J. Phys. Chem. 1995,99, 11889-11895
11889
Chemical Resonance and Chaotic Response Induced by Alternating Electrical Current A. Fdrster, K.-P. Zeyer, and F. W. Schneider" Institute of Physical Chemistry, University of Wiirzburg, Marcusstrasse 9-11, 0-97070 Wiirzburg, Germany Received: March 28, 1995; In Final Form: May 23, 1 9 9 9
We investigate sinusoidal perturbations by an alternating electrical current imposed on focal steady states close to a Hopf bifurcation in the enzymatic peroxidase-oxidase (PO) and in the organic/inorganic BelousovZhabotinsky (BZ) reaction. Dynamic responses such as periodic, complex periodic, and chaotic resonances are observed. For low perturbation amplitudes both systems show period-1 resonances whereas for higher current perturbation amplitudes a period doubling route to chaos is found in the BZ reaction when the perturbation frequency is increased. Chaotic responses have also been found in the PO reaction when oxygen flow perturbations were applied to a focus. Model calculations using the Aguda-Larter PO model and the four-variable Gyorgyi-Field BZ model are in good agreement with the experiments.
1. Introduction
2. Experimental and Numerical Methods
Electrical perturbations using electrodes have been applied in biological systems like neurons, algae, and cardiac cells, where entrainment, period doubling sequences, and chaos have been observed as response behavior.'-9 In contrast to biological systems, perturbations of nonlinear chemical systems with electrical current received so far little attention. Crowley and FieldIo-l2 investigated the electrical coupling of two periodic Belousov-Zhabotinsky (BZ)I3 oscillators in order to study entrainment behavior, using an amplified potential between the cells. Electrical coupling of p e r i ~ d i c ' ~and . ' ~ chaotict6 BZ oscillators without amplification has also been studied. By applying external electrical potentials, switching between bistable steady states and shifting of monostable steady states of the BZ reaction occur.17 This effect was used to construct efficient chemical logical gates.'* The measurement of phase response curves by electrical pulse perturbation^'^ and the comparison with Ce4+ perturbations using Ce4+ solutions support the conjecture that the electric current mainly affects the Ce4+/Ce3+redox couple,'0-'2,20which, in turn, affects the kinetic behavior of the system. The effects of electrical current on the peroxidase-oxidase reaction (PO)21are yet unknown. A single-pulse perturbation of a focal steady state near a Hopf bifurcation declines in a damped oscillation.22 Buchholtz, and Schneider22demonstrated chemical resonance in the BZ reaction. Perturbing a focal steady state of the BZ reaction with flow rate variations at different frequencies leads to response amplitudes which pass through a maximum when the perturbation frequency is similar to the eigenfrequency of the focus (scan method).22 The resonance frequency of a chemical focus can also be determined in a single multiplex experiment, as demonstrated by Munster and S ~ h n e i d e r .Perturbations ~~ with high amplitudes may lead to various periodic and chaotic states. The complex response is due to the interaction of the external perturbation frequency with the system's autonomous dynamics. This was shown in the PO reaction by imposing flow rate perturbations on a focus.24 Electrical perturbations are expected to produce a response behavior in a noninvasive manner by mainly affecting certain redox couples in the chemical reaction. In the present study we impose sinusoidal electrical perturbations (alternating current) instead of flow rate perturbations on focal steady states of the P024925and the BZ2*reaction.
2.1. PO Reaction. Materials. Peroxidase from horseradish (HRP, RZ 3.05, activity 270 units/mg of solid) was purchased from Sigma as a salt-free powder. Glucose-6-phosphate dehydrogenase (DH) from Leuconostoc mesenteroides (540 units of B-NAD+/mg of protein) was purchased from Sigma as a biuret suspension. B-NAD' from yeast (99%) and ~-glucose-6phosphate disodium salt (G6P, 99%) were purchased from Sigma. 2,4-Dichlorophenol (DCP) and methylene blue (MB) were purchased from Aldrich. CSTR (Continuous Flow Stirred Tank Reactor). A Plexiglas CSTR (Figure 1) with two identical chambers is used, with an oxygen electrode fitted on one side of each chamber. The chambers are separated by a Teflon membrane (0.2-0.5pm pore size) which guarantees electrical conductivity between the two reactors. The reactors contain two Pt electrodes (2.3 cm2 surface each) which are connected to a potentiostat supplying a given potential. A computer controls the frequency and the amplitude of the perturbing electrical potential. Each chamber has a square base of 18 mm width and 18 mm height with an effective liquid volume of 4.3 mL. The gas volume above the liquid is 2.0 mL. We used an unsymmetric magnetic stirrer of about 1.5 mL self-volume driven by a small motor with a stirring rate of 750 rpm in each chamber. Preparation of Inflow Components. All reactants were dissolved in aqueous 0.1 m phosphate buffer (pH 5.8) containing 1 yM MB and 50 yM DCP. All solutions were prepared in an argon atmosphere before each measurement. We used two gastight syringes (Hamilton) filled with NADf/G6P solution and DWHRP solution with the following concentrations:
* To whom correspondence should be addressed. @
Abstract published in Advance ACS Abstracts, July 1, 1995.
