limit cycle to exist,the chemical mechanism should have at least one auto-catalytic stage and a negative feedback (13). Auto-catalysis provides accelerating growth of one species, andthe negative feedback terminates the autocatalytic ex-plosion. In oscillations this sequence repeats periodically.More complex dynamics is possible in systems with threeor more variables, including strange attractors and theassociated phenomenon ofdeterministic chaos (2225).Strange attractors are geometrical objects with dimen-sion that is strangely non-integer.

An attractor may lose its stability suddenly when acontrol parameter is changed. This event is called bifur-cation. Figure 1 is a so-called response diagram, which

represents the dynamic response (e.g., the value of a vari-ablexi) as a function of control parameter. Solid lines inthis figure show stable manifolds; dotted lines representunstable manifolds. Beyond the bifurcation point, the sys-tem diverges from the fixed point that has just becomeunstable and usually goes to an alternate attractor. Inthe cases illustrated in Figure 1, this new attractor is alimit cycle. Periodic and aperiodic oscillations may be bornthrough a variety of bifurcations. Periodic oscillations(limit cycles) may be born through a Hopf bifurcationwhere the stable fixed point loses its stability and thesystem is forced to a new attractor: the limit cycle. TheHopf bifurcation can appear supercritically or subcriti-cally as shown in Figure 1. In the supercritical case, thelimit cycle has vanishing radius and the oscillations are

born with zero amplitude. The amplitude grows mono-tonically with the distance from the bifurcation point anddoes not depend on the direction of the parameter change(Fig. 1a). In the subcritical case, illustrated by Figure1b, the oscillations are born suddenly with finite ampli-tude at one critical parameter value. When the param-eter is scanned in the opposite direction, the oscillationsdisappear at another critical parameter value: hysteresisoccurs. The bifurcation may be supercritical or subcriti-cal, depending on the values of the other parameters.

This is illustrated by the bifurcation diagram in Fig-ure 2, which represents a map of the dynamic domainsin the parameter plane {p1,p2}. It shows three domains,which correspond to steady state, periodic oscillations,and chaotic oscillations. The boundaries of these domains

are the sets of bifurcation points. In a closed system thereactants are frequently consumed on a time scale thatis long compared to the time scale of the oscillations.While their concentrations are, strictly speaking, slowlychanging dynamic variables whose evolution is describedby the appropriate component rate equations of eq 1, oneoften refers to these slow variables as parameters andconsiders them as constant on the time scale of interest(e.g., of oscillations). The quotation marks draw atten-tion to the approximate nature of this terminology. Thecoordinate axes of Figure 2 are such parameters. Asthe system drifts through parameter space along a tra-jectory that is determined by initial conditions and thestoichiometry, it encounters different dynamic domains.For instance, when the system evolves from the initial,

non-oscillatory conditions (in the BZ system,p1 = [BrO3]

andp2 = [MA]), it undergoes a Hopf bifurcation and per-forms almost regular oscillations until it encounters theexit-Hopf bifurcation according to the trajectory 1 in Fig-ure 2. To match the present experiments, assume thatthe latter is subcritical. For a different initial conditionillustrated by trajectory 2, the system begins to oscil-late, then encounters a strange attractor where chaotic

oscillations exist. As the system leaves the oscillatorydomain, the character of the Hopf bifurcation may havechanged from subcritical to supercritical, as indicatedby the dotted line.

The concept ofseparation of time scales, which al-lows one to eliminate the slowly varying parametersfrom the set of dynamic variables, is fundamental to alldynamics. For example, when studying a dynamic pro-cess, be it the internal motion of a molecule, the expan-sion of a gas into a vacuum, or a chemical reaction, oneis generally only interested in phenomena on a certaintime scale tobs. Slow variables that change on a much

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Figure 3. Time dependence of bromide electrode (a) and Pt-elec-

trode (b) potentials. Initial concentrations are: [NaBrO3]0 = 0.118 M;

[malonic acid]0 = 0.44 M; [Ce2(SO4)3]0= 0.00133 M; [H2SO4]0 = 1 M.

