J. Phys. Chem. 1995,99, 13173-13180
13173
Quasiperiodic Forcing of a Chemical Reaction: Experiments and Calculations K.-P. Zeyer, A. F. Miinster, and F. W. Schneider" Institut f i r Physikalische Chemie, Universitat Wiirzburg, Marcusstrasse 9-11, 97070 Wiirzburg, Germany Received: April 17, 1995; In Final Form: June 15, 1995@
We present experimental and theoretical investigations of the nonlinear resonance behavior of a focal steady state in the Belousov-Zhabotinsky (BZ) reaction which is perturbed by its own dynamics. For this purpose the outputs of two independent BZ-oscillators of incommensurate frequencies are linearly combined in analogy to a simple feed-forward net to achieve the quasiperiodic driving of a third BZ-reactor which is in a focal steady state. For low amplitudes of the quasiperiodic flow-rate perturbations the response is also quasiperiodic in the focal BZ-reactor. Above a threshold value of the quasiperiodic amplitudes, however, single bursts are observed at small phase differences between the two perturbing oscillators. A further increase in the amplitude leads to groups of bursts with small oscillations in between. Aperiodic stochastic bursting in the BZ-reactor is observed when the flow-rate into the focal reactor is perturbed by Gaussian distributed white noise of sufficiently high amplitude and pulse length. The experiments are shown to agree with computer simulations using the Showalter-Noyes-Bar-Eli (SNB) model. At high amplitudes of quasiperiodic forcing the SNBmodel gives evidence for the existence of an attractor which is strange but nonchaotic (information dimension = 2.0).
1. Introduction
2. Experimental Setup
Quasiperiodic forcing has hardly received any experimental attention in chemistry' while extensive theoretical work2-' has been done on this subject. In quasiperiodically forced theoretical models a number of interesting phenomena have been calculated, among them two-frequency quasiperiodicity, three-frequency quasiperiodicity, chaos, and in particular, strange nonchaotic attract or^.^-' A strange nonchaotic attractor is a fractal object in phase space, but in contrast to chaos, its trajectories do not diverge exponentially. Therefore, a positive Lyapounov exponent does not exist, and the Lyapounov dimension is equal to 2.0 just as in quasiperiodic motion on a two-frequency torus. In this work we give evidence of a strange nonchaotic attractor in the SNB-model. Focal steady states in the BZ-reaction have been perturbed sinusoidally at different frequencies by flow-rate variationsSor by imposing flow-rate noise (multiplexing m e t h ~ d ) .Sinusoi~ dallo and complex" waveforms were also used to perturb a focal steady state in the open PO-reaction and to study the dissipation. In the methylene-blue-oscillator Resch et a1.I2 experimentally studied the influence of flow-rate noise imposed on a focus near a subcritical Hopf bifurcation. An optimal way of coupling an extemal force to a nonlinear oscillator is to use its own dynamics as a forcing function as demonstrated in theoretical work on conservative oscillators by Hubler et al.I3 and Chang et al.I4 In this work we investigate quasiperiodic flow-rate perturbations of a focus in the BZreaction using the system's own dynamics. Two oscillating BZreactors provide the quasiperiodic forcing function which perturbs a BZ-focus contained in the third reactor. For comparison we also investigate the effects of simple periodic perturbations and random flow-rate noise imposed on the same focal steady state. In the BZ-reaction the reaction products are not uniquely characterized. Therefore, the dissipation cannot be determined in this case.
The experimental setup for the periodic and quasiperiodic flow-rate perturbations (unidirectional flow-rate coupling) consists of two and three separate continuous flow stirred tank reactors of 7.6 mL volume each (Figure l), respectively, resembling a simple feedforward net.I5 Feedforward nets are arranged in at least two layers (input and output layer). The activation of the input layer neurons (Le., chemical reactors) is processed forward to the next layer (hidden or output layer) using defined connectivities wo which correspond to the axons with dentrites and synapses in biological neural networks.I5The output signals of reactors 1 and 2 (input layer) are transformed into a flow-rate which is applied to reactor 3 (output layer) using the following linear equation:
* Author to whom correspondence should be addressed. @
Abstract published in Advance
ACS Abstracts.
August 1, 1995.