0022-3654/95/2099-11889$09.00/0
syringe 1:
[HRP] = 11.9 pM, [DH] = 0.47 p M
syringe 2:
[NAD'] = 3.0 mM, [G6P] = 50 mM
To obtain reactor concentrations, divide by two. All experiments were done at pH = 5.40. Continuous Flow Conditions. A self-designed high-precision syringe pump driven by a computer-linked stepping motor was employed to control the flow rate of the reactants into the CSTR. The stepping frequency was 15 Hz in all experiments, producing a constant flow rate of the two inflowing solutions at kf (liquid) = 2.52 x min-' (residence time t = 40 min). Only one of the chambers is supplied with the reactant solutions. The other chamber is constantly rinsed with a 0.1 m phosphate buffer 0 1995 American Chemical Society
Forster et al.
11890 J. Phys. Chem., Vol. 99, No. 31, 1995
Figure 1. PO reaction: CSTR with two identical chambers. Each chamber has two inlets for the reactant solutions, a gas inlet, a gasfliquid outlet, and a 0 2 Clark electrode (1). A potentiostat (2)applies the current perturbations to the platinum electrodes of 2.3 cm2 surface each (3). Electrical conductivity between the two chambers is permitted by a Teflon membrane (4)of 0.2-0.5 p m pore size. Two unsymmetric stirrers (5)are used (stirring rate of 750 rpm).
at the same flow rate mentioned above. The flow rate of the gaseous mixture 02/N2 was constant in both reactors at kf (gas) = 0.7 mL/s in all experiments. The 0 2 content in the 02/N2 mixture was used as the bifurcation parameter and adjusted identically (12.8% 0 2 in the 02/N2 mixture) in both reactors. Temperature. The syringes containing the input chemicals, the tubing, and the reactor were thermostated at 25 "C for all experiments . Detection. The time series of the oxygen electrode potential were measured with an oxygen-selective Clark electrode and a microprocessor oximeter (WTW Oxi 96) with a sampling rate of 1 Hz. All figures of the PO measurements show the oxygen potential time series. Oxygen Flow. Two computer-linked mass flow controllers (MKS Type 1259C) for 0 2 and N2 were employed to mix and regulate any chosen gas flow and 0 2 content. During the perturbation experiments using the electrical current the oxygen and liquid flow rates through the reactors remained constant. 2.2. BZ Reaction. Materials. Malonic acid was purchased from Merck and recrystallized twice from acetone to remove trace impurities.26-28Potassium bromate (Merck), cerous sulfate (Riedel-de Haen), and sulfuric acid (Riedel-de Haen) of analytical grade were used without further purification. The water was purified by ion exchange (specific resistance 2 10 MQ.cm, purification system Milli-Q, Millipore). All solutions were equilibrated with air. CSTR. The experimental setup is identical to ref 17. It consists of two identical reaction chambers of 3.4 mL volume each and a middle chamber in between (volume 0.48 mL). The reaction chambers are separated from the middle chamber by Teflon membranes (1-2 pm pore size). To minimize mass exchange, sulfuric acid with the same concentration as in the reactors (0.375 mom) flows through the middle chamber. Each reactor contains a platinum working electrode (3.0 cm2 surface each) which is connected to a potentiostat. A computer controls the frequency and the amplitude of the perturbing electrical potential. Both reactors are equipped with a measuring WAg/ AgCl electrode (Ingold) to monitor the redox potential. An interference between the perturbing electrodes and the measuring electrodes is excluded for both systems by the use of optocouplers.' '.I2 The data are recorded with a sampling frequency of 2 Hz. The reaction solutions are mixed with Teflon stirrers at
1100 rpm. Each reactor is fed with three syringes containing the following solutions: syringe 1:
0.42 moVL KBrO,
syringe 2:
1 .S x io-, mom ce3+ from c~,(so,),
0.90 m o w HOOCCH2COOH syringe 3:
1.125 moVL H2S04
To obtain reactor concentrations, divide by three. Continuous Flow Conditions. The three feed lines of each reactor are driven by syringe pumps at constant flow rates of kf = 2.43 x lo-, s-I (residence time z = 6.9 min). A flow rate s-I (residence time z = 245.9 min) is used of kf = 6.78 x for the sulfuric acid flow through the middle chamber. Two electrically coupled reactors are used in order to investigate the phase relationship between their repsonses. Temperature. The syringes, Teflon tubes, and the reactor setup are thermostated at 25.0 "C. 2.3. Periodic and Stochastic Electrical Perturbations. Current perturbations were imposed on the steady states of the PO and the BZ reaction, causing redox processes on the surface of the platinum electrodes. These redox processes produce or remove the perturbing species. Stochastic Fluctuations Imposed on a Focus in the PO Reaction. The variations in the amplitudes of the electrical current were chosen to be statistical (f1.5 V, f 0 . 