J. Phys. Chem. 1993,97, 398-402
398
Chaos Induced by Delayed Feedback Peter W. Roesky, Sory I. Doumbouya, and Friedemann W. Schneider' Institut fur Physikalische Chemie, Universitiit Wiirzburg, Marcusstrasse 9- 1 1 , 0-8700 Wurzburg, F.R.G. Received: August 5, 1992; In Final Form: October 13, 1992
The output redox potential of the chemical oscillations of the Belousov-Zhabotinskii (BZ) reaction was wed as the input for the flow rate with a built-in time delay between output and input. A variety of periodic states were observed when the time delay and the coupling strength were increased. Chaotic states are produced by delayed feedback which are normally not observed in the free-running oscillator at the same residence time, A BZ model containing at least two cycles is supported by the delay experiments: the BZ-limit cycle and the inflow cycle. All states are analyzed and characterized in terms of their power spectra, attractors (reconstructed by singular value decomposition), PoincarC sections, return maps, Lyapunov exponents, and the generalized Renyi dimensions.
Introduction
Delayed feedback processes are important in many nonlinear dynamical systems. For example, the human ventilatory response, and probably that of other mammals, depend on the carbon dioxide partial pressure sensed by the nervous system. The output is sent to chemoreceptors in the brainstem after some delay, which may be of the order of a few milliseconds. The brain on its own then causes the lungs to ventilate faster or more slowly.1 The application of feedback multilayer perceptrons (FMLP) in neural nets offers the possibility to detect words in a sequence of letters.* Several functional delay differential equations with delayed arguments have been used to model phenomena such as the regulation of enzyme synthesis (Goodwin model),3the regulation of respiration,' T3/T7 phase infection: bursting in an autocatalytic reacti~n,~ and the response of photoilluminated thermochemical reactions.6 Calculating the influence of linear feedback on the reduced Oregonator, Epstein et al.' found with an appropriate choice of the delay time that their reduced model behaves very much like the full system. In some of these models chaos has been observed.Ie6 In general, the delay can be treated in the following form dx(t)/dt = f l ~ ( t ) , ~ ( t - D ) ] (1) where D is the delay time. The first experimental chemical oscillator that was used with delayed feedback on itself is the minimal bromate oscillator (MBO).8,9 Weiner, Schneider, and Bar-Eli8 observed that the period of the controlled oscillations increases and decreases as a function of delay time in a "sawtooth" fashion with some overlapping regions of birhythmicity. Chevalier, Freund, and Ross used nonlinear delayed feedback to obtain chaos9 in the MBO. Earlier, Zhabotinskii et al.IOemployed feedback without delay in the same system where the bromide ion flow changed in proportion to the instantaneous concentration of ceric ions, while the flow of bromate and ceric ions was kept constant. In this paper we present experimental evidence that simple linear delayed feedback can generate complex dynamic behavior, including period doubling, P3 sequences, and deterministic chaos in the BZ reaction in a continuous flow stirred tank reactor (CSTR) where the oscillating redox potential was measured and stored digitally in a computer. Thus, the flow rate was modulated by the redox potential of the reaction itself with or without time delay. In the present experiments we chose a residence time ( 7 ) of 22 min (ko= 7.6 X 10-4 s-l) in the reactor, where theuncoupled Author to whom correspondence should be addressed.
Figure 1. Experimental setup. The CSTR is fed by a piston pump. The ion potential measured by a platinumelectrode at time t is used to control the flow rate at time t
B = 0.8 ; B = 0.5
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Figure 2, Experimental bifurcation diagram for different coupling strengths (8)as a function of delay time (D).
system shows exclusively periodic PI oscillations.*I On the other hand, the free-running BZ reaction exhibits chaos for low flow rates ( T = 67 min)11-14only, and high flow rate chaos ( T = 5-7 min) was shown to be absentl1JS The observed changes between different states as caused by delays have been treated theoreticallyI6in a generic mathematical model. The experimental time series are analyzed and characterized by their power spectra, reconstructed attractors, PoincarC sections, return maps, Lyapunov exponents, and the generalized Renyi dimensions.
0022-3654/58/2097-0398$04.00/0 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 2, 1993 399
Chaos Induced by Delayed Feedback Delay: 40 s
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Roesky et al.