0022-3654/95/2099-13173$09.00/0
where the values of @ I [mV] and @ 2 [mV] are proportional to the redox potentials measured by a Pt-electrode in each reactor. @la. [mV] and [mV] are the arithmetic mean values of the oscillating potentials measured in the input reactors 1 and 2, respectively. These values are controIled and adjusted every 500 s during measurement. ~ 1 [dimensionless] 3 and ~ 2 [dimensionless] represent the coupling strengths between each input reactor and the output reactor. kf3,, is the flow-rate into reactor 3 without coupling. The reactors 1 and 2 operate in oscillatory period-1 states at almost equal flow-rates (reactor s-l, residence time t = 1953 s; reactor 2, 5.03 1, 5.12 x x s-I, t = 1988 s). The oscillatory periods of oscillators 1 and 2 at these flow-rates are 29.5 and 27.5 s, respectively. At a flow-rate kf, = 2.01 x s-l (t= 498 s) a Hopf bifurcation occurs using the parameters in reactor 3 and the system adopts a focal steady state at higher flow-rates. An oscillatory period-1 state with very small amplitudes is found just below kf, in a narrow interval of flow-rates. A secondary Hopf bifurcation is observed at kf = 1.89 x s-l (t = 529 s) where complex periodic oscillations of high amplitudes occur. For the system to settle in a focal steady state the flow-rate of reactor 3 is
0 1995 American Chemical Society
3
13174 J. Phys. Chem., Vol. 99, No. 35, 1995
v A Reactor 3
W
Figure 1. The experimental setup consists of three separate reactors,
7.6 mL volume each. The electrochemicalpotentials are measured with Pt-electrodes against Ag/AgCl in all reactors. The potentials of the reactors 1 and 2 are transformed to a flow-rate via eq 1 that is applied to reactor 3. The setup is similar to a simple neural feedforward net where the input layer consists of the reactors 1 and 2 and the output layer of reactor 3. adjusted at 1.03kfc,i.e., 3% above the Hopf bifurcation (kf30 = 2.07 x s-’). The resonance frequency of the focus at this particular flow-rate was determined separately (section 3.1). By the application of different bromate concentrations, the frequencies of the oscillations in the input reactors are chosen in such a way that they are near resonance with the focal steady state in the output reactor 3. The redox potentials @pi are mainly related to the momentary cerickerous ion ratio.I6-l8 They are recorded every 0.5 s and simultaneously converted according to eq 1. The recorded potentials are transformed to analog signals via a 12-bit digitauanalog converter card and then transformed to a frequency applied to the stepping motor of the linear feed pump for reactor 3. In our investigations of quasiperiodic perturbations we used equal values of ~ 1 and 3 ~ 2 3 . For periodic perturbations of the focus ~ 2 = 3 0. Each reactor is fed with three syringes (a,b,c) containing the following solutions: (a) 0.70 m o m NaBrO3 for reactors 1 and 2 and 0.42 m o m -1-03 for reactor 3; (b) 1.5 x mom Ce3+ from Ce2(S04)3 and 0.90 m o m HOOCCHzCOOH; (c) 1.125 m o m H2S04. To obtain reactor concentrations divide by 3. The malonic acid was recrystallized twice from acetone to remove trace i m p ~ r i t i e s . ’ ~ -All ~ ~ other chemicals are of analytical grade and used without further purification. All solutions are prepared from water purified by ion exchange (specific resistance > 10 M R cm; purification system Milli-Q, Millipore). The reactors are stirred magnetically with tefloncoated stirrers at 13.3 Hz. The flow-rates of the above solutions into the three reactors are regulated separately with three precise linear pumps. The reactors, feed-lines, and syringes are thermostated at 25 f 0.2 “C. For experiments on random forcing of a steady state close to a Hopf bifurcation the experimental setup described in refs 22 and 23 was used. Here, a shortened spectrophotometric cell of 1.4 mL volume serves as a continuous flow stirred tank reactor (CSTR). The reactant solutions are fed into the CSTR by a computer-controlled three-channel piston pump. Random shot noise is superimposed on the flow-rate by randomly changing the speed of the syringe pump in fixed time intervals of 2 s according to eq 2.
where R(6) are approximately Gaussian distributed random numbers in the range from - 1 to 1, a is the white noise
+
Zeyer et al.
flow-rate jump
400
0
250
500
time
[SI
750
Figure 2. At a low flow-rate of kf = 7.56 x
s-I (t = 1323 s) the BZ-reaction shows period-1 oscillations while at a high flow-rate of kr = 2.07 x s-I (t = 483 s) the system stabilizes on a focal steady state showing damped oscillations. The period of these damped oscillations is equal to the resonance period of the focal steady state.
amplitude, and kfo and kd6) are the unperturbed and perturbed flow-rates, respectively. The standard deviation of the flowrate noise is called the “amplitude” of the fluctuations, and the time between two successive flow-rate changes is designated as the “pulse length” of the shot noise.