2 mA) according to a Gaussian distribution law. The stochastic amplitude perturbations were imposed at a constant frequency of 1.26 rads. Periodic Perturbations of a Focus in the PO and the BZ Reaction. The amplitudes a of the electrical current perturbations were a 5 f 1 . 5 V (f0.2 mA) for the PO reaction and a 5 f0.82 V (f0.49 mA) for the BZ reaction. Due to the low conductivity of the PO reaction solution, current amplitudes not higher than 0.2 mA were obtained. Electrolysis of water was not observed at all potentials and currents applied. In the BZ reaction we did not observe any evidence of oxygen consumption on the Pt electrodes at all potentials under investigation. In fact, the oxygen contained in the aerated solutions has a
J. Phys. Chem., Vol. 99, No. 31, 1995 11891
Chemical Resonance and Chaotic Response
TABLE 1: Values of the Parameters and Initial Conditions of the Variables Used in the Model Calculations of the AL ModeP [coI] [coII] + [coIII] + [Per2+]+ [Per3+]= 2.7 x mom Initial Values of the Variables [Per3+] = 0.76 x moa [NADH] = 2.09 x mom mom [NADa] = 3.18 x lo-* molL [Per2+]= 0.5 x [coI] = 0.02 x mofi [NAD’] = 1 mom mom [02’-] = 4.84 x mom [coII]= 0.02 x mofi [ 0 2 ] = 0.758 x mom [coIII] = 1.4 x [H202] = 1.27 x mom flow rate kfcol)= 1.16 x lo-’ mol/(L s) (bifurcation parameter) flow rate kf(NADH)= 1.3 x IO-’ mol/(L s)
+
(I
TABLE 2: Values of the Parameters and Initial Conditions of the Variables Used in the Model Calculations of the Four-Variable Gygrgyi-Field ModeP [Celt,, = [Ce3+l+ ice4+]= [H+] = 0.26 mol/L 7.90 x mom [HzC(COOH)2]= 0.25 m o m [BrO3-] = 0.10 m o m Initial Values of the Variables [Ce4+]= 1.95 x molL [Br-] = 1.21 x 10-6moVL [HBrOz-] = 1.59 x moa [BrHC(COOH)2]= 1.24 x lo-) mom flow rate k, = 10.39 x s-l (bifurcation parameter) a All rate constants are taken from Gyorgyi and Field (ref 33). All inflow concentrations of the variables are 0.
All rate constants are taken from Aguda and Larter (ref 30).
neglegible influence on the CSTR kinetics in the BZ reaction using similar concentration^.^^ The time-dependent perturbations U(t) have been applied according to U(t) = a sin(wt)
(1)
where o is the perturbation frequency. 2.4. PO Model. The Aguda-Larter (AL) modePo represents a mechanistic model based on some elementary steps of the PO reaction; it consists of 10 variables and 13 steps. All rate constants and starting concentrations used in the calculation of the focus are given in Table 1 (bifurcation parameter kf(0,) = k9, = 1.16 x lo-’ mol/(L s)). All integrations were done using the Gear a l g ~ r i t h m . ~A” ~perturbation ~ term is added to the differential equation of the variable y(5), which represents the NADH species:
The expression f ( c ) represents the differential equations of the model mechanism. [NADH10 is the inflow concentration of NADH at a given flow rate kf(NADH). In the model a’ denotes the amplitude in mol/(L s) and w the frequency of the perturbation in r d s . In the calculations we added the perturbation term to the differential equations of y(5),because the total redox potential is mainly due to the substrate (02heduced oxygen species, NADH/NAD+) species in the experiment. For the calculation of the resonance curve y(5) was perturbed sinusoidally with different frequencies (0.002 5 w 2 0.15 r a d s). For the simulation of the electrical multiplexing the variable y(5) was perturbed with Gaussian distributed fluctuations at a frequency of w = 1.26 rads. Therefore, a’ is set equal to 0 (eq 2), and a term p’R(6)is added. R(6) is a generator of equally distributed random numbers in the interval from - 1 to +1, and [mol/(L s)] is the noise amplitude. The perturbation amplitudes were arbitrarily set equal for both types of perturbations at a’ = p’ = 1 x IO-* mol/& s). 2.5. BZ Model. Model calculations were performed with the four-variable G~orgyi-Field~~ model (model D e in ref 33). The model treats Br-, HBrO2, Ce4+, and BrHC(COOH)2 as variables. Br03-,H+, H2C(COOH)2, and the total amount of cerium species (Ce4+ and Ce3+) are kept constant. For the eight elementary steps of the model we used the rate constants given by Gyorgyi and Field.33 All parameters, rate constants, and initial values of the variables are given in Table 2. To model the effects of electrical current perturbations, we added a sinusoidal perturbation term to the Ce4+ differential equation:
0
0.03
0.06
0.09
0.12
0.i5
~mtent
Figure 2. PO reaction: schematic bifurcation diagram (NADH absorption versus 0 2 content of the 0 2 / N 2 mixture) for the free running PO reaction with the constraints given in the text. The focus (at 12.8%) is located 5% above the supercritical Hopf bifurcation HBI (02 content = 12.2%).
