J . Phys. Chem. 1993,97, 12244-12248
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Continuous Control of Chemical Chaos by Time Delayed Feedback F. W. Schneider,' R. Blittersdorf, A. Forster, T. Hauck, D. Lebender, and J. Muller Institute of Physical Chemistry. University of Wiirzburg, Marcusstrasse 9/11, 0-97070 Wiirzburg, Germany Received: May 26, 1993; I n Final Form: September 2, 1993"
Continuous time delayed feedback, based on a method by Pyragas, is applied to the Belousov-Zhabotinsky (BZ) reaction to stabilize unstable periodic orbits embedded in the chaotic attractor. Our experimental results a r e compared with numerical calculations of the four-variable model (Montanator) for the B Z reaction. The Pyragas method is also applied to calculations of a mechanism (Aguda-Larter model) for the peroxidaseoxidase reaction. In addition to the stabilized orbits in the strange attractor, new periodic, chaotic, and steady states are produced for large values of the feedback function in both models. The possible role of chaos and feedback in nature is discussed.
Introduction
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The control of chemical chaos involves the stabilization of unstable orbits embedded in a chaotic attractor. A method of discontinuous control of chaotic states has been proposed by Ott, Grebogyi, and Yorke (OGY)' for stabilizing unstable periodic orbits contained in a chaotic attractor by using a single small perturbation when the system is close to a fixed point on the strange attractor.2 This method requires a simultaneous on-line computer analysis to determine when the single perturbation must be applied. It has been used to control chaos in a number of physical3-9 and chemical systems including the BelousovZhabotinsky"JJ1 (BZ) reaction. Continuous methods of controlling dynamic states inluding chaos appear more convenient from an experimental standpoint. The continuous control of chaos by time delayed feedback or by an external periodic source has been recently formulated by Pyragas.12 The Pyragas method used in our experiments and calculations is based on continuous self-controlling delayed feedback. This method continuously applies small amplitude perturbations to stabilize unstable periodic orbits contained in the chaotic attractor. A feedback function F(t,T)12 is defined as the difference between a time delayed signal y(t--7) and the actual signal y ( t ) F ( f , T ) = K b ( t - 7 ) - y ( t ) ] = KD(t,T) (1) in which y is a time dependent species concentration, T is the delay time, Kis the feedback strength, and D(t,7) is the difference between the delayed and the actual signal (Figure 1). F(t,r) is used as an additive term in the differential equations. Due to greater experimental convenience, in the present BZ experiments the feedback function acts on the inflow rate kf of all reactants, whereas in our calculations the feedback function is allowed to act only on the flow rate of Z. In the Aguda-Larter model for the peroxidase-oxidase (PO) reaction the feedback function acts on the inflow rate of gaseous 02.When T is chosen to be equal to the period of an unstable orbit in the strange attractor, F(f,T)may tend to zero, and the particular orbit becomes stabilized, Le. T = Ts(oscillation period of the stabilized orbit). For other choices of 7 , where F ( ~ , Tmay ) become larger, new dynamic states are generated by this time delayed feedback, as illustrated by Pyragas for the Rassler model." In earlier work, we have applied a similar method of continuous time delayed feedback to the minimal bromate (MB) oscillatorI4l6 and to the BZ r e a ~ t i o n ~in~ aJ *CSTR (continuous flow stirred tank reactor). For experimental reasons an average value for the Ce4+ concentration bav) was used in placeofy(t) to implement the delayed feedback. A saw tooth behavior of the oscillation periods and ~~
Abstract published in Aduance ACS Abstracts, November 1, 1993.
0022-365419312097- 12244$04.00/0
CHEMICAL REACTION
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Figure 1. Diagram of continuous time delayed feedback control with a delay line D and delay time T after ref 12. The output is the observed species y ( t ) which is equal to Ce4+ in the BZ experiment, Z in the Montanator model?' and 0 2 in the PO model.23 In the BZ experiment the inflow rate is kf = kf, + F ( t , T ) . In the Montanator model the flow rate of Z is kf(m = kf,(q + F(r,s), and in the PO model the inflow rate refers to the oxygen flow rate u9 = ~9~ + F ( t , T ) .
also birhythmicity were observed in the MB reaction, which does not show chaotic behavior for a linear time delay. Using nonlinear delayed feedback, chaotic states were observed by Chevalier et al.19 in the MB oscillator with manganese. When we applied time delayed linear feedback in the BZ reaction using the y,,, method, a number of new periodic and chaotic states were generated experimentally.17 In the following we report experiments and calculations of continuous time delayed feedback in the BZ reaction and its four-variable model (Montanator), respectively. The application of the Pyragas method to the biochemical peroxidase-oxidase reaction in the Aguda-Larter model is also shown.
