Indian lournal of Chemistry VoI.39A, lan-March 2000, pp. 345-355

Evidence for low dimension chaos in electrically coupled chemical oscillator in batch condition

K Narayanan, R B Govindan & M S Gopinathan *

Department of Chemistry, Indian Institute of Technology, Madras, Chennai 600 036, India

Received 13 December 1999; accepted 27 December 1999

In the last few years, the study of coupling of chemical reactions has gained immense importance because of the dynamics arising from such a set up is analogous to that of biological processes. In the present study we have made an attempt to characterize the dynamics observed during the electrical coupling of two Belousov - Zhabotinskii (BZ) oscillators in batch condition. One BZ oscilla­tor is catalysed by cerium and the other by manganese. The coupling strength is varied by using external resistances. The possibility of existence of detenninistic chaos in this reaction is examined by nonlinear dynamical techniques such as the estimation of Lyapunov exponent and correlation dimension, fol lowed by false nearest neighbourhood method, surrogate data and predictability analyses. All the above methods of analysis show that due to coupling the oscillator catalysed by cerium undergoes more changes in its dynamics from the limit cycle behaviour than the other oscillator, i .e., the fast oscillating manganese system affects the dynamics of the slow oscillating cerium system. Chaos is observed in time series representing the potential difference between the two coupled oscillators. The degree of chaos is a function of the external resistance.

Introduction Oscillating phenomena are ubiquitous in physics, as­

tronomy and biology. They range from the familiar motion of pendulum and the orbits of planets to the com­plex biological clocks that govern the daily and seasonal behaviour of living organisms. Chemists have found that the reactions in their test tubes and beakers can also show similar oscillating behaviour observed in other areas of science. Modem research on oscillating reaction dates from the experimental work carried out by Belousovl and independently by Zhabhotinsky2 which is basically the oxidation and bromination of an organic species cata­lyzed by metal ions like Ce4+, Fe3+ or Mn3+ in acid me­dium. Hence t h i s oscillatory reaction is called Belousov­Zhabhotinsky (BZ) reaction. The reaction involves au­tocatalysis, which is a form of positive feedback. In such processes, rate of the reaction depends on the concen­tration of an intermediate product. As the product is produced, the reaction accelerates. Thus the concentra­tion of the species varies and gives rise to the oscilla­tions3 • The sequence of the historical events in this con­nection is recounted by Winfree4 .

Coupling of the limit cycle oscillators provides a simple but powerful mathematical model for simulating * Correspondence to be sent: msgopi @hotmai\ .com

the collective behaviour of a wide variety of systems that are of interest in physics, chemistry and biological sciences5• In this paper, the effect of electrical coupling introduced between two non-identical BZ reactors is studied experimentally. The effect of coupling is stud­ied in terms of dynamical parameters. Amongst many different types of chemical oscillators, the BZ system is chosen for its easy experimental conditions and more­over this system is probably the most widely studied oscillating reaction both theoretically6 and experimen­tally?

In a closed system there is no inflow of reactants or outflow of products and such a set up is called a batch condition. The oscillatory behaviour may last for hours depending on the concentration chosen. However, the reactant consumption and the approach to the chemical equilibrium must eventually take their toll . Oscillations can only be a transient phenomenon in a thermodynami­cally closed system and must eventually die out.

On the other hand, a flow reactor (open system) is provided with continuous and often separate inflow of fresh reactants with the facility for stirring. So the flow reactor is called continuously stirred tank reactor (CSTR). In a CSTR condition, oscillations can persist

346 INDIAN J CHEM, SEC. A, JAN - MARCH 2000

for any amount of time by keeping the system far from equilibrium by the constant supply of the required reac­tants8 .

An oscillator must be perturbed in some way in order to exhibit complicated dynamical behaviour. Forcing, coupling and controlling flow are the physical mecha­nisms by which chemical chaos has been produced9.

Coupling of oscillators The situation of coupling is analogous to forced sys­

tems except that coupling is bi-directional . This is achieved by allowing the two separate reactors to inter­act with each other in some way. The interacting system may possess complex dynamics even when the individual oscillators are simple. The coupling of BZ oscillators in particular has been the subject of a number of experi­mental investigations. Generally, these couplings are performed in CSTR conditions but that is not manda­tory.