A c ) contains the chemical kinetics of the model. [Ce4+]ois the inflow concentration of Ce4+, and [Ce4+] is the outflow concentration at the flow rate kf.
3. Results 3.1. PO Reaction. A focal steady state was established in the CSTR at a flow rate of 2.52 x min-’ (t = 40 min). The oxygen content in the gas mixture was adjusted to 12.8% by constant gas flow, which fixed the focal steady state at 5% above the superciritical Hopf bifurcation (HBI in Figure 2). During the electrical current perturbations the observable 0 2 oscillates. With the chosen set of parameters simple period- 1 relaxation oscillations occur in the region between the two supercritical Hopf bifurcations (Figure 2). A maximum amplitude of a = f 1 . 5 V (f.0.2 mA) was used in order to avoid any electrolytic decomposition of water. After turning off the electrical perturbation, the reaction retums to the focus with damped oscillations. Periodic Electrical Perturbations of a Focus. The electrical current perturbations were applied at about 30 different frequencies (0.01 5 w 5 0.15 rads). In all cases, the response frequency wres was equal to the modulation (perturbing) frequency wPr, Le., w,,:oPr = 1:l. The maximal response amplitude (arbitrarily adjusted to unity) occurred at a frequency of w = 0.063 rads (TFr = 100 s). The response frequency was w = 0.063 f 0.003 rads (T,,, = 100 f 5 s). Figure 3 shows a time series where the focal steady state has been perturbed with an altemating electrical current of w = 0.314 rads (TPr = 200 s). After starting the perturbations at t = 500 s, a period-2 pattern arose with a frequency of w FZ 0.0314 f 0.0003 rads (Tres= 200 & 2 s), which was in a 1:l ratio with the perturbation frequency. The term “period-1, period-2, etc.” refers to the oscillation pattern, and it defines the number of maxima per one oscillation period. A given oscillation pattern may occur for different response ratios between the response
11892 J. Phys. Chem., Vol. 99, No. 31, 1995
Forster et al.
L
220
140
'
3.5
I
lo00 Time [SI
0
0
2000
Figure 3. PO reaction: oxygen concentration versus time for a = 0.031 rads (T,,, = 200 s, a = h1.5 V, perturbation frequency oper f0.2 mA). The perturbed focus shows a period-2 response pattern in a 1: 1 frequency ratio with the perturbation. The distance between the points is equal to the perturbation period. The location of the points shows the phase shift between the perturbing and response periods, where the points represent the maxima in the anodic current.
0.02
0.04 0.06 0.08 0.1 perturbation frequency [rad/s]
0.12
0.14
Figure 5. PO reaction: normalized Fourier spectrum obtained by the
multiplex method. ' 1
I
1
-32
0.8
5
0.6
'c)
0.4
3
xE 8
0 0
e!
0.02
0.04
0.06
0.08
0.1
0.12
0.14
perturbationfrequency [rad/s]
-3
Figure 6. PO reaction: resonance curve calculated by the AL model using the scan method. At oFr= 0.031 rads the response of a period-2 pattern is observed.
0.2 4
0' 0
,
0.02
k, 0.04 0.06 0.08 0.1 perturbation frequency [ W s ]
0.12
0.14
Figure 4. PO reaction: resonance curve of the focus by the scan
method. Current perturbations were applied with an amplitude of a = h1.5 V (h0.2 mA). Only the largest peaks were used for the = 0.031 calculation of the amplitudes and their square values. At oper rads the response of a period-2 pattern is observed. frequency and the perturbing frequency, for example 1:1 or 1:2. For a perturbation frequency of 0.021 rads (T,,, = 300 s) period-3 oscillations with a response period of Tres= 300 & 5 s were obtained (not shown), whose amplitudes were ~ 0 . 2 9 arbitrary units. The square of the response amplitudes A2 versus the perturbation frequency represents the power spectrum of the focus (Figure 4) normalized to the maximum response. A broad maximum was observed at w x 0.065 rads, the resonance frequency of the focus. A smaller maximum was obtained at w 0.03 1 rads, with a period-2 pattern. For large perturbation frequencies (w = 0.12-0.14 rads) the response amplitudes tend to zero, since the focus is not capable to follow the fast perturbations. At lower amplitudes than f 1 . 5 V the response behavior remained in the period-1 state over the whole range of perturbation frequencies leading to smaller resonance amplitudes (not shown). The phase shift between the forcing current and the 0 2 response oscillations depends on the perturbation frequency. For high perturbation periods the phase shift was %180° (half a perturbation period). It increased to %360° (full period) as the perturbation period was lowered. This behavior is qualitatively similar with that of the BZ reaction (Figure 8). Electrically Multiplexing a Focus. The focal resonance frequency may also be determined by imposing stochastic potential fluctuations (multiplexing) as described in ref 23. The same focus was chosen (02content 12.8%) as described above.