3. Experimental Results 3.1. Determination of the Resonance Frequency of the Focus. To yield optimal response signals in the perturbed focus, it is necessary that the perturbing frequency is similar to the resonance frequency of the focus. Therefore, we performed a flow-rate jump experiment in a single free running reactor. At a flow-rate of kf = 7.56 x s-l (z = 1323 s) the BZ-reaction shows period- 1 oscillations with the concentrations given above ([BrO3-] = 0.42 M) (Figure 2). At the point marked with an arrow the flow-rate was suddenly increased to kf = 2.07 x s-! (t = 483 s), which is located 3% above the Hopf bifurcation. Thus, the BZ-system relaxes to the focus with damped oscillations (Figure 2). The resonance period of the focus was found to be 27 s. We used an increased bromate concentration of 0.7 M in both input reactors in order to adjust the periods of the input reactors to be close to the resonance period of the focus. 3.2. Periodic Forcing of the Focal Steady State. At low coupling strengths the focus shows a period-2 response in the output reactor (Figure 3). The total periods of the perturbation and the period-2 response are entrained fundamentally (1:1) with each other. The amplitude of the response increases with increasing coupling strength until the focus shows bursting behavior at a threshold perturbation amplitude (Figure 4). Every burst is followed by small oscillations that increase in amplitude before the next burst occurs. Generally the number of bursts increases with increasing coupling strength in a given time interval. The number of small oscillations between the bursts varies due to experimental noise and shows the great sensitivity of the focal response. The focus responds immediately after applying the flow-rate perturbation, In control experiments using computer-controlled sinusoidal forcing of the flow-rate the above observations were confirmed. If the experimental system was driven across the Hopf bifurcation by a sufficiently high forcing amplitude, no bursts were found at high forcing frequencies where the focus responds with small amplitude sinusoidal oscillations. One observes responses of period-2 when the forcing frequency is decreased. As the perturbation frequency approaches the resonance frequency of
J. Phys. Chem., Vol. 99, No. 35, 1995 13175
Quasiperiodic Forcing of a Chemical Reaction coupling strength 4.00
-.--
1200
-
"Outputreactor"
v1
C
0 3000
3250
3500 time [SI
3750
4000
Figure 3. Periodic perturbation of the focal steady state at the coupling strength ~ 1 =3 4.00 and w23 = 0. The perturbation consists of period-1
relaxation oscillations. The perturbed focus displays period-2 oscillations. The frequencies of the driving period-1 state and the period-2 response are equal. coupling strength 5.50 1600
I
I
1 couyling
"Outputreactor"
1
"Inputreactor" 0 0
1000
2000 time
[SI
3000
4000
Figure 4. At a coupling strength of ~ 1 =3 5.50 and w23 = 0 the focal steady states responds with bursting behavior. The focus reacts immediately when the coupling is tumed on (see arrow). The forcing function is identical to that for Figure 3.