There are three different types of coupling used for studies of coupled reactors. They are:

(i)

(ii)

(iii)

Coupling by a direct convective exchange of mass between the reactors through a porous wallio-ix - By varying the coupling strength, various stationary and oscillatory regimes are observed. Coupling studies have also been mod­eled which explained various dynamical phe­nomena like birhythmicity, period doubling and chaos when one subsystem is in the bistable and the other in the oscillatory region of the param­eter spacel9-22 . Coupling by an indirect convective exchange of mass between the reactors via controlled pump­ing between the reactors23-26 - Variation of coupling strength shows different behaviour, which include stable steady states where oscil­lations cease. Electrical coupling -The third type of coupling, i .e . , the electrical coupling of two CSTRs was carried out by Crowley and Field27•28• A large­area platinum electrode was placed in each of the reactors and connected through an electrical circuit and an internal ion bridge. If the [Moxi]/ [Mred] ratio is different in the two reactors, the resulting potential difference drives a current through the system, causing the electrical cou­pling.

In the present investigation, an electrical coupling be­tween two non-identical BZ reactors has been carried out in batch condition. The electrical coupling is pre­ferred over mass coupling for the following reasons : Thl'rc are some practical problems involved in the pro­cess of mass coupling. All the reactants, products, cata­lyst and intermediate species are exchanged in the case of mass coupling and this makes qualitative mathemati­cal analysis of the coupling more difficult. There is a direct physical connection between the experiments in which the coupled oscillators have significantly differ­ent chemical composition28 • To avoid these problems, an electrical coupling is used in this study. In the case of electrical coupling, electrons flow between the systems without exchanging reaction mixtures. In this study, the experiment is performed in batch condition which avoids the mixing effects between inflow reagents and the bulk reaction mixture of the CSTR29 • But the prob­lem with the batch experiment is that we cannot get sus­tained oscillations for long time, as initial concentrations of the reactants are not maintained constantly through­out the reaction.

Experimental set up The experimental set up implemented by Crowley and

Field28 is followed for the present study (Fig. I ) . There are two BZ reactors, which are non-identical in the sense that they are catalyzed by different metal ions. One BZ system is catalyzed by cerium and the other by manga­nese. Both the reactors contain sulphuric acid, malonic acid and potassium bromate as the common constituents in same conditions. The large-area platinum electrodes are placed in each of the reactors to be coupled and the circuit is completed by a wire between the electrodes and by an ion bridge between the reactors. S ince the metal ion redox couples are different in the two reac­tors, the resulting potential difference drives a current through the system, causing the electrical coupling. The current is carried by reaction of metal ions M(n+ I )+ and Mn+ at the electrodes, which causes the potential to ap­proach the same value in both the reactors.

Each reactor is equipped with two platinum electrodes. One platinum electrode monitors redox potential (mainly of metal ions) and the second platinum electrode (Iarge­area electrode or working electrode) is used to control the coupling circuit. The potential difference between the platinum electrodes (which monitors the redox po­tential) is amplified and applied to the large-area or work­ing electrodes. The degree of coupling is varied by the

NARAYANAN et al. : LOW DIMENSION CHAOS IN ELECTRICALLY COUPLED CHEMICAL OSCILLATOR 347

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amplifier using variable external resistance. Separate and isolated power supplies are used for the analog to digital converter and for the input and output sides of the am­plifier to prevent the development of potential among the various electrodes. Further, direct electrical connec­tion between the input and output sides of amplifier is avoided by using opto-coupler, which converts an input potential to light whose intensity is proportional to the input potential .

The two reactors are placed as close as possible to maintain minimum distance between the platinum foils. The reactors are well stirred to avoid striking of gas bubbles on the surface of electrodes. The temperature of the reactors is maintained at 30.0 ± 0. 1 0 C. Since in a batch system, the oscillations die after a short time pe­riod, the concentrations of the reactants have been so chosen as to obtain oscillations for at least 30 to 45 min. The concentrations of the reactants used for Mn and Ce sy stem in coupled and uncoup led states are [Mn(II)]=0.002M, [Malonic acid]=0.05M, [KBr03

]=

O. l 5M, [H2S04]= 1 .3M and [Ce(IV)]=0.002M, [Malonic acid]=0. 1 6M, [KBr03]=0. l M, [H2S04]=2.0M respec­tively. We have studied the dynamics at three different values of resistance, namely, 3k, 6k and 1 0k ohms (n).

Method of analysis

The effect of coupling has been studied through non­linear time series analysis tools such as estimation of correlation dimension (D2) and maximum Lyapunov ex­ponent (Amax) followed by phase space dimension analy­sis by false nearest neighbourhood, short-term predic­tion and surrogate analysis from the reconstructed attractor3° at three different resistances, namely, 3k, 6k and 10kn. D2 is related to the minimum number of de­grees offreedom necessary to describe the system at any time and hence the degree of complexity.