It is located 5% above the critical 02 content (12.2%) describing the Hopf bifurcation (Figure 2). The variations of the imposed electrical potential were statistical (f1.5 V, f 0 . 2 mA) according to a Gaussian distribution. The system responded with forced irregular oscillations, which were nonchaotic (not shown). In order to obtain the resonance frequency of the focus, a Fourier spectrum of the time series has been calculated. A normalized Fourier spectrum (Figure 5) is obtained by dividing the Fourier spectrum of the response oscillations by the Fourier spectrum of the pure perturbations. The resonance frequency (w = 0.063 rads) was the same as in the scan method (w = 0.063 rads). Due to the finite number of response oscillations used in the time series, the muliplex method does not provide any precise information about the damping constant.23 3.2. Calculationsof the AL Model. Using the scan method, a period-1 response is found over a wide range of perturbation frequencies (w W 0.035-0.140 rads). A maximum in the response amplitude occurs at w = 0.062 rads ( T = 101 s) (Figure 6), which denotes the resonance frequency. A small resonance is observed at the perturbation frequency of w % 0.031 rads (T % 204 s). In the AL model multiplexing was performed (not shown) in analogy to the experiments using 6000 pulses of pulse length 5 s, yielding a resonance frequency of w = 0.062 rads as in the scan method. The experimental results are in good agreement with the Aguda-Larter (AL) model. 3.3. BZ Reaction. Periodic Electrical Perturbations of a Focus. Identical focal steady states were established in both reactors at a flow-rate of kf = 2.43 x s-' (z = 6.9 min). The focus has been located 17% above the supercritical Hopf b i f ~ r c a t i o n . ~We ~ investigated the effects of an imposed sinusoidal alternating current at three amplitudes. At a potential
J. Phys. Chem., Vol. 99,No. 31, 1995 11893
Chemical Resonance and Chaotic Response
permrbation frequency 0.116 d
s (Tper=54.0 s)
1 .o c-4
s3
1
1
3
4
1200 0.8 0.6
0.4
0.2 0.0
0.1 0.2 0.3 0.4 perturbation freqwncy [Wal
0.0
0.5
Figure 7. BZ reaction: resonance curve of the focus obtained by the scan method. At uper = 0.066 rads the response of a period-3 pattem is observed.
'
0
0
500
lo00
1500 time [SI
ZOO0
2500
I 3000
Figure 9. BZ reaction: time series measured in either reactor at a perturbation frequency of upsr= 0.116 rads (TPr = 54 s) (a= 50.82 V). The period-2 pattem is in a 1: 1 frequency ratio with the perturbation current. The points indicate the perturbation period (see Figure 3).
1
I
0.4
0 '
I
50
100
150 200 250 300 perhubation period [SI
350
400
Figure 8. BZ reaction: phase shift between the sinusoidally forced focus and the perturbation current versus the perturbation period obtained in either reactor.
amplitude of f 3 5 0 mV (f0.07 mA) the perturbed focus responded with period-1 oscillations from 0.016 to 0.898 rads. Due to the low response amplitudes the resonance curve is not shown. The frequencies of the response oscillations were always equal to the perturbation frequencies. The individual oscillations in the two reactions were 180" out-of-phase (except in the case of chaos where the phase is not defined and in the case of the 1:2 response where the phase shift is 90"), since one reactor contains the anode when the other reactor contains the cathode. We also determined a power spectrum of the perturbed focus at a potential amplitude of f 5 9 0 mV (f0.25 mA). The square of the response amplitude in units of the maximum amplitude is plotted against the perturbation frequency (Figure 7). The focus did not respond to perturbation periods shorter than 7 s. Resonance was found at a perturbation frequency of w = 0.155 rads (TWr= 40.5 s). Small peaks were also observed at lower frequencies. A period-1 response was found at perturbation frequencies higher than w = 0.233 rads (Tper = 27.0 s). This pattem changed to period-2 when the perturbation frequency was lowered from ores = 0.155 rads to wres= 0.093 rads. A period-3 signal was obtained between wres= 0.078 and 0.058 rads (TFr = 81.0 and 108.0 s), respectively. At still lower frequencies a period-2 response again emerged. All response ratios were 1:1. The phase shift between the focal response and the current perturbation is defined as the time interval between the maximal perturbing cathodic current and the minimum potential of the response normalized with the perturbation period. Figure 8 shows that the experimental response signal of the focus is delayed by half of period at high perturbation periods. This delay increased slightly when the perturbation period was