Shown in (c)(h) are the parts of system evolution presented in

Figure 2(a) that correspond to different transient behavior.

Figure 4. The phase portraits reconstructed from the time series

presented in Figure 3. Part (a) illustrates the limit cycle. Insert shows

the level of noise in signal. Part (b) corresponds to the intermit-

tently chaotic state.

longer time scale ts>>tobs may be taken as approximatelyconstant and are treated as parameters. On the otherhand, variables that evolve much faster, on a time scaletf

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Figure 5. Time series and phase portraits of supercritical (a, b)

and subcritical (c, d) death of the limit cycle. Initial concentrations

for a and b are given in Fig. 3. For c and d, the subcritical Hopf

bifurcation, only the fuel concentration was changed to [malonic

acid]0 = 0.44 M.

Figure 6. (a) Time series of bromide electrode potential at high

initial concentration of malonic acid: [NaBrO3]0 = 0.118 M; [malonic

acid]0 = 0.7 M; [Ce2(SO4)3]0 = 0.00133 M; [H2SO4]0= 1 M. Shown in

(b) is the portion that corresponds to the transient period doubling

phenomenon. In (c) the corresponding phase portrait is recon-

structed from Br- and Pt- electrode time series.

(10 14-) operational amplifiers (Analog DevicesAD549LH or Keithley 616 Digital Electrometers) as im-pedance matchers. For digital recording (see Figs. 36)the impedance-matched electrode signals were fed intoa Data Translation A/D converter board and a 486-typepersonal computer. Labtech Notebook was used as thedata-logging software. Figures 3, 4, and 6 were derivedfrom a single experiment and were printed on a laser

printer. If digital equipment is unavailable, a two-chan-nel stripchart recorder may be used in conjunction withan X-Y plotter, the latter for the phase portraits.

The data shown in Figures 36 are presented as theywere collected, except for Figures 6a and 6b, which arethe results of a smoothing operation to reduce the levelof resolution noise. An 8-bit interface was used for dataacquisition. Its resolution is 0.0039 V (1/28) for the 01-Vinput range, a value comparable with the oscillationamplitude near the end of oscillations. To reduce the levelof digitization noise, data were collected in the 00.1-Vrange, enhancing the resolution by 10. Furthermore, datacollected every 0.01 s were subsequently averaged over100 adjacent points. This smoothing procedure does notdistort the signal, since its oscillation period is an order

of magnitude higher.

Results and Discussion

Results from typical experiments are presented inFigures 36. Figures 3a and 3b are the overviews of a5- h time series of the Br

-sensitive electrode and the Pt-

microelectrode potentials, respectively. These recordsshow the sudden emergence of oscillationsan oscillat-ing time series with rich internal structure that rangesfrom high-amplitude and relatively high-frequency regu-

lar oscillations and a subsequent domain of highly ir-regular intermittent oscillations, to low-amplitude os-cillations and the gradual vanishing of their amplitude,and finally to the cessation of oscillations.

The reaction starts with an induction period duringwhich no oscillations take place, followed by the suddenbirth of a limit cycle with high-amplitude, period-1 oscil-lations (Fig. 3c). While the induction period is typical for

the BZ system studied here, it does not occur in all oscil-lating reactions. In the BZ reaction the actual reactantthat is oxidized by Ce4+ is not malonic acid butbromomalonic acid. The latter is synthesized from thereactants, and when its concentration reaches a criticalvalue, bifurcation to a limit cycle occurs. The oscillationsare born with a fully developed amplitude, through ahard or subcritical Hopf bifurcation, as illustrated byFigure 1b. The period of these almost regular oscillationsincreases very slowly; on the time scale of the oscilla-tions it is virtually constant. This illustrates the separa-tion of time scales, central to dynamic thinking: whereasoscillations take place on the scale of 30 s (Fig. 3c), theparameters drift so slowly that the oscillation attributes(period, amplitude) are nearly constant over a few peri-

ods, and successive bifurcations (Fig. 3c,d,f,h) occur on amuch longer time scale of ~5,000 s.