Materials and Methods Malonic acid (Merck) was twice recrystallized from acetone; KBr03 (Merck) contained less than 0.02%bromide. Ce2(SO4)3 (Fluka, 99.99%) and H2S04 (Merck, 97%) were used without further purification. The three feed solutions consisted of 0.60 M sulfuric acid (syringe l ) , 0.30 M potassium bromate (syringe 2), and 0.75 M malonic acid with 2.5 X I t 3 M cerium sulfate (syringe 3). To obtain reactor concentrations, divide by 3. The CSTR, as described earlier,20is a spectrophotometric cell with 1.92-mL volume. A magnetic bar stirred at 1200 rpm by a small motor below the reactor ensured effective mixing of the reactants. A self-designed high-precision pump driven by a stepping motor directed the flow of solutions into the CSTR at a controlled rate.20 The syringes containing the inflow solutions, the connecting tubes, and the CSTR were thermostated by a circulating water bath, held at a constant temperature in all of the experiments (T = 28.0 f 0.1 "C). The time dependent concentration of the aqueous Ce(1V) complex was monitored with a UV/vis spectrophotometer (Beckmann Lambda7) at a wavelength of 350 nm. The sampling rate was 1 Hz. 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 47, 1993 12245
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Figure 2. (a) Experimental time series of the BZ reaction showing chaotic oscillations and a stabilized P2 pattern after the start of the feedback control (at t = 7200 s) for a delay time of T = 230 s. The reactor concentrations, after mixing, are [malonic acid] = 0.25 M, [(Ce2(SO4)3] = 8.33 X 10-4 M, [KBrOa] = 0.10 M, and [H2S04] = 0.20 M. The temperature is constant at 28.0 0.1 OC. The residence time for the free running system is 63 min (kf = 2.65 X 10-4 s-1). (b) Dispersion curve of the above time series. The dispersion for the chaotic part Dc2 and the staiblized P2 part Dpz2 of the time series are Dc2= 5.66 X 10-4and Drq2 = 4.66 X 10-5, respectively. (c) SVD-reconstructed attractor of the chaotic time series. (d) Attractor
of the stabilized P2 orbit. (e) 1-D map, calculated from the Poincard section (not shown) of the attractor (c), indicating deterministic chaos. Experimental Results Figure 2a shows a chaotic time series (residence time of 63 min) for the BZ reaction. With the Ce4+ concentration as the controlling species, the continuous feedback algorithm of Pyragas is applied to the inflow rate kf of all reactants in our BZ experiments. It is initiated at 7200 s with a delay time T = 230 s and a coupling strength K = 0.30 M-1 s-1, Following a 800-s transience time, a stabilized period-2 (P2) time series is obtained, as shown in the dispersion curve (Figure 2b). The experimental chaotic and stabilized P2 attractor are shown in Figure 2c,d, respectively, together with the one-dimensional map of the chaotic attractor (Figure 2e). We also show a stabilized period-3 (P3) time series at T = 371 s and K = 0.22 M-1 s-1 (Figure 3a) which was obtained from the same chaotic state as in Figure 2. Here thefeedbackalgorithmwasalsostartedat 7200s,andstabilization occurred after a 600-s transience time. The corresponding dispersion curve (Figure 3b) and attractors (Figure 3c,d) are shown. It should be noted that within experimental error the strange attractors of Figures 2c and 3c are identical but their viewing angles are different.
Model Calculations and Discussion The experimental control of chaos in the BZ reaction using the Pyragas method (eq 1) was simulated with the four-variable model (Montanator) of Gyargyi and Field,21*22where the observed species y ( t ) -2,andtheflowrateiskfcz) = k f N a + F ( t , ~ )Thedispersion . 0 2 represents the average square value of D ( ~ , Tcalculated ) for a time interval of 10 000 s a t a given delay time. 0 2 becomes small (Figure 4a) when r is close to the period of an unstable
periodic orbit in the chaotic attractor. For the Montanator a bifurcation diagram has been calculated for K = 100 M-1 s-1 with values of the delay time ranging from 0 to 400 s (Figure 4b). The rate constants and concentrations were those of Gyorgyi and Field.21 The unstable P2 orbit is stabilized at T = 346 s, where 0 2 shows a local minimum under the set of conditions used (Figure 4a). Other stabilized orbits for T < 340 s represent new periodic states (e.g. a P1 state at T = 100 s), since their 02 values are relatively high or they are not part of the original strange attractor. The calculations show the stabilization of an unstable P2 orbit embedded in the chaotic attractor with good experimental agreement (Figure 2). The values of K are different for experiment and simulations since the respective input parameters are different. Parts a and b of Figure 5 show the chaotic attractor of the free running Montanator and the stabilized P2 orbit embedded in thechaotic attractor, respectively. We haveobserved experimentally that the use of F ( ~ , Ton) the total flow rate of all species leads to similar results in stabilizing an unstable periodic orbit as the use of the flow rate of a single species. We also applied eq 1 to a biochemical model by Aguda and Larter23for the horseradish peroxidase (PO) catalyzed oxidation of NADH. Here the controlling species y(r) is represented by the 02 concentration. During control, the oxygen inflow is the sum ug = u9, F(t,r). Our numerical calculations for K = 0.02 s-1 show the stabilization of P1 and P2 orbits at T = 156 and 276 s, respectively, as well as three stabilized steady states (Figure 6b). Corresponding minima are seen in the 02 plot (Figure 6a). New periodic and chaotic states are generated in this particular T range (Figure 6b). Parts a-c of Figure 7 show the chaotic attractor of the free running PO model and the stabilized P1 and P2 orbits embedded in the chaotic attractor, respectively.