lowered. At low perturbation periods (< 150 s) the phase shift increased, and it reached one period for fast perturbations (T,,, 7 s). For shorter perturbation periods than ~ 5 s7 the response amplitude was within the experimental noise. We further investigated the effects of perturbations with potential amplitudes of f 8 2 0 mV (f0.49 mA). As in the cases of low amplitudes, the focus is insensitive toward very fast perturbations. At perturbation frequencies w 1 0.233 rad/s (TFr I27.0 s) a period-1 response was observed. For frequencies between w = 0.155 rads (TFr= 40.5 s) and w = 0.116 rads ( T p = 54.0 s) a period-2 pattern was obtained which was in a 1:l ratio with the perturbation (Figure 9). A region of chaotic responses was found, when the perturbing frequency was further lowered. Figure 10a shows a chaotic time series obtained for a perturbation frequency of 0.101 rads (TWr= 62.5 s). We c o n f m the deterministic origin of the measured aperiodicity as follows: The Fourier spectrum of the chaotic response (Figure lob) is broad, and it is dominated by a sharp peak (w = 0.101 rads) with its overtones, which corresponds to the frequency of the sinusoidal perturbation. The attractor reconstructed by the SVD method35has the characteristics of a chaotic attractor (Figure 1Oc). The one-dimensional map (Figure 10d) displays an extremum, which is typical for deterministic chaos emerging from a period doubling sequence. The spectrum of generalized Renyi dimensions36 (not shown) was obtained by a nearneighbor a n a l y s i ~using ~ ~ . ~an~embedding dimension of 10 and loo0 reference points. It yields the Hausdorff (DO= 2.9), the information (Dl= 2.7), and correlation ( 0 2 = 2.6) dimensions which are fractal and above 2.0, as required for deterministic chaos according to the Kaplan-Yorke c ~ n j e c t u r e .Period-4 ~~ oscillations were obtained in a broad range of perturbation frequencies from about w = 0.070 rads ( T p = 90 s) to w = 0.021 rads (Tper = 300 s) (Figure 11). For the period-4 pattem the response ratio of the response frequency wres to the perturbation frequency wperis 1:2. The period-4 oscillations originate through period doubling from a period-2 state which is in a 1: 1 frequency ratio with the perturbation frequency. The period-2 state is found for perturbation frequencies below w = 0.021 rads (TWr= 300 s) (Figure 12). 3.4. Calculations with the Four-Variable GyiSrgyi-Field Model. A resonance diagram of the periodically driven focal steady states has been calculated (Figure 13). For low perturbamoV(L s)) only fundamental tion amplitudes (a' 5 7 x period-1 responses are observed over the whole range of applied frequencies. For higher perturbation amplitudes (7 x 5 a' 5 2 x moV(L s)) and low perturbation frequencies a region of period-2 pattern which is in a 1:l frequency ratio with the perturbation is found as in the experiments. A region of
11894 J. Phys. Chem., Vol. 99, No. 31, 1995
Forster et al.
perturbation frequency 0.101 rad/s (Tper= 62.5 s)
S 1200 1000 800
.....
'5
m
200 0
4Ooo
0
2000
6Ooo
8000
time [SI
I
perturbation frequency 0.101rad/s (TPr = 62.5 s)
lo 5-
0.008
00.006
-5
-
-10
-
-15
-
0.004
0.002 *
4
O.OO0
0.0 0.1 0.2 0.3 0.4 0 . 5 0.6 0.7 0.8 0.9 1.0
-%o
frequency [rad/sl
-15
-10
-5
0
5
3
x (nl
Figure 10. BZ reaction: (a) Chaotic time series obtained in either reactor at a perturbation frequency of uper= 0.101 rads (TFr = 62.5 s, a = f0.82 V). (b) Fourier spectrum of the time series shown in (a). (c) Attractor reconstructed by the SVD method, using 25 columns. The three dimensions (X,Y,Z) correspond to the largest singular values. (d) A one-dimensional map of the attractor of (c) shows an extremum (solid curve) indicating deterministic chaos.
1400
I
perturbation frequency 0.016 rad/s (TPr= 400 s)
perturbation frequency 0.070 rad/s (Tper= 90.0s)
1200
'
200 0 ' 4Ooo
I 5000
6Ooo time [SI
7000
8000
0
0
2000
4Ooo time [SI
6OoO
8000
Figure 11. BZ reaction: period-4 response obtained in either reactor at a perturbation frequency of uper= 0.070 rads (TFr= 90 s, a = h0.82 V), where u,,,:uper is 1:2. Still smaller peaks occur at half of the perturbation period to produce period-4 oscillations. The points indicate the perturbation period (see Figure 3).