The first step towards this methodology is construc­tion of multidimensional state space3 1 from single-di­mensional time series. Usually, this is done by the method of delay coordinates3o . Let x(t) be the time se-

348 INDIAN J CHEM, SEC. A, JAN - MARCH 2000

ries (a variable measured as a function of time). In the present case it is the potential difference between the platinum electrode (Pt) and standard calomel electrode (SCE). The tth point in the d-dimensional state space is given as follows, X(t)= x(t), X(tH) . . . x(t+(d- l )'t), where 'd' is the dimension in which state space is reconstructed and 't is the lag time between the successive components of the vector. Usually, 't is fixed from autocorrelation function31 . This is to ensure that state space vectors are linearly independent. This process of constructing mul­tidimensional attractor from time series is called recon­struction of attractor30 . Usually, the dimension is fixed from false nearest neighbour method32 .

In this study, the Grassberger and Procaccia33.34 algo­rithm with Theiler correction35 is used. A. has been max calculated according to the method of Wolf et al.36, which gives the average rate at which the nearby trajectories of an attractor diverge from each other. The existence of positive Amax is supposed to denote the chaotic behaviour of the system. The short-term prediction analysis gives the time up to which the future dynamics of the system can be predicted. The method of Lefebvre et aP7, is followed in this study. The surrogate data analysis de­veloped by Theiler et al.38, is utilized to prove the pres­ence of nonlinear structure underlying dynamics of the system. In this study, both phase randomized and Gaussian scaled surrogates are used to detect the pres­ence of nonlinearity. If the statistical deviation (S) in the discriminating statistics calculated for the original data and surrogate data is greater than 2 then we can say that the time series has got nonlinear structure in it39 . In this case we use D2 as discriminating statistic. Further, the false nearest neighbourhood analysis is also per­formed to get the optimum embedding dimension (OED) at which the attractor has to be reconstructed. The OED is chosen at the embedding dimension at which the false nearest neighbours percentage (FNNP) first falls below 2 (ref.32) . All these methods are collectively used to study the effect of coupling of the BZ oscil lators. The Unstable Periodic Orbit (UPO) spectrum of these coupled systems has been also investigated by Govindan40 .

The nonlinear time series analysis has been performed on the time series of redox potential of the reactor cata­lyzed by cerium (this is denoted as cerium system), re­dox potential of the reactor catalyzed by manganese (this is denoted as manganese system) and also to the time series of potential difference between the two oscilla­tors under coupling (this is denoted as 11£).

All these signals were sampled at 1 00 Hz and down sampled to 1 0 Hz after subjecting to low pass fi lter. In all the cases, the time series is chosen from a region where there is a steady state oscillation and there is no signifi­cant decrease in the amplitude of the oscillations because of the depletion of the reactants. This is to ensure that the difference in the oscillatory behaviour obtained at different coupling strengths is due to the change in the dynamics brought about by coupling and not because of the depletion of the reaction constituents.

Dynamics of oscillators without coupling

When there is no coupling, then there exist periodic oscillations in both the cerium and the manganese sys­tems which give rise to limit cycle behaviour. The time series (Fig. 2a) gives a single frequency of 0.07 Hz in the power spectrum (Fig. 2b) and the limit cycle in the phase space (Fig. 2c) . This clearly establishes that we have a regular oscillator. S imilar results are obtained for the cerium system also, the only difference being that its frequency is less (0.045 Hz).

For a perfect limit cycle, D2 is unity and Amax should be zero and it should be possible to predict the system for any length of time. These expectations are borne out in both these cases of Ce and Mn oscil lators.

The limiting value of D2 is close to unity and in the case of short-term prediction, p does not fall apprecia­bly with increasing time (T ) as it would fall for a cha­

p otic or stochastic signal4 1 . Surrogate analysis for the uncoupled oscillators was not carried out. The A value max should be zero for a perfect limit cycle. Since noise is always a part of any experimental signal (even after employing proper fi ltering techniques) we were unable to get a value of zero for A in this case. max

Dynamics of oscillators at resistance 3 kn When the coupling i s introduced into the system, at

the external resistance of 3kn, both the systems are per­turbed from their limit cycle behaviour, but the extent of perturbation is not the same. The redox potential of the cerium system is perturbed more from the limit cycle behaviour than that of the manganese system in terms of its dynamical parameters.