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Experimental Section The experimental setup is shown in Figure 1. The CSTR volume is 10.0mL, and the reactor is stirred by a propeller rotating at 600 rpm. The CSTR and the input tubing are thermostated at 28 i 0.05 OC. Malonic acid was purified by recrystallization from acetone1' to remove impurities. A self-designed high-precision piston pump driven by a stepping motor is used to control the flow of reactants into the CSTR. Three gas-tight Hamilton syringes (50 mL each) serve as the reagent reservoirs. The first syringe contains 0.75 M malonic acid and 0.025 M cerous sulfate, the second 0.3 M potassium bromate, and the third 0.6 M sulfuric acid. Each step of the
motor advances the pistons by 100 nm into the syringes. A practically constant flow of solutions into the CSTR is ensured with a stepping rate of 35 Hz. The electrochemical potential of the reaction is monitored with an Ingold platinum electrode (PT4800). The signal from the electrode is digitized by a 12-bit A/Dconverter interfaced with an IBM W/XT computer. The data are stored in a computer file, and points are read at corresponding delay times in order to calculate the new flow rate which is proportional to the redox potential. The flow rate is controled according toa method d by Weiner, Schneider, and Bar-Eli:* in each experiment the oscillationsare started from the same initial conditions. In order
The Journal of Physical Chemistry, Vol. 97, No. 2, 1993 401
Chaos Induced by Delayed Feedback
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to obtain a stable P1 limit cycle, the oscillations are run for the duration of 3 4 residence times; then feedback is started at a random point on the PI limit cycle. Short transients are observed (1-2 residence times) which are not used for data analysis. The flow rates (k)are calculated according to the following equation
k = k~ + k & ( [ ~ t ( t - ~ -, I b t a v l ) / [ ~ t a v l
(2) in which is a constant representing the average flow rate (ko = 7.6 X 10-4 s-l; corresponding to 'T = 22 min) and [pot,v] is the average redox potential of the free running oscillator. The term [poqr-~)] is thedelayed potentialused for a delay time of Dseconds. The coupling strength is denoted by 8. For B = 1 the amplitude
of the flow rate oscillates in the range between 3.5 X 10-4 and 1.17 X lt3s-l. The period of the free running oscillator is 69 5.
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The free-running oscillator has been coupled to the flow rate at five different values of the coupling strength (0 = 0.5,0.8,1.0, 1.2,and 1.5) and at delay times D varying from 0 to 240 s (in steps of 3-5 8). The results are shown in Figure 2. An increasing number of states are observed as B increases from 1.0 to 1.5. Variations of B without delay ( D = 0) have no effect on the period,but increasing fluctuations in the amplitude are seen when B is increased. For B < 1, only PI oscillationsare observed. There
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402 The Journal of Physical Chemistry, Vol. 97, No. 2, 1993
is no change of the periods in a 'sawtooth" fashion as in the MBO system.8 For (3 = 1.0, a number of states are observed for delay times between 50 and 60 s including the Feigenbaum route to chaos18via period doubling (PI:Pz:P~).A PSstate is not observed. We measured two chaotic windows, CI and CZin this range. For (3 = 1.2, period doubling is observed at D = 48 s followed by deterministic chaos C1 at D = 51 s and a P2 state at D = 53-59 s. Following the PI state another chaotic region (C2) occurs, and another periodic state (P3) is observed. The theorem of Li and Yorke19 (period three implies chaos) leads us to postulate the existenceof a subsequentchaoticstate (C3), which was unobserved in our experiments, since its range is too narrow. The largest useful valueof thecoupling strength was@= 1S, becausecoupling strengths exceeding this value would generate negative values of the flow rate k. At (3 = 1S, the aperiodic windows appear in the same region as for (3 = 1.2, with the former regions tending to be broader for (3 = 1.5. In addition, a P3 pattern is observed at D = 160 s. Again following Li and Yorke, we postulate the existence of chaotic states bordering the left (C4) and the right (C,) sides of the P3 region. Finally, another chaotic state (C6) is observed at D = 180 s. The tools of nonlinear dynamics have been used to confirm the characteristics of all states designated in Figure 2. The time series and power spectra for (3 = 1.2 are shown only for the states P2 (D = 48 s), P4 ((3 = l.O), P3, CI, and C2 (Figure 3a-c to Figure 5a-e). The chaotic states CI and C2 were characterized further by their reconstructed attractors, Poincarb sections, return maps, Lyapunov exponents, and the spectra of the generalized Renyi dimensions (Dq, as calculated with the nearest-neighbor method1ssz0.21)for k = 100 in linear scaling (Figures 4f,g and 5f,g).