At 5,250 s (Fig. 3d) the simple oscillations give wayto aperiodic oscillations. Each pair of large-amplitudepeaks is separated by one or more small-amplitude spacerpeaks, whose number appears to be randomly distrib-uted. At 12,500 s a similar time series is seen in re-

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verse order: aperiodic oscillations are followed by oscil-lations that are nearly periodic. This part of the timeseries is qualitatively similar to the type ofdeterminis-tic chaos known as intermittency. A quantitative proofthat this is indeed the case can only be obtained by cal-culating quantitative measures of the fractal attractor(Lyapunov exponents, fractal dimension, etc.) (2225)from long and stationary time series that require CSTR

experiments. However, the qualitative similarity of thenonstationary time series in Figure 3 with the establishedlow flow rate chaos in a CSTR (31, 32) must suffice ascircumstantial evidence that one deals here with a caseoftransient intermittency. The physical basis of intermit-tency is well understood(2225) and will not be discussedhere. There are different routes to chaos. The anaerobicbatch experiment of Wang et al.(21) is suggestive of de-terministic chaos reached by a period-doubling route,which is well established under CSTR conditions (3336). The dependence of the chaotic state on oxygen is an-other example of the sensitivity of the BZ reaction(2).This may be understood in terms of the scavenging ofradicals that are formed in the course of the oxidativedegradation of the organic substrate.

Transientperiod halving occurs in a different regionof initial malonic acid concentration. Figure 6a showsthe corresponding time series and Figure 6b the detailof double-peaked oscillations that last for 1000 s. At theend the oscillation period is two times shorter than atthe beginning of this segment. The phase portrait, Fig-ure 6c, shows the alternation of small- and large-ampli-tude oscillations.

Chaotic behavior is characterized by the systemssensitivity to the initial conditions (2225). This makes achaotic time series irreproducible in the following sense:when the experiment is repeated under identical condi-tions, unavoidable differences in concentrations and inexternal perturbations would be amplified so that thedetailed appearance of the intermittent time series will

be quantitatively different while their qualitative appear-ance is preserved. At the same time, the periodic oscilla-tions and associated bifurcations may be reproducedquantitatively.

The Death of Limit Cycles

The chaotic domain is followed by simple, periodicoscillations (Fig. 3f,g,h). While their period is nearly con-stant, their amplitude is now quite irregular. On the av-erage, the oscillation amplitude decreases steadily untilit finally vanishes and merges into the background noise( 1 mV) of the digital detection system. These are thesignatures of a supercritical Hopf bifurcation (Fig. 3c;Fig. 1a,b)in sharp contrast to the sudden birth of thecycle through a subcritical Hopf bifurcation (Fig. 3c). Fol-

lowing the Hopf bifurcation, the system continues to drifttoward its asymptotic equilibrium point.

Phase Portraits

The different dynamic domains are most vividly char-acterized by their phase portraits. These are constructedby plotting the values of two or more dynamic phase vari-ables over an appropriate time interval in the correspond-ing phase space. An alternative, so-called time-delay,method (37) starts from the timeseries of a single dy-namic variablexi(t) and plots it against one or more time-delayed signalsxi(t + j), where j are constants. The re-sulting phase plot is known (38) to be topologicallyequivalent to the phase portrait obtained by the directmethod. Here we used the direct method and plotted the

signal of the Br-sensitive electrode (a measure of

log[Br]) against that of the Pt electrode (a measure of

log[Ce4+]/[Ce3+]). Figure 4 shows two phase portraits re-constructed from portions of the time series presentedin Figure 3. Figure 4a shows five cycles of the regularperiod-1 oscillations in the early phase of the reaction(Fig. 3c). One notes that the scatter of the trajectoriesdepends strongly on the oscillation phase, indicating that

the response to noise is similarly phase-dependent. Thisbehavior is related to the variability of the local rate ofconvergence to the limit cycle, also called the local sta-bility (39). The insert in Figure 4a is a detail of the phaseplane where the trajectories are particularly noisy andfar apart. In Figure 4b is plotted one large-amplitudecycle from the intermittently chaotic time-series. It showsin the bottom-left corner the phase portrait of severalsmall oscillations that are inserted between the large am-plitude peaks (Fig. 3e).