The time series, power spectrum and the attractor of redox potential of the cerium system (reactor catalyzed by cerium), manganese system (reactor catalyzed by manganese) and the !:iE between the reactors (coupled system) are given in Figs 3, 4 and 5, respectively. Broad

NARAYANAN et at. .' LOW DIMENSION CHAOS IN ELECTRICALLY COUPLED CHEMICAL OSCILLATOR 349

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Fig. 2 (a)- Oscillatory potential profile, (b) power spectrum (power is in arbitrary units) and (c) attractor reconstructed in an embedding dimension 3 from the potential of the Mn(I1) under uncoupled condition. [Mn(II)]=O.002M; [Malonic acid]=O.05M; [KBrO)) = O. 1 5M; [H2S04]= 1 .3M. x is the time series (voltage at the Pt electrode) in arbitrary units. X

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Fig. 3- Oscillatory potential profiles obtained under coupling at the resistance of 3kQ for (a) Mn(II) system, (b) Ce(IV) sys­tem and (c) M between the two reactors.

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350 INDIAN J CHEM, SEC. A, JAN - MARCH 2000

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peaks in the power spectrum (Fig. 4) and the broad band of trajectories in the attractors (Fig. 5) clearly show that the dynamics is deviated from the limit cycle behaviour.

In the case of manganese system, the D2 and the Amax deviate from 1 and zero, respectively to 1 .48 ± 0.05 (solid circles) and 0.07 ± 0.01 bits/sec (Table 1 ). A is chosen max from the region where their values are constant with re-spect to evolution time36.

The FNNP falls below 2 .for tolerance equal tq 30 (see Abarbanel et al.32, for details of tolerance) at an embedding dimension of 3 shown as solid circles in Fig. 6 (b) . From the short-term prediction analysis it can be observed that the p falls down as the prediction time (T ) is increased and the dynamics of the system cannot

p be predicted after 2 1 sec (Fig. 6(c) - open circles). This is denoted as TO (time at which p first falls to near zero).

p The gradual fall in p with respect to Tp is characteristic of chaotic systems. The predictability decay rate (PDR) is calculated from the linear portion of the curve of p versus Tp and it is found to be 0.04 sec- I (Table I ) . The higher the value of PDR, the faster the loss in predict­ability.

The presence of nonlinearity is tested by surrogate data analysis, and the statistical significance S obtained by D2 as discriminating statistic as greater than 2 proves the nonlinear structure in a system39 . In this study, D2 is used as the discriminating statistic and S is calculated. In both the types of surrogates, the number of surrogates are increased until the saturation of S is observed and this is shown in Fig. 6 (d and e) as a solid line.

In the case of redox potential of cerium system, the deviation from the limit cycle behaviour is clearly vis­ible. D2 (Fig. 6(a) - solid squares) and Amax are 2. 1 7 ± 0.25 and 0. 1 2 ± 0.02 bits/sec, respectively. FNNP gives an OED of 4, which is shown in Fig. 6(b) as solid squares. The system can be predicted only up to less than 8 sec (Fig. 6 (c) - open squares). PDR is obtained as 0.08 sec- I . All these analyses show that the cerium system undergoes more change than the manganese system at the resistance of 3kn. S ince Mn is the faster oscillating system of the two, it affects the dynamics of slower os­cillating Ce system. This is also reflected in the power spectrum where the frequency of the Mn remains almost the same value as that of its uncoupled state whereas frequency ofCe system is shifted to higher value of 0.09 Hz from 0.045 Hz. This phenomenon is called "drift­ing" of frequency28 . The surrogate data analysis proves the existence of nonlinearity in the cerium system also (Fig. 6 (d and e) - dotted l ines).

NARAYANAN et a!' .. LOW DIMENSION CHAOS IN ELECTRICALLY COUPLED CHEMICAL OSCILLATOR 3S 1

Table I - The values of correlation dimension (D2) , largest Lyapunov exponent O"m,)' predictability decay rate (PDR), the first time at which the predictability (p) falls to zero (To ), the optimum embedding dimension (OED) from FNNP and Significance (S) using D2 as discriminat-ing statistic for manganese (Mn), c:rium (Ce) and the M between the reactors (coupled system) for uncoupled state and at the resistance values of 3 , 6 and 10 kQ. PRS, GSS represent Phase randomized surrogates and Gaussian scaled surrogates, respectively.

