Experimental observations of periodic and chaotic regimes in a forced chemical oscillator

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6 February 1984

EXPERIMENTAL OBSERVATIONS OF PERIODIC AND CHAOTIC REGIMES IN A FORCED CHEMICAL OSCILLATOR M. DOLNIK, I. SCHREIBER and M. MAREK Department of Chemical Engineering, Prague Institute of Chemical Technology, 166 28 Prague 6, Czechoslovakia

Received 31 August 1983

Experiments showing periodic and chaotic behavior in a Belousov-Zhabotinski (B-Z) reaction mixture in a stirred flow reactor with periodic addition of bromide ions are presented. An alternating sequence of periodic and aperiodic regimes as a function of the period of Br- pulses is observed in experiments. This behavior is well described by a simple model based on the results of single-pulse experiments.

Our purpose is to report measurements of the dynamical behavior of a periodically stimulated B - Z reaction. Similarly as in biological oscillators [ 1 - 3 ] , the phase o f the B - Z chemical oscillator can be reset by a single brief pulse [4]. Periodically repeated series of pulses can considerably affect the natural rhythm of oscillators [3,4]. One interesting application of this approach is the problem of cardiac arrhythmias [ 5 - 7 ] . The use o f the B - Z oscillator with periodic addition of B r - ions as a model system is substantiated by the fact that biological and/or physiological systems have a complex (often unknown) chemical nature. A continuous flow stirred reactor was used in the experiments. The reactor has a cooling jacket, stirrer, thermometer and is provided with measuring electrodes and a capillary serving for the introduction of the inlet concentration perturbations. Two separate streams of inlet solutions (solutions of Ce 4÷ ions + sulphuric acid and solutions o f BROW-ions + malonic acid) were delivered by a peristaltic pump to the bottom of the reactor. The solution of Br- ions used fo~ stimulation was delivered through a capillary located close to the liquid level in the reactor and the flow was controlled by a solenoid valve. The temperature in the reactor and the frequency of the stirrer were kept constant. The inlet concentrations o f the reaction components were: Ce 4+ ions [Ce(SO4)2] : 0.001 M, KBrO3: 0.05 M, malonic acid: 0.05 M, sulphuric acid: 1.5 316

M; the stirring intensity was 500 rpm, the residence time in the reactor was 18 min (reactor volume was 100 ml) and the volume of the pulse was 1 ml. We used a Pt electrode-calomel electrode couple (K2SO 4 salt bridge was used with a calomel electrode) to follow the course of oscillations. The potential of the Pt electrode is proportional to ln([Ce 4+ ] / [Ce 3÷ ] ). First a steady oscillatory regime with period T B has been set up in the reactor (the regime was considered steady when fluctuations in T B were less than one percent). The magnitude of T B was controlled by variation of temperature in the reactor. Then concentration forcing was started by the addition of 1 ml of a solution containing B r - ions of deflmed concentration, the residence time being kept constant. Concentration pulses were either periodically repeated with period T F or a single pulse applied at a specified phase of the oscillation was used and the dynamical response of the system to both kinds of perturbations has been followed. (1) Periodic repeating of pulse additions with period

rv. The Pt electrode records oscillations which are of relaxational character, i.e. they consist o f very fast jumps o f the potential followed by a slow potential decrease. This forms a basis for the measurement of the phase of the oscillations. The sequence of response "periods" {5 Tn } is defined as a sequence of time inter0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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vals separating two successive jumps of the potential. The phase ~ at the instant of the stimulation (considered modulo TB) is a time interval between a pulse and the preceding jump of the potential. It is convenient to measure all quantities in units of the natural period T B (i.e. r F = TF/T B , Ar = 8TIT B , ~ = ~/TB). Several examples of experimental results as a function of r F are shown in table 1 and in figs. l a - h . When during q perturbations the number of responses of the system is p, i.e. P

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55 45

50

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qrr= ~+IT., then a periodic regime with phaseqocking ratio p = p/q is established in the system. There is a sequence of increasing ratios p/q corresponding to periodic motions observed in experiments with increasing values of the external parameter ZF, see table 1. One example of a sequence of response "periods" for ~ resonance together with the corresponding power spectrum is shown in figs. la, b. However, for many values of~"F there are no distinct periodic regimes between resonances. The response sequences {STn} are aperiodic, cf. figs. lc, e, g, and power spectra show broad band noise, cf. figs. ld, f, h. There are at least two kinds of these motions: developed chaos (figs. l c - f ) and intermittency (figs. lg, h), as will be shown later.

Table 1 Examples of dynamical behavior of B-Z reaction periodically stimulated by Br- ions for severalvalues of rF. Regimes are either phase-locked (i.e. periodic) with the ratio p/q or aperiodic (eventually with some p/q cycles interspersed). rF

character of behavior

2.51 2.50 2.39 2.31 2.24 2.17

apenod,c (+ ~) 5

1.77

y

1.66 1.46 1.18

~periodic (+ 3)

.

.

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Fig. I. Sequences of response "periods (a, e, e, g) and corresponding power spectra (b, d, f, h), experimental conditions are described in the text, the amplitude of Br- pulses - [Br- ] = 5 × 10 -s M, (a) ~F = 2.5, periodic pattern with ~ phaselocking, (c) 7F = 2.24, chaotic oscillations,(e) rF = I..66, chaotic oscillations,(g) rF = 1.18, intermittent chaos.