the focus, one large amplitude burst occurs in two sinusoidal forcing cycles. At resonance and at lower forcing frequencies large amplitude bursting occurs once per forcing cycle. 3.3. Quasiperiodic Perturbation of the Focal Steady State. When ~ 1 and 3 ~ 2 are 3 finite and the periods in the input reactors 1 and 2 are slightly different, the focal steady state is perturbed quasiperiodically. At the coupling strength w13 = w23 = 2.50 the focus responds quasiperiodically (Figure 5a). The Fourier analysis shows two main frequencies, their difference, and harmonics. These frequencies are equal to the main frequencies of the period-1 oscillations in reactors 1 and 2 (Figure 5b). We reconstructed an attractor (Figure 5c) according to the method of singular value decomposition ( S V D - m e t h ~ d which ) ~ ~ shows a torus-like structure. Figure 5d shows a Poincark section. The points on the PoincarC section and on the one-dimensional map (not shown) form closed curves in accordance with quasiperiodicity. When both input reactors are in phase, the output reactor shows a maximum in response amplitude. At this moment the flow-rate amplitudes applied to the output reactor are largest according to the coupling equation (eq 1). The flowrate amplitudes are smallest when both input reactors are 180" out-of-phase and the response of the focus is lowest. Therefore, one can observe the perturbations due to the individual input
reactors. At w13 = ~ 2 =3 2.75 the output reactor 3 shows bursts when both oscillations of the input reactors are in phase (Figure 6). These bursts are almost identical in amplitude. At all other phase angles of the input oscillations the output shows small amplitude oscillations, with frequencies corresponding to their linear combinations. The applied coupling strength is slightly above the threshold for the onset of bursting, and bursting is occasionally absent. At wl3 = ~ 2 =3 3.25 the bursts occur in groups of two separated by four small oscillations (Figure 7). Bursting occurs when both inputs are in phase as in the former case. After one burst the system needs a certain time for recovery before another burst is observed, although the oscillations in the input reactors are still almost in phase. When the coupling strength is further increased to ~ 1 =3 ~ 2 =3 3.50, bursting occurs in groups of three when both inputs are almost in phase (Figure 8). As in the former case the system still needs a certain recovery time after one burst. Figure 8 also shows the fast response of the focus when the coupling is tumed on. 3.4. Random External Forcing of the Focal Steady State. Random forcing of the focus using external shot noise was performed by changing the actual flow-rate through the reactor after constant time intervals according to a Gaussian distribution of random numbers. Fluctuations of small amplitude lead to irregular small amplitude oscillations around the steady state without any large amplitude bursts. At perturbation amplitudes large enough to drive the system across the Hopf bifurcation into the oscillatory region, however, large amplitude bursts eventually occur. The length of the time interval between two successive flow-rate changes is of particular importance. At a relative perturbation amplitude of 10% of the actual flow-rate value no bursts were observed if the above defined fluctuation pulse length was smaller than 5 s. The system does not respond strongly toward fast flow-rate changes. Slower perturbations of the same amplitude, however, lead to large bursts. These bursts are related to the large amplitude oscillations observed within the oscillatory region of Farey-ordered p a t t e m ~ . * ~ . ~ ~ Figure 9 shows a time series of the Ce4+concentration recorded at an average flow-rate located 3% above the critical value and at a random perturbation amplitude of 10% with a pulse length of 5 s. Bursts eventually occur, and they are separated by irregular trains of small amplitude oscillations (not shown). If the fluctuation amplitude is increased to 20%, the bursts occur more frequently as seen from Figure 10. The occurence of bursts may be viewed as an amplification of the imposed fluctuations. If the time scale of the noise allows resonance of the fluctuations with the focus, relatively small perturbations may lead to large excursions through phase space.22 In contrast, fluctuations occuring on a very fast time scale do not couple effectively to the system's dynamics, and amplification cannot be observed.
4. Calculations with the SNB-Model The SNB-mode126(Table 1) is an extension of the Oregonator of the BZ-reaction. In contrast to the Oregonator reverse reactions are considered for the inorganic part. Moreover, the autocatalytic formation of HBr02 is split into two steps, introducing BrO2' as an additional variable. We use the rate constants given by Field and Forsterling2*except for k2. We set k2 = 200.0 as proposed by Schneider and Miinster22(Table 1). The initial values of the variables and parameters are given in Table 1. The SNB-model shows complex periodic states at high flowrates26 as observed e ~ p e r i m e n t a l l y . ~By ~ - use ~ ~ of the given constraints, these complex oscillations terminate at kf = 3.6565 x s-I bifurcating to period-1 oscillations of low amplitudes
13176 J. Phys. Chem., Vol. 99, No. 35, 1995 coupling strength 2.50
(a) 1600
"Outputreactor"
coupling
-
Zeyer et al.