Figure 12. BZ reaction: period-2 response obtained in either reactor at a perturbation frequency of uper= 0.016 rads (Tper= 400 s, a = f0.82 V) in the experimental BZ system. The observed pattern shows a 1:l frequency ratio with the perturbation current. The points indicate the perturbation period (see Figure 3).
chaotic states C is reached through a period doubling sequence, and a broad period-3 state is surrounded by the chaotic states. For illustration the positions of two response curves are also shown at two perturbation amplitudes.
frequencies and perturbation amplitudes (Figure 13). The calculated resonance diagram of the four-variableGyorgy-Field model represents an excerpt of the location of various dynamic states. For example, if one traces the dynamic states (Figure 13) at a constant perturbation amplitude of a' = 2.0 x mol/(L s), starting with a low reduced perturbation frequency, one observes period- 1 response oscillations in a 1:1 frequency ratio with the perturbation. Then the response passes through subsequent regions of period-2, period-3, period-2, and period4. At a reduced frequency of 0.38 a narrow region of chaos is encountered followed by period-3. At increasing frequency another region of chaos (C) occurs, followed by period-4, period-
4. Discussion and Conclusion Whereas a linear oscillator will show a simple Lorentz-type resonance curve, a nonlinear oscillator may display deviations from the Lorentz form even at relatively low perturbation amplitudes. The reason is the variety of dynamic states into which the focus may be excited at various perturbation
J. Phys. Chem., Vol. 99, No. 31, I995 11895
Chemical Resonance and Chaotic Response
Dechert, M. H. Watzl, A. F. Munster, and M. J. B. Hauser for valuable discussions. We further thank the Volkswagen Stiftung, the Deutsche Forschungsgemeinschaft, and the Fonds der Chemischen Industrie for partial support of this work.
References and Notes (1) Hirsch, H. R. Nature 1965,208, 1218. (2) Guttman, R.; Feldman, L.; Jakobsson, E. J. Membr. Biol. 1980, 56,9. (3) Guevara, M. R.; Glass, L.; Shrier, A. Science 1981,214,9. (4) Hayashi, H.; Ishizuka, S.; Otha, M.; Hirakawa, K. Phys. Lett. 1982, 0.25 0.50 0.75 1.00 88A,435. reducedperturbationfrequency (5) Hayashi, H.;Nakao, M.; Hirakawa, K. Phys. Lett. 1982,88A,265. Figure 13. Resonance diagram (perturbation amplitude versus reduced (6) Glass, L.; Guevara, M. R.; Shrier, A.; Perez, R. Physica 1993,7 0 , perturbation frequency in units of 00, where 00 = 0.0395 rads) for 89. (7) Glass, L.; Guevara, M. R.; Belair, J.; Shrier, A. Phys. Rev. 1984, the periodically perturbed focus of the four-variable Gyorgyi-Field 29A,1348. whereas the P-i denote model. The brackets denote the ratio cores:uper, (8) Aihara, K.; Matsumoto, G.; Ichikawa, M. Phys. Lett. 1985,IIIA, the response oscillation patterns. Higher perturbation amplitudes ('2 251. x moV(L s)) lead to closely spaced complex response patterns (9) Hanyu, Y.; Matsumoto, G. Physica 1991,490, 198. (not shown). Resonance curves at perturbation amplitudes of 0.50 x (10) Crowley, M. F.; Field, R. J. In Nonlinear Phenomena in Chemical moV(L s) (lower curve) and 1.25 x lo-* mol/(L s) (upper curve) Dynamics; Vidal, C., Pacault, A., Eds.; Springer: Berlin, 1981; p 147. are plotted at the right margin (square of the Ce4+ response amplitude). (11) Crowley, M. F.; Field, R. J. In Lecture Notes in Biomathematics, The upper curve shows the influence of the period-2 pattern. Nonlinear Oscillations in Biology and Chemistry; Othmer, H., Ed.; Springer: Berlin, 1986; p 68. (12) Crowley, M. F.; Field, R. J. J. Phys. Chem. 1986,90,1907. 2, and finally period-1 regions. This complex response at (13) Belousov, B. P. Sb. Ref. Radiat. Med. 1959,145. constant perturbation amplitude cannot be acommodated by a (14) Botrk, C.; Lucarini, C.; Memoli, A.; D'Ascenzo, E. Bioelectrochem. simple resonance curve, which in fact uses only the amplitude Bioenerg. 1981,8,201. (15) Schneider, F. W.; Hauser, M. J. B.; Reising, J. Ber. Bunsen-Ges. of the major peak in each time series. Thus, the nonlinear Phys. Chem. 1993,97,55. response curves display complex shapes due to the variety of (16) Zeyer, K.-P.; Munster, A. F.; Hauser, M. J. B.; Schneider, F. W. J. dynamic states encountered at high perturbation amplitudes. Chem. Phys. 1994,101,5126. (17) Dechert, G.; Schneider, F. W. J. Phys. Chem. 1994,98,3927. Resonance in chemical systems has already been demon(18) Zeyer, K.