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Figure 4. (a) Dispersion 02 and (b) Z versus the delay time T for the Montanator model for K = 100 M-' s-I and kf+-) = 5.9375 X 1 P s-1.
Allother parameters aregiven in ref 21. Maxima and minima areplotted. P2 = period-2 oscillation.
Figure3. (a) Experimental timeseries of the BZ reaction showing chaotic oscillations and a stabilized P3 pattern after the start of feedback control (at r = 7200 s) for a delay time of 7 = 371 s. (b) Dispersion curve of the above time series. The dispersion for the chaotic part Dc* and the stabilized P3 part Dp,Zof the time series are Dc* = 1.13 X 10-3and Dp+ = 5.58 X 10-s,respectively. (c) SVDravlnstructedattractorofthechaotic time series. (d) Attractor of the stabilized P3 orbit. The discrimination between stabilized and newly created periodic orbits is particularly well demonstrated in Figure 6a whenever 02 is close to zero. This is the case, for example, for P1,P2,and three steady states. The range of possible K values is relatively narrow, as exemplified for the PO model for 7 = 276 s (P2orbit) in Figure 8, which represents a small portion of the total K range used (0-100 s-1). In general, starting in a chaotic region, low feedback strengths will tend to stabilize periodic orbits in narrow periodic windows. At higher feedback strengths the contribution of the newly created periodic states increases and fewer chaotic states are found as a function of delay time in both models. In experiment and calculation the range of K is limited, since negative flow rates are not possible. We would like to note that the stabilization of specific periodic orbits in the strange attractors of the models takes considerable numerical and experimental effort, since the precise values of K are unknown. In an experiment it is relatively difficult to acheive control of chaos due to the narrow range of the stabilized periodic solutions and the experimental fluctuations in the oscillation periods. The delayed feedback methods are approximate methods for the
Figure 5. (a) Original attractor of the free running chaotic time series of the Montanator model ( T = 0). (b) Attractor of the stabilized P2 orbit of the Montanator model for 7 = 346 s and K = 100 M-1 s-1. stabilization of unstable orbits. Measures such as the maximum Lyapounov exponent or the Haudorff dimension may also be used to characterize the stabilized motion, but they are not able to distinguish between a stabilized orbit and a newly created
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Figure 6. (a) Dispersion 0 2 and (b) [02] versus the delay time T for the PO model for K = 0.02s-1 and ugo = 1.25 X IO-' M s-l, where the total enzyme concentration is 2.7 X I@ M. All other parameters are given in ref 23. Maxima and minima are plotted. PI = period-1 oscillation; SS = steady state. For simplicity some calculated bistabilities are not shown.
orbit. There are three factors which support the stabilization of an unstable periodicorbit. The stabilized orbit must be contained in the strange attractor, the dispersion D2 must show a minumum, and the delay time 7 must correspond to the period of the stabilized orbit Ts. Compared with the discontinuous OGY method of controlling chaos, the continuous Pyragas method does not require any online analysis. Periodic solutions may also be obtained from chaos by external entrainment using a periodic external function.12Work along these lines is in progress for the BZ reaction and the PO reaction.