1200
800 A
.-w
P)
a 0
400 -
0.003 -
"Outputreactor"
I
0.002
0.001
Y 0.000
0.0
0.2
0.4
0.6
1.o
0.8
x
frequency [rad/s]
Figure 5. (a) Quasiperiodic flow-rate perturbation of the focus results in aquasiperiodic response of the focus at low coupling strengths. (b) The Fourier spectrum of the time series shown in part 1 shows two main frequencies that are equal to the main frequencies of the two perturbing input reactors. The difference between the two main frequencies is also observed. (c) The attractor was reconstructed using the SVD-method from the time series shown in part a. 100 columns were used for constructing the trajectory matrix and a maximum embedding dimension of 10 for the reconstruction. The dimensions x, y, and z are shown which correspond to the largest singular values in decreasing order. (d) A PoincarC section of the attractor shown in part c shows points forming an approximately closed curve. coupling strength 3.25
coupling strength 2.75
1600
lGGG "Outputreactor"
coupling
O L
0
"Outputreactor"
I
2000
4000 time
Figure 6. At a coupling strength of ~
[SI
=3 ~
GOO0
2 = 3 2.75
0' 5000
I
5500
time [SI
6000
6500
the output reactor shows large bursts when both inputs are in phase. The bursts are almost equal in amplitude.
Figure 7. The bursts occur in groups of two when the coupling strength is set to w13 = W23 = 3.25. Bursting is only possible when the phase
(Ce4+amplitude 5 1 x 10-l2 m o m ) . At kfc = 3.6620 x s-I (t= 27.3 s) a supercritical Hopf bifurcation occurs and gives way to a steady state at higher flow-rates. In our model calculations we set the flow-rate of reactor 3 to k30 = 1.03kfc= 3.7719 x lo-* s-l (t= 26.5 s) at zero coupling strength. The resonance period of the focus was determined to be =23.5 s.
The flow-rates of the input reactors 1 and 2 were kept constant s-I (t= 111.1 s) and k~ = 0.5 x s-l at kfl = 0.9 x (t= 200.0 s) where the SNB-model shows period-1 oscillations. As in the experiments, the bromate inflow concentrations of the input reactors are chosen to be higher than that in the output reactor (0.1900 and 0.0875, respectively) to establish oscillation
1
shift of the inputs is small.
J. Phys. Chem., Vol. 99, No. 35, I995 13177
Quasiperiodic Forcing of a Chemical Reaction
TABLE 1: SNB-Model"
coupling strength 3.50
1600 coupling
"Outputreactor"
reaction
rate
+Y +2 H e X +P X +Y +H=2P A +X +H s 2 W W+ C +H t X + Z ' 2X A +P +H Z'-gY + c
kl = 2.0; kz = 200 k3 = 3 x lo6; k4 = 2 x IO+ ks = 42; kg = 4.2 X 10' k7 = 8 x IO4; ks = 8.9 x lo3 k9 = 3 x lo3;klo = 1 x klI = 0.2
A
m
initial values
I
5
I
a 0
A(0) = 0.087 3 1 Y(0) = 2.6 x X(0) = 9.687 x
400 -
0'
I
0
2000
6000
4000 time [SI
Figure 8. The number of bursts increases to three in each group at a coupling strength of 3.50. The bursting is also phase dependent in this case.
-
0.25
I
i
sm c
x .* 2
0.15 -
-0
-
H
.a
0.10
I
1
0.05
0.00
0
500
I ' 1000 1500 2000 2500 3000 3500 4000 time [SI
Figure 9. Random forcing of the experimental focus. The forcing amplitude is 10% and the pulse length 5 s. 0.30
r
I
I1
0.25
0.00
'
0
inflow concentrations
Ao= 0.0875 co=I x 10-3 H+ = 0.7466 (constant)
" A = BrO3-, Z' = Ce4+, C = Ce3+, P = HOBr, W = Br, X = HBr02, Y = Br-, and g = 0.45 (stoichiometric factor).