Stochastic Aspects: Noise and F luctuations

The irregularities in both time series and phase por-traits of these regular oscillations point to interestingstochastic and stability aspects of the cycles. These de-

viations from perfect periodicity arise from external per-turbationsfor example, in the present aerobic experi-ment, from the exchange of reactive gases (e.g., O2, Br2)between the liquid and gaseous phasesand they reflectthe response of the dynamic system to these perturba-tions. Figure 4a shows that the local appearance and thedeviation of the trajectories from their average variesgreatly with oscillation phase, due to two principal causes:the strong variability of the velocity of flow in phasespace, which is not represented in the timeless phaseplot, and the phase dependence of the cycles local stabil-ity(Ali, F.; Menzinger, M., unpublished results) .

The fast phase of the cycles is shown by the branchesat the right and bottom of the phase portraits. It extendsfrom the onset to the extinction of autocatalysis (2), cor-

responding to the rapid rise and subsequent drop of thePt-electrode potential in Figures 3c and 3d. During thefast phase, the trajectories form nearly parallel laminarbundles, whereas during the slow phase (the left-handbranch of the phase plot; here autocatalysis is switchedoff) the trajectories are closely entangled and show pro-nounced fluctuations. The single trajectory shown in Fig-ure 4b illustrates fluctuations in the Pt-potential andtheir pronounced phase-dependence. The latter signalsa similar phase dependence of the noise susceptibility,or the systems local stabilitythat is, its response to apresumably constant external noise.

As Figure 2 illustrates, the system may drift throughdifferent dynamic domains and experience the associatedbifurcations, depending on its initial conditions. For in-

stance, the transient chaos reported here is accessiblefrom a (yet unknown) range of initial conditions. Out-side this range the reaction exhibits only the standardscenario, namely the essentially periodic limit cycle os-cillations (trajectory 1 in Fig. 2). The character of theterminal Hopf bifurcation also depends on the initial con-ditions. If the malonic acid concentration is decreasedfrom [MA]o = 0.44 M to [MA]o = 0.3 M, then the limitcycle dies suddenly with finite amplitude through a sub-critical Hopf bifurcation. This is illustrated in Figures5c and 5d by the time series and phase portrait. Forcomparision, the supercritical death of the limit cycle atthe original, higher [MA]o is once more illustrated in Fig-ures 5a and 5b.

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Conclusion

Simple batch experiments readily display a varietyof dynamic phenomena and thereby awaken theobservers appreciation of the richness of nonlinear dy-namic systems.

The qualitative study of transient bifurcations andchaos in the BZ reaction may be achieved in the physical

chemistry laboratory by time-series analysis of simplebatch experiments. The experiment offers a self-organizedwalk through different domains in parameter space. Itteaches the fundamental concept of separation of timescales and makes the student familiar with the notion ofdynamics on a given, relevant time scale. During a singlewalk, the observer becomes familiar with the concepts oftransient bifurcations and deterministic chaos. A closerlook at the irregularities and fluctuations of the observ-able alerts the experimenter to the stochastic nature ofdynamic phenomena, that is, the response of the nonlin-ear dynamic system to external perturbations.

Not all of this dynamic territory has yet been fullycharted. The experiment lends itself to numerous modi-fications, from routine ones dictated by teaching labora-

tory practice to issues that are more challenging to bothstudent and instructor. A few suggestions for modifica-tions and further inquiry are:

Consider the dependence of the dynamics on the mul-tidimensional parameter space. Different groups ofstudents could be given well-considered initial con-ditions with the goal of using this time-share ap-proach to construct the bifurcation diagram illus-trated in Figure 2.

Consider different substrates of the BZ system (dif-ferent fuels and catalysts) (2).

Compare the dynamics under aerobic and anaerobicconditions to appreciate the effect of reactive gaseson the dynamics.

Construct phase diagrams both by the direct methodused here and by the time-delay method(37) and com-pare the results.

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