Discussion Deterministic chaos was generated when a linear feedback was imposed on the BZ-reaction. A period-doubling bifurcation describing the Feigenbaum route to chaos was observed (Figure 3, a and b). The return maps for Pz and P4 show two and four point clouds, respectively, scattered by experimental noise. A return map reconstructed for the P3 state shows three discrete point clouds. The chaotic time series show broad-band power spectra (Figure 3d,e) and are characterized by a "strange" attractor (Figures 4a and sa). Onecan clearly observe continuous curves in the Poincarb sections and extrema in the return maps. For C, we obtained bell-shaped curves in the return maps for all coupling strengths ((3 2 1.O) (Figure 4c), typical of deterministic chaos.22 For C2 ((3 = 1.2) a bell-shaped curve in the return map was not observed because the route to chaos seems to be a tangent bifurcation instead of period doubling. This observation,together with a screw-type form of the attractor (Figure 5a), points to the existence of homoclinic chaos that has already been found and analyzed theoretically and experimentally in the BZ reaction by Argoul, Arneodo, and R i ~ h e t t i . ~ ~ For the chaotic time series the Lyapunov exponents (Figures 4e and 5e) are positive, zero, and negative where A, = 0.014 bits/s for CI and ,A, = 0.09 bit+ for C2 which supports chaos. From the spectra of the generalized Renyi dimensions, one can obtain the Haussdorff dimension DO(=2.33), the information dimension Dl(12.27) and the correlation dimension D2 (=2.05)
for CI (Figure 4d). For CZthe dimensions are higher: DO= 3.40, D1 = 2.98, and Dz = 2.55 (Figure 5d), respectively. All these values exceed 2.0, as expected for systems, which exhibit deterministic chaos according to the Kaplan Yorke conjecture.u A general trend was observed: the dimension of chaos also increases when the delay time is increased at constant coupling strength. Such behavior has been predicted by Farmer25and Le Berre et a1.26 Chaotic behavior is not observed when the flow rate is altered by feedback without delay. We have shown experimentally that the number of aperiodic states increaseswith increasing coupling strength in the presence of delay. Thus, time delay combined with a minimal coupling strength seems to be responsible for the occurrence of chaos under the present conditions. In a recent model of the BZ reaction (M~ntanator~~), the occurrence of chemical chaos was explained by interactions between two frequency sources. Following Gy6rgyi, Rempe, and Field:7 one can conclude that chemical chaos results from the coupling of two independent cycles in the homogeneous system. One of these cycles is the BZ cycle, while the other is represented by the delayed flow cycle.
Achwkdgment. We gratefully acknowledge helpful discussions with C. Doona, F. Buchholtz, D. Lebender, and Th. M. b e l . We also thank the Volkswagenstiftung and the Fond der Chemischen Industrie for partial financial support. Refere" and Notes (1) Mackey, M.C.; Glass, L. Science 1977,287. (2) Bauer, H.-U.; Geisel, T. Phys. R N . 1990, 42, 2401. (3) Goodwin, B. C. Adu. Enzyme Regul. 1966, 3,425. (4) Buchholtz, F.; Schneider, F. W. Eiophys. Chem. 1987.26, 171. (5) Babola, P.; Rajani, P.; Kumar, V. R.; Kulkami, B. D. J. Phys. Chem. 1991, 95, 2939. (6) Schell, M.;Ross, J. J. Phys. Chem. 1986,85, 6489. (7) Epstein, I. R.; Luo, Y. 1. Chem. Phys. 1991, 95, 244. (8) Weiner, J.; Schneider, F. W.; Bar-Eli, K. J . Phys. Chem. 1989,93, 2705. (9) Chevalier, T.; Freund, A.; Ross, J. J. Phys. Chem. 1991, 95, 308. (10) Zhabotinskii, A. M.; Zaikin, A. M.;Rovinskii, A. B. React. m e r . Catai. Len. 1982, 20, 29. (11) Schneider, F. W.; MiInster, A. M. J. Chem. Phys. 1991.95, 2130. (12) Tumer, J. S.; Roux, J. C.; McCormick, W. D.; Swinney, H. L. Phys. Lett. 1981, 85A, 9. (13) Simoyi, R. H.; Wolf, A,; Swinney, H. L. Phys. R N . Len. 1982,19, 245. (14) Swi~ey,H.L.,Roux, J.C.InNonEqw'libriumDyMmlcsinc~micu/ Sysrenw; Vidal, C., Pacault, A,, Eds.; Springer: Berlin, 1984; pp 124-140. (15) Blittersdorf, R.; Mhster, A. F.; Schneider, F. W. 1.Phys. Chem. 1992, 96, 5893. (16) Cookc, K. N.; Grossman, Z. J . Math. A M / . Appl. 1982,127,592. (17) Gy(kygi,L.;Field,R.J.;Nosaiaius,Z.;McCormick,W.D.;Swinaey, H. L. J . Chem. Phys. 1992.96, 1228. (18) Feigenbaum, M. J. J . Slut. Phys. 1978, 19, 25. (19) Li, T.; Yorke, J. A. Am. Murh. Monthly 1975, 82, 985. (20) Renyi, A. Probubiliry Theory; North-Holland: Amsterdam, 1970. (21) Badii, R.; Politi, A. Phys. Rev. Lerr. 1W,52, 1661. (22) Argoul, F.; Ameodo, A,; Richetti, P.; Roux, J. C. J. Phys. Chem. 1987, 86, 3339. (23) Argoul, F.; Arneodo, A.; Richetti, P. Phys. Letr. A 1987, 120, 269. (24) Kaplan,J. L.; Yorke, J. A. In Lccrure Notesin Mathemarlcs;hitgen, H. O., Walther, H. O., Eds.;Springer: Berlin, 1980; Vol. 730, p. 204. (25) Farmer, J. D. Physica D 1982, 4, 366. (26) Le Bene, M.;Resayre, E.; Tallct. A.; Gibbs, H. M.;Kaplan, D. L.; Rose, M. H.Phys. Rev. Lett. 1987, 35, 4020. (27) Gybrgyi, L.; Rempe, S. L.; Field, R. J. J. Phys. Chem. 1991, 95, 3 159.