I
0.30 1
E
P(0) = 5.129 x
w(o)= 5.931 x 10-7 C(0) = 9.0122 x zyo) = 9.87 x 10-5
I
500
1000
1500 time [SI
2000
2500
3000
Figure 10. Random forcing of the experimental focus. The forcing amplitude is increased to 20% while the pulse length is held constant at 5 s. The perturbed focus shows aperiodic bursting behavior.
periods similar to the resonance period of the focus in the output reactor. The Fourier spectra of these oscillations show main frequencies at wl = 0.269 25 rads and w 2 = 0.281 50 rads and their harmonics, which correspond to oscillatory periods of 23.3 and 22.3 s, respectively. We calculated the electrochemical potentials in all three model reactors by the use of the Nernst equation under the assumption that only the CeJ'/ Ce3+redox couple determines the potential. A standard potential
of Eo = 1.44 V was used.29 We determined the average values of the potential oscillations in both input reactors at the given flow-rates to be 1.369 29 and 1.365 21 V, respectively. In our calculations we applied the same coupling equation as in the experiments (eq 1). 4.1. Periodic Forcing of the Focus. In a first series of calculations we set ~ 2 =3 0 to investigate the effects of periodic perturbations of the focus. At low coupling strengths (wl3 5 2.0) a period-1 state emerges which is fundamentally entrained with the perturbation. Periodic and aperiodic bursting are observed when the coupling strength is Z 2.1. A period-2 state as in the corresponding experiments is obtained when the period of the perturbing oscillator is higher than the resonance period of the focus. A perturbation period of 38.3 s, which is obtained at a bromate inflow concentration of 0.0875 and a coupling strength ~ 1 53 5.0, leads to a response of period-2 as in the experiments. 4.2. Quasiperiodic Forcing of the Focus. We further investigated the effects of quasiperiodic perturbations applied to the focus. The coupling strengths ~ 1 and 3 ~ 2 were 3 set equal. At relatively low coupling strengths ( ~ 1 = 3 ~ 2 53 1.4) the output responds quasiperiodically as in the experiments. In Figure 1l a the output reactor at ~ 1 = 3 ~ 2 = 3 1.20 is depicted. The corresponding Fourier spectrum (not shown) displays two main frequenciesthat are equal to the main frequencies of the period-1 oscillations of the input reactors and their harmonics. Additionally, the Fourier spectrum shows a low-frequency signal equal to the difference of the two main frequencies. The original attractor has a torus-like structure (Figure 1lb). The Poincark section (Figure 1IC) and one-dimensional map (not shown) display closed curves in accordance with quasiperiodicity. The response amplitude of the focus increases with increasing coupling strength. At coupling strengths higher than ~ 1 =3 w,, 2 1.5 bursting occurs when both input oscillators are in phase. The bursts occur in groups and are separated by small amplitude oscillations as in the experiments (Figure 12a). This demonstrates that the perturbed focus needs some recovery time before new bursts can take place in accordance with the experiments. The number of bursts increases with increasing coupling strength as was found experimentally. In contrast to the experiments the individual group pattern of the bursts is aperiodic. In quasiperiodically forced systems, such as in a variety of maps, differential and an experiment with a magnetoelastic ribbon,30 strange nonchaotic attractors are known to be a generic phenomenon. It has been proposed that strange
, Zeyer et al.
13178 J. Phys. Chem., Vol. 99, No. 35, 1995 coupling strength 1.20
(a)
1.42
(a)
"Outputreactor 3"
1.38
w
F
coupling strcngth 3.00
1.42
"Outputreactor 3" -
-
-
-5;
1.38
#
9
u
S?
1.34
a
i
1.30
201000
(c)
0.150
r10
-3
i--------
O' l 2 0.10
0
time [SI
202000
7 I
J 0.125
201000
200000
[SI
time
A
I
202000
200000
0.08
-
0.06
-
0.04
-
-
0 . i
0.100
I I I 0. 2 -8.4564
LAI_/_
0.0558.3
0.4
0.5
ti
0.6
L
7
0-6
X Figure 11. (a) Quasiperiodic flow-rate perturbationin the SNB-model is shown. At the coupling strength w13 = w23 = 1.20 the focus shows
quasiperiodic behavior. (b) The corresponding original attractor shows a torus structure typical for quasiperiodicity.The variables Ce4+,Ce3+, and HOBr are plotted on the x, y, and z axes, respectively. (c) A Poincare section of the attractor in part b shows a closed curve.