-P.; Dechert, G.; Hohmann, W.; Blittersdorf, R.; Schneider, strated by flow rate perturbations in the BZ,22in the methylene F. W. Z. Naturforsch. 1994,49A,953. blue oscillator$0 and in the PO24system by varying the oxygen (19) Dechert, G.; Lebender, D.; Schneider, F. W. J. Phys. Chem., in gas flow. All these results confirm the occurrence of resonance press. (20) Schmidt, V. M.; Vielstich, M. Ber. Bunsen-Ges. Phys. Chem. 1992, for perturbation frequencies near the natural frequency of the 96,534. chemical focal steady states as exemplified by the present PO (21) Yamazaki, I.; Yokota, K.; Nakajama, R. Biochem. Biophys. Res. and BZ experiments, which use the method of electrical Commun. 1965,21,582. perturbation. By perturbing the oxygen inflow sin~soidally,~~ (22) Buchholtz, F.; Schneider, F. W. J. Am. Chem. Soc. 1983,105,7450. (23) Munster, A. F.; Schneider, F. W. Ber. Bunsen-Ges. Phys. Chem. chaotic oscillations were observed in the PO reaction. In the 1992,96,32. present experiments a chaotic response occurs in the BZ reaction (24) Forster, A.; Hauck, T.; Schneider, F. W. J . Phys. Chem. 1994,98, when the focus is electrically perturbed. This chaotic state is 184. (25) Samples, M. S.; Hung, Y.-F.; Ross, J. J. Phys. Chem. 1992,96, not identical with the low flow rate chaos in the BZ 7338. r e a ~ t i o n . ~ ~Instead, * ~ ' - ~ the ~ chaotic response corresponds to a (26) Noszticzius, Z.; McCormick, W. D.; Swinney, H. L. J. Phys. Chem. new dynamic state characteristic of the perturbed chemical 1987,91,5129. system, which exceeds the autonomous system by one degree (27) Coffman, K. G.; McCormick, W. D.; Noszticzius, Z.; Simoyi, R. H.; Swinney, H. L. J. Chem. Phys. 1987,86, 119. of freedom, namely, the perturbation frequency. On account (28) Gyorgyi, L.; Field, R. J.; Noszticzius, Z.; McCormick, W. D.; of the low electrochemical conductivity of the PO solutions, Swinney, H. L. J. Phys. Chem. 1992,96,1228. only low concentrations of species are turned over at the (29) Munster, A. F. Doktorarbeit, Universitiit Wurzburg, 1992. (30) Aguda, B. D.; Larter, R. J. Am. Chem. SOC. 1991, 113,7913. electrode. Therefore, we suppose that chaos was not observed (31) Gear, C. W. Numerical Initial Value Problems in Ordinary when electrical perturbations were applied to 'the focus in the Differential Equations; Prentice-Hall: Englewood Cliffs, NJ, 1971; p 209. PO reaction. (32) Hindmarsh, A. C. Gear: Ordinary Differential Equations System Simultaneous experimental detection of NADH, COIII,0 2 Solver; VCID 2001, rev 3, Dec 1974. (33) Gyorgyi, L.; Field, R. J. J. Phys. Chem. 1991,95,6594. concentration, and the total redox potential measured with a (34) Zeyer, K.-P.; Munster, A. F.; Schneider, F. W. J. Phys. Chem., in platinum electrode against a Ag/AgCl standard showed that the press. oscillations of NADH, 0 2 , and the redox potential are ap(35) Broomhead, D. S.; King, G. P. Physica 1986,2 0 0 , 217. proximately in phase. Therefore, it is assumed that the total (36) Renyi, A. Probability Theory: North-Holland: Amsterdam, 1970. (37) Badii, R.; Politi, A. Phys. Rev. Lett. 1984,52, 1661. redox potential is mainly due to the substrate (NADH"+, (38) van de Water, W.; Schram, P. Phys. Rev. 1988,37A,3118. 02heduced oxygen species) and not due to the enzymatic species (39) Kaplan, J. L.; Yorke, J. A. In Lecture Notes in Mathematics; Peitgen, (Per3+,Per2+, COI,COII,COIII),which are present in very small H. O., Walther, H. O., Eds.; Springer: Berlin, 1979; Vol. 730, p 228. (40) Resch, P.; Munster, A. F.; Schneider, F. W. J. Phys. Chem. 1991, concentrations. In the BZ reaction the redox processes at the 95,6270. electrodes involve mainly the Ce3+/Ce4+ redox couple as (41) Turner, J. S.; Roux, J. C.; McCormick, W. D.; Swinney, H. L. Phys. supported by comparison with the results of pulsed additions Lett. 1981,85A,9. of known Ce4+ solutions.19 (42) Simoyi, R. H.; Wolf, A.; Swinney, H. L. Phys. Rev. Lett. 1982, 49,245. (43) Schneider, F. W.; Munster, A. F. J. Phys. Chem. 1991,95,2130. Acknowledgment. We thank J. Stephan for valuable ex-
perimental assistance in part of the work. We also thank G.
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