System D2 A bits sec· 1 PDR secl T " sec OED S max p PRS GSS

Uncoupled state Mn 1 .0S O.OS±O.O I 0 Ce 1 . 1 0.04±O.02 0

Resistance 3 kQ Mn 1 .48±O.OS 0.07±O.0 1 0.04 2 1 3 3S.38±0.2S 3 .23±0.03 Ce 2. 1 7±0.2S 0. 1 2±0.02 0.08 08 4 2 1 . 1 2±O. 1 1 9 .9S±0.08 Llli 2.3S±0.0 1 0. 1 2±0.01 0.04 I I 4 I S .7 I ±O.03 1 3 .78±0.06

Resistance 6 kQ Mn 2.08±O.OS 0.09±O.O I 0.04 1 8 3 1 9.7 I ±0.36 8.6S±O.08 Ce 2.4S±O.OI 0. I S±0.02 0. 1 3 07 S 1 9.07±0. I S 8.6S±0.06 Llli 2.4S±0.0 1 O. l l±O.OI 0.04 1 0.S 4 I S .7 1±0.03 9 .6S±O.04

Resistance 1 0 kQ Mn 1 . 8S±O. 1 1 0.08±0.02 0.04 23 Ce 2.20±0.07 0. 1 7±0.02 0. 1 1 08 Llli 2.32±O.01 0. 1 0±0.0 1 0.07 1 8

These analyses are also carried out on the coupled system time series (/1£ between the reactors) at this re­sistance. In all the analysis (see Fig. 6) except for the value of PDR the dynamics is similar to that of the cerium sys­tem. The PDR value is close to that of the manganese system. The surrogate data analysis reveals the presence of nonlinearity. The above analyses show that the effect of coupling is to change the limit cycle behaviour (which occurs when there is no coupling) and supports the hy­pothesis of deterministic chaos in all the three cases at the resistance of 3kQ. Dynamics of oscillators at resistance 6kQ

As the resistance is increased 10 6 kQ, (the coupling strength is decreased), the deviation from the limit cycle behaviour increases further in both the reactors. The com­plexity is also increased from what is observed at the re­sistance of3kQ in terms of the dynamical parameters (see Table I ). Once again, cerium system (slower oscillator) is the one, which is more perturbed than the manganese (faster oscillator) reactor. It is interesting to note that there is no significant change in the dynamics of the /1£ be­tween the reactors as the resistance is increased.

In the case of manganese system, as the resistance is increased from 3kQ to 6kQ, the D2 value increases from 1 .48 ± 0.05 to 2.08 ± 0.05 (Table I ) and there is a slight

4 24. 8 1±0.23 2.S6±0.04 4 1 8.42±0. 1 6 8 .2S±O.03 4 I S .7 1±0.03 1 2.09±O.OS

increase in the value of A. from 0.07 ± 0.01 to 0.09 ± max 0.01 bits/sec (Table I ) . TO decreases from 2 1 sec to 1 8

p sec (Table O. All these results indicate the increase in the complexity of the dynamics as the resistance is in­creased. The values of OED and PDR do not show any significant difference (Table 1 ) . In the case of cerium system also, the increase in the complexity from 3kQ to 6kQ is observed in terms of all the dynamical param­eters estimated in this study. At this resistance value, the limiting values of dynamical indices like D2, A.max' PDR, OED (Table 1 ) increase from 2. l 7 ± 0.25 to 2.45 ± 0 .01 , 0. 1 2 ± 0.02 to 0. 1 5 ± 0.02 bits/sec, 0.08 to 0. 1 3 sec l and 4 to 5, respectively. The decrease in TO from 8 to 7 sec

p (Table 1 ) shows the decrease in the predictability result-ing from increase in the complexity.

As already mentioned, there is no significant change in the dynamics of the /1£ between the reactors because of decrease in the coupling strength. All the above analy­ses once again support the hypothesis of chaos in all the three cases at the resistance of 6kQ.