Figure 7. (a) Original attractor of the free running chaotic time series of the P O model ( T = 0). (b) Attractor of the stabilized P1 orbit of the P O model for T = 156 s and K = 0.02 s-l. (c) Attractor of the stabilized P2 orbit of the P O model for 7 = 276 s and K = 0.02 8. 5e-13
Role of Chaos and Spatial Feedback Chaos may occur in dynamic systems containing sufficiently nonlinear interactions. A possible role of chaos in nature may be to act as an efficient absorber of local fluctuations due to the robustness of a chaotic attractor. This latter property has been named the spectral flexibility of chaos. A further role of a strange attractor may be to act as a fractal storage of information in the form of a large number of unstable periodic orbits. The feedback calculations and experiments suggest that this information may be retrieved as periodic signals by time delayed feedback (eq 1) or by other external periodic perturbations yi(t) acting continuously on the strange attractor where y(t-r) is substituted by yi(r) in eq 1.12 Chaotic attractors and delayed feedback mechanisms are likely to be of importance in the retrieval of periodic information contained in neural nets. The higher the dimensionality of a chaotic attractor, the higher is its information content. Chaotic attractors of high dimensionality have been observed in electroencephalogram measurements on the human brai11.2~ In the layered structure of neural networks or in cardiac tissue spatial patterns are known to beof importance. Wesuggest
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that time delayed spatial feedback may produce new spatial patterns, as already demonstrated in calculations using a onedimensional ionic Brusselator m0de1.2~ Finally, the mutual coupling of two or more chemical oscillators represents a form of distributed delayed feedback which has also been shown to produce a variety of chaotic and periodic states in the BZ reaction, depending on the type and strength of coupling.18926
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Acknowledgment. We thank the Volkswagen Stiftung, the Deutsche Forschungsgemeinschaft, and the Fonds der Chemischen Industrie for partial support of this work. We thank M. Hauser and A. Miinster for discussions. References and Notes (1) Ott, E.; Grebogi, C.; Yorke, J. A. Phys. Rev. Lett. 1990, 64, 1196. (2) Romeiras, F. J.; Grebogi, C.; Ott, E.; Dayawansa, W. P. Physica 1992. 58D. 165. (3) Roy, R.; Murphy, T. W.; Maier, T. D.; Gills, Z.; Hunt, E. R. Phys. Rev. Lett. 1992, 68, 1259. (4) Gills, Z.; Iwata, C.; Roy, R.; Schwartz, I. B.; Triandaf, I. Phys. Rev. Lett.' i992, 69, 3 169. ( 5 ) Hunt, E. R. Phys. Rev. Lett. 1991, 67, 1953. (6) Singer. J.; Wang, Y. Z.; Bau, H. H. Phys. Rev. Lett. 1991,66,1123. (7) Ditto, W. L.; Rauseo, S.N.; Spano, M. L. Phys. Rev. Lett. 1990,65, 3211. (8) Garfinkel, A.; Spano, M. L.; Ditto, W. L.; Weiss, J. N. Science 1992, 257, 1230. (9) Hiibinger, B.; DBrner, R.; Martienssen, W. Z . Phys. 1993,590, 103. (10) Peng, B.; Petrov, V.; Showalter, K. J . Phys. Chem. 1991,95,4957. (1 1) Petrov, V.; Gaspar, V.;Masere, J.; Showalter, K. Nature 1993,361, 240.
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Schneider et al. (12) Pyragas, K. Phys. Lett. 1992, A170, 421. (13) RWler, 0. E. Phys. Lett. 1976, A57, 397. (14) Weiner, J.; Schneider, F. W.;Bar Eli, K. J . Phys. Chem. 1989, 93, 2704. (15) Weiner, J.; Holz, R.; Schneider, F. W.; Bar Eli, K. J. Phys. Chem. 1992. 96, 8915. (16) Holz, R.; Schneider, F. W., in print. (17) Roesky, P. W.; Doumbouya, S.I.; Schneider, F. W. J . Phys. Chem. 1993, 97, 398. (18) Zeyer, K. P.; Holz, R.; Schneider, F. W. Eer. Bunsen-Ges. Phys. Chem. 1993, 97, 11 12. (19) Chevalier, T.; Freund, A,; Ross, J. J . Chem. Phys. 1991, 95, 308. (20) Schneider, F. W.; Miinster, A. F. J . Phys. Chem. 1991, 95, 2130. (21) GyBrgyi, L.; Field, R. J. J. Phys. Chem. 1991, 95, 6594. (22) GyBrgyi, L.; Field, R. J. Nurure 1992, 355, 808. (23) Aguda, B. D.; Larter, R. J . Am. Chem. SOC.1991, 113, 7913. (24) Babloyantz, A.; Dcstexhe, A. In From Chemical t o Biological
Orgunisution, Muller, S., Markus, M., Nicolis, G., Eds.; Springer: Berlin, 1987; p 307. (25) Miinster, A. F. Private communication. (26) Doumbouya, I. S.;Miinster, A. F.; Doona, C. J.; Schneider, F. W. J . Phys. Chem. 1993, 97, 1025.