(2) Single pulse addition. In contrast to the previous case a single pulse addition does not change the natural period TB but shifts the phase. The "periods" before the pulse are 8T = TB and after the pulse 8 T converges quickly to TB . It was found that within experimental errors only the first response "period" (ST) I (i.e. the period in which the perturbation by Br- ions occurred) was affected. The phase transition curve [1,2] (PTC), 0 = @+ A~(@), where Ar = [TB - (ST)I ] [TB can be easily constructed on the basis of experimental results. The shape of this curve depends on the perturbation amplitude, i.e. on the concentration of Br- ions in the pulse Three examples of the PTSs are shown in figs. 2a, 317

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As the response "periods" following after the stimulated period (6 T) I are almost unaffected by the pulse a PTC can be used to model dynamical behavior of the phase in the periodically stimulated experiments. The phase is governed by a simple difference equation

1

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-

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known as an equation for the Poincar6 map of the phase or phase advance map [8,9]. Eq. (1) can be successfully used to predict the experimental results given in table 1 using the experimentally measured PTC from 2b (we have used a continuous and piecewise differentiable approximation of the PTC by two polynomials of degree 3). The dynamical behavior of the model (1) is very complicated. According to theoretical results [ 10] we can expect quasiperiodic or phase-locked (i.e. periodic) solutions for monotonous PTC and chaotic or phase4ocked solutions for PTC with extrema. A suitable criterion for distinction among periodic, quasiperiodic and chaotic solutions is the Lyapunov number N ), = lira N -1 ~ lnld~k(~bk)/d~l. N~** k

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Fig. 2. Experimentally measured PTC in three qualitatively different situations described in text, (a) [Br-] = 1.43 X 10-5 M,(b) [Br-] = 5 X 10-s M,(c) [Br-] = 1.43 X 10-a. b, c. Initially the nondecreasing curve (fig. 2a) becomes nonmonotonous with increasing perturb ation amplitude (fig. 2b, c). The curves have average slope either 1 (fig. 2b) or 0 (fig. 2c), this corresponds to type 1 or type 0 resetting [2]. 318

For the PTC from fig. 2b, X is either positive or negative, which corresponds to chaotic or phase-locked regimes. A positive Lyapunov number for a different kind of periodic forcing of the B - Z reaction was already reported [11 ]. The dependence of X on r F is not simple, however, this is not of great importance in our case because inevitable noise always present in experiments of this kind destroys the observability of any existing fine structure. Another possibility of comparison of the model (1) with experiments is to evaluate a sequence (¢tc} directly from experimental data and plot Ck+ 1 against ¢k" Figs. 3 a - d (corresponding to figs. la, c, e, g) show consistency between data and the model function ft. In fig. 3a there are two clusters of points corresponding to s phase-locking, in figs. 3b, c the experimental points are more or less uniformly distributed along the curve qJ which corresponds to chaotic oscillations. Fig. 3d is an example of intermittent chaos. The experimental conditions are very close to -I phase-locking which forms a basis of laminar phase. In fig. l d a laminar phase between two turbulent bursts is shown, the approach to laminar behavior is illustrated again in fig.

Volume 100A, number 6

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6 February 1984

I @k+1

b

@k*1

@k

•

@k 6

@k+1

1

d

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Fig. 3. Comparison of experimental data from figs. la, c, e, g with the function ~ constructed from PTC in fig. 2b, (a) rF = 2.5, s2 phase-locking, (b) eF = 2.24, chaos, (c) 7F = 1.66, chaos, (d) TF = 1.18, intermittency. 3d. Numerical simulation reveals that an alternation of turbulent and laminar phases is almost regular (weak chaoticity) or quite regular (overall periodic motion with phase-locking ratio p slightly above one). However, added noise makes the alternation more irregular which corresponds better to experimental results. Our experiments show that the behavior of a periodically stimulated B - Z reaction is essentially governed by one-dimensional deterministic dynamics. The construction of maps of higher dimension would require a consideration of the effects of a single perturbation on additional response "periods"; this is not substantiated by the present accuracy of experiments. The model can be improved b y considering the effects o f noise [4]. This case may correspond better to the experimental situation where fluctuations due to stirring and the use of the peristaltic pump are inevitable. More complete results will be presented elsewhere.

[ 1] T. Pavlidis, Biological oscillators. Their mathematical analysis (Academic Press, New York, 1973). [2] A.T. Winfree, The geometry of biological time (Springer, Berlin, 1980). [3] R.M. Guevara, L. Glass and A. Shrier, Science 214 (1981) 1350. [4] M. Dolnlk, D. Suehanov~, M. Marek and P. Liehnovsk~, Sci. Papers of the Prague Institute of Chemical Technology, K (1983); M. Doin&, Thesis, Prague Inst. of Chem. TechnoL (1981).

[5] [6] [7] [8] [9] [10]

M.R. Guevara and L. Glass, J. Math. Biol. 14 (1982) 1. LP. Keener, J. Math. Biol. 12 (1981) 215. A.T. Winfree, Scientific American 248 (1983) 118. M. Zaslavsky, Phys. Lett. 69A (1978) 145. L. Glass and R. Petez, Phys. Rev. Lett. 48 (1982) 1772. M.T. Herman, in: Geometry and topology, Lecture Notes in Mathematics, VoL 597 (Springer, Berlin, 1977) pp. 271-293; S. Newhouse, J. Palis and F. Takens, Stable families of dynamical systems, I: Dfffeomorphisms, PubL IHES, to be published. [11] H. Nagashima, J. Phys. Soc. Japan 51 (1982) 21. 319