-0. 4562
-0.4560
r10-3
X
Figure 12. (a) The output reactor 3 is shown at a coupling strength of ~ 1 = 3 w23 = 3.00,where the SNB-model shows aperiodic bursting behavior. (b) The original attractor corresponding to the time series of part a shows a complex structure which is completely different from
the appearence of the quasiperiodicattractor obtained at lower coupling strength (part c). The variables Br-, HBr02, and Br03- are plotted on the x , y , and z axes, respectively. (c) A Poincark section of the attractor in part b does not show a closed curve which would be expected for a quasiperiodic attractor (part c).
nonchaotic attractors emerge through torus-doubling followed by collision of the doubled torus with its unstable parent tablishes a relation between the Lyapounov exponents and the Strange nonchaotic attractors are fractal objects in phase space. Lyapounov dimension DL: In contrast to deterministic chaos the trajectories do not diverge on the attractor, and therefore, no positive Lyapounov exponents exist. Since the quasiperiodic forcing introduces two degrees of freedom into the system, the geometric dimension cannot be i= 1 D,=k+(3) less than 2.0. Assuming that the Kaplan-Yorke-c~njecture~~~~~ I4S-l I is applicable to our present quasiperiodically forced system, the Lyapounov dimension DL must be 2.0 for a quasiperiodic as The A; denote the Lyapounov exponents that are ordered with well as for a strange nonchaotic attractor and higher than 3.0 decreasing values where k is an index number such that for deterministic chaos.30 The Kaplan-Yorke-conjecture es-
J. Phys. Chem., Vol. 99, No. 35, 1995 13179
Quasiperiodic Forcing of a Chemical Reaction k
k tI
i
I
5. Discussion
The Lyapounov dimension DL is an upper estimate of the information dimension DI.
D, 2 D,
(4)
N
-CP;In pi D,= -1im 6-0
i= I
log(€)
The attractor is covered with a net of n-dimensional cubes of side length E . n is the dimension of the state space. Pi is the probability of finding a point of the attractor in a given cube. D, describes the gain of information when the length E of cubes covering the attractor is decreased. We performed a dimensional analysis of the original attractor made up of the seven variables of the SNB-mode126underlying the time series in Figure 12a using the near-neighbora l g ~ r i t h m . ~For ~ . the ~ ~ analysis we used 100 000 data points, separated by constant time intervals of 0.5 s, up to 5000 reference points and up to a maximal order of 2000 neighbor points. We obtained an information dimension of DI = 2.0 despite the aperiodic appearence of the time series. Furthermore, the appearance of the original attractor (Figure 12b) and its PoincarC section (Figure 12c) excludes a quasiperiodic behavior (compare parts b and c of Figure 11 with parts b and c of 12). Thus, our analysis may be viewed as a hint for the existence of a strange nonchaotic attractor in the quasiperiodically forced SNB-model. Any further analysis of the attractor regarding Lyapounov exponents or the Hausdorff dimension (above 2.0) tumed out to be problematic due to its heterogeneity. 4.3. Random Forcing of the Focus. In the SNB-mode126 we superimposed Gaussian distributed random fluctuations on a stationary state located 3.0% above the Hopf bifurcation point by randomly changing the model flow-rate in a way similar to that in the experiments. The subroutine mnoa from the IMSL library was used to perturb the flow-rate, and an explicit RungeKutta scheme was utilized to solve the set of differential equations including noise. Our numerical simulations showed good agreement with the experimental findings. Small amplitudes of noise imposed upon the focus lead to small amplitude nonperiodic oscillations while irregular bursts occured at higher noise amplitudes. In a typical calculation of the Ce'" variable at a relative noise amplitude of 15% and a perturbation pulse length of 2 s single bursts occur occasionally on a background of irregular small oscillations similar to that in the corresponding experiments (Figure 10). The amplitudes of the bursts are equal to the amplitudes of the regular period-1 oscillations observed in the SNB-model inside the oscillatory region. As demonstrated in previous work,9 the small oscillations between the bursts show a distribution of frequencies around the resonance frequency of the focus. 4 comparison of the standard deviations of the perturbation and the response signal demonstrates that the imposed noise is amplified if the perturbation amplitude exceeds a threshold value. In other words, the standard deviation of the response time series from its mean value is smaller than the imposed noise amplitude for small perturbations and larger than the noise for large perturbations. The latter case represents an amplification of large fluctuations due to the frequent occurence of large amplitude bursts.