Dynamics of oscillators at the resistance lOkQ As the resistance is still further increased from 6kQ

to I OkQ (coupling strength still further decreased) , there is no increase in the complexity. This may be due to the

352

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INDIAN J CHEM, SEC. A, JAN - MARCH 2000

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NARAYANAN et al. : LOW DIMENSION CHAOS IN ELECTRICALLY COUPLED CHEMICAL OSCILLATOR 353

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Fig. 7- Periodic orbits of Mn and Ce systems in uncoupled state. (a) Mn system and (b) Ce system. Amplitude is in arbitrary units. Inset in each Fig. represents the attractor from which the UPOs are extracted.

fact that the individual oscillators retain their identity as no significant disturbance is caused on them in this low �oupling regime. Moreover, in many calculations, the deviation from the limit cycle behaviour decreases when compared with that of the values at 6 kQ. The decrease in the complexity when compared to 6kQ is observed in D2 and TO values of all the three cases and A in the

p max case of manganese and coupled cases and their values are given in Table 1 . In the FNNP method, the OED increases from 3 to 4 for manganese system and PDR increases from 0.04 to 0.07 sec-1 in the case of coupled system (Table 1 ) . These are the only instances, which show a higher complex nature at this value of .resistance. Even though some decrease in the complexity is ob­served, all the analyses still support the hypothesis of deterministic chaos in all the three cases at the resis­tance of 1 0kQ.

UPO Analysis Apart from the above dynamical indices, existence of

finite number of unstable periodic orbits (UPOs) is a direct evidence42 for presence of deterministic chaos. We have extracted the periodic orbits (POs) for the iso­lated oscillators and UPOs for coupled oscillator at all the three resistance values, on the lines of the methodol­ogy suggested by Lathrop and Kostelich43 .

The number of periodic orbits turns out to be unity for the isolated oscillator indicating the limit cycle dy­namics (Fig. 7).

It can be readily understood from Fig. 7 that Mn oscillator is the faster oscillator compared to the Ce os­cil lator as the recurrence time of the Mn oscillator is less than that of the Ce oscillator. This is due to the higher reduction potential of Mn2+/ Mn system compared to that of Ce4+/ Ce3+. Thus, periodic orbit analysis reflects the dynamics underlying the system.

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Fig . 8- Comparison of UPO's of 1'1£ between the two oscillators (a) at resistance 3kQ , (b) at resistance 6kQ , and (c) at resis­tance 10 ill . Amplitude is in arbitrary units. Inset in each Fig. represents the attractor from which the UPOs are extracted.

354 INDIAN 1 CHEM, SEC. A, 1AN - MARCH 2000

At low resistance values, 3 and 6 kQ, the dynamics of the coupled oscillator (Fig.8 a and b) is significantly dif­ferent from that of the individual oscil lators (Figs 7 a and b). The coupled oscillator oscillates at higher fre­quency than the individual oscillators. At resistance 1 0 kQ the dynamics of the coupled oscillator (Fig. 8c) is similar to that of Ce oscillator (Fig. 7b).

Thus at high coupling strength the dynamics of the coupled oscillator is dominated by the faster oscillator whereas at low coupling strength its dynamics is con­trolled by the slower oscillating system.

At all the resistance values the coupled oscillator has got more than one UPO indicating the chaotic dynamics underlying the system.

Conclusions The nonlinear time series analysis of the coupled sys­

tem of two non-identical BZ reactors clearly shows the effect of coupling at different coupling strengths. Ex­cept a few cases (D2 of manganese system at 3k and 10kQ, see Table 1 ), all the results of the dynamical analysis are consistent with the hypothesis of deterministic chaos for the coupled BZ oscillators. Because of coupling, the limit cycle behaviour of the uncoupled oscillators is per­turbed and deterministic chaotic dynamics is obtained. As the external resistance is increased from 3k to 6kQ, (coupling strength is decreased), an increase in the com­plexity in all the three cases (manganese, cerium and M between them) is observed in terms of its dynamical pa­rameters. But there is no significant difference observed in the dynamics of the oscillators as the coupling strength is decreased further (resistance increased to 10kQ) .

These results suggest that the faster oscillating man­ganese system affects the slow oscillating cerium sys­tem without itself undergoing much change. This type of electrical coupling enables one to observe chemical chaos even in batch conditions which, to our knowledge, has not been observed before through coupling. Cou­pling of two oscillators introduces a higher dimension­ality for the phase space compared to the uncoupled os­cillators and induces chaos. The interpretation of generic and robust results of ex­perimental observations in coupled chemical systems can be helpful in the description of various far more com­plex biological excitable systems. For example, this type of study can serve as a model to understand the dynam­ics as a consequence of coupling between cardiac sys­tem (low frequency) and brain system (higher frequency). [See Narayanan et al44. , and Govindan et a1.45, for dy-

namical study of the human cardiac and brain systems] . This potential significance makes the study of coupled chemical oscillators an area of growing interest. More experimental as well as modeling studies need to be un­dertaken in future.

Acknowledgement

We gratefully acknowledge the help from Prof. R Ramaswamy and Dr. V Sridevi in the experimental set up.

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