Three reactors have been arranged like a simple neural feedforward nett5 to investigate the effects of quasiperiodic perturbations using the system's own dynamics. The perturbation is performed unidirectionally from the input layer (reactors 1 and 2) to the output (reactor 3). The input reactors 1 and 2 work in oscillatory period- 1 states at incommensurate frequencies for the quasiperiodic perturbation. The output reactor contains an excitable focal steady state. In nonlinear chemical reactions perturbations of focal steady states can lead to various periodic and chaotic states when the periods and amplitudes of the perturbations are varied.I0 This complex response is due to the interaction of the external perturbation frequency and the inherent frequency of the focus. In our case simple periodic perturbations lead to oscillations of period-2 in the output reactor at low coupling strengths (Figure 3) while strong coupling results in bursting behavior (Figure 4). When the perturbations are shaped quasiperiodically the focus also responds quasiperiodically at weak coupling strengths as expected (Figure Sa). The Fourier spectrum of this quasiperiodic experimental response contains mainly two incommensurate frequencies equal to the main frequencies of the driving period- 1 oscillators in the input reactors. Quasiperiodic perturbation also leads to bursting behavior (Figures 6-8) provided that two conditions are fulfilled: the phase shift between the oscillations in both input reactors must be sufficiently small and the effective perturbing amplitude must be above threshold. A refractory time exists during which the perturbed system recovers to be ready for another bursting excursion. All bursts are almost equal in amplitude. On the basis of the above control experiments using sinusoidal forcing generated by a computer, one may conclude that there is a qualitative similarity between using the system's own dynamics and using the corresponding computer-generated forcing function. When Gaussian distributed random noise is imposed on the focus, the system oscillates irregularly around the steady state with low amplitudes at weak perturbation strengths (Figure 9). Above a certain noise amplitude the focal steady state responds with aperiodic bursts (Figure 10) when the system is driven stochastically across the Hopf bifurcation. The focal response is determined by the perturbation amplitudes as well as the pulse lengths of the applied fluctuations. At short pulse lengths the dynamics of the system cannot couple efficiently with the flowrate fluctuations. Bursting behavior has been known in the BZ-reaction through early CSTR s t ~ d i e s . ~ ~Bursting -~O can be modeled when a slow changing variable moves the system periodically across a However, in the present subcritical Hopf bif~rcation.~' -45 experiments and in the SNB-mode126 the investigated focal steady state loses stability via a supercritical Hopf bifurcation upon decreasing the flow-rate and giving way to period-1 oscillations of small amplitudes. In the BZ reaction bursting occurs through the presence of an additional secondary Hopf bifurcation which is located at a slightly lower flow-rate.22The observed bursts originate from a combination of these two Hopf bifurcations when the perturbation drives the system into the lower flow-rate ranges for a sufficiently long time. Striking phenomenological analogies exist between burstlike behavior in open nonlinear chemical systems and neural signals which suggest that bursting is a universal phenomenon. Neural action potentials resemble the observed large amplitude excursions of the b ~ r s t s . ~ ~To . ~ achieve ' a neural response, an excitation threshold has to be exceeded; the amplitude of the triggered response is independent of the excitation strength.
13180 J. Phys. Chem., Vol. 99, No. 35, I995
Zeyer et al.
In our numerical calculations of the quasiperiodically forced SNB-model we could find indications of a strange nonchaotic attractor. From the shape of the original attractor and its Poincart section one may observe that the attractor is more complex (Figure 12b) than a quasiperiodic attractor (Figure 1lb) which is found at lower forcing amplitudes. On the other hand, the value of the information dimension indicates that the attractor is not chaotic since DI = 2.0. For a chaotic attractor DImust be larger than 3.0 if the system is quasiperiodically driven. Thus, strange nonchaotic attractors also exist in realistic chemical models such as the SNB-model. In the experiments our dimensional analysis turned out to be inconclusive due to the stiffness of the observed motion and the experimental noise. Therefore, one should look for experimental strange nonchaotic attractors in those chemical oscillators whose attractors are sufficiently homogeneous to allow a reliable statistical analysis.
Acknowledgment. We thank S. Gunther for valuable experimental assistance and M. J. B. Hauser for discussions. We further thank the Volkswagenstiftung and the Deutsche Forschungsgemeinschaft for partial financial support. References and Notes .~.._ . . ~ ~ . ~ ~
~
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