The Journal of
PhysicaI Chemistry VOLUME 98, NUMBER 3, JANUARY 20,1994
0 Copyright 1994 by the American Chemical Society
LETTERS Transient Period Doublings, Torus Oscillations, and Chaos in a Closed Chemical System Jichang Wang, P. C. S~rensen,’and F. Hynne’ Department of Chemistry, H. C. 0rsted Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark Received: October 6, 1993; In Final Form: December 6, 1993
We report the first experimental observation of successive transient period doublings and torus oscillations adjacent to transient chaos in a closed Belousov-Zhabotinsky (BZ) system. Our observations support the view that genuine chemical chaos exists in the BZ system.
Introduction Oneof thebest understood routes to chaosof dynamical systems is through a sequence of period doubling bifurcations.lS2 Such sequence has been observed in a wide variety of dynamical system^,^ including chemical reactions in open systems.&’ Another route to chaos is through a bifurcation from a torus.* In closed chemical systems, only transients exist, and the concepts of period doubling and chaos do not apply, strictly speaking. Nevertheless, these concepts can be used to understand and approximately describe the time evolution of a complex system.* Sometimes a dynamical variable may be viewed as a parameter that slowly changes with time, giving rise to a “bifurcation” at a definite instant.8 Almost all chemical oscillations observed in batch reactors have been simple. Irregularities have been seen:-” but deterministic chaos has not been demonstrated in a closed system.12J3 Chemical oscillations in open systems are best understood in terms of geometrical objects in the concentration space such as fixed points, limit cycles, invariant tori, and chaotic attractors, as well as stable and unstable manifolds associated with them.2 Such “dynamical structures” are permanent (i.e. independent of time) but may depend on controlled parameters such as rates of inflow of chemical species in a continuously stirred tank reactor (CSTR) . In closed systems, only one fixed point, the equilibrium point, can be permanent. Motion is alway transient. But often it is e
Abstract published in Advance ACS Absrracrs, January 1 , 1994.
possible to view transients as composed of motions of widely different characteristic times. Such transients may then be understood as dynamical structures slowly changing in time. For example, decaying oscillations may be described as motion on a “limit cycle” that slowly shrinks. Below we report motion on a shrinking torus and transient chaos, which can be understood in a similar way.14 In closed systems, transitions between different types of behavior similar to bifurcations in open systems may take place in the course of their time development. Such qualitative change may be particularly easy to recognize. We report the first observation of successive transient period doublings in a closed chemical system. We find that for a small change of initial conditions, the oscillations gradually transform into what appears to be transient chaos.
Experimental Results In this paper, we study the cerium catalyzed BelousovZhabotinsky (BZ) reaction, i.e. the oxidation and bromination of malonic acid by bromate, catalyzed by Ce ions. The reaction is carried out in a glass cell thermostated at 25 f 0.1 OC. The reaction mixture has a volume of 45 cm3 and a free surface. It is kept free of oxygen by a slow flow of nitrogen above the surface; the flow rate of Nz was so low that it hardly could remove significant amounts of any volatile species such as Br2. Thus the system can be considered closed. The mixture is stirred with a three-bladed glass impeller at 600 RPM. Figure 1 shows the time evolutions of closed BZ systems with
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726 The Journal of Physical Chemistry, Vol. 98, No. 3, 1994
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2 3 L timeIhours Figure 1. Time evolutionsof closed BZ systems showing complex transient oscillations and bifurcations: (a) period doubling; (b) two period doublings leading to four peak oscillations; (c) complex nonchaotic oscillations. The three systems have the same initial concentrations of malonic acid (0.44 M),Ce3+(0.001 33 M),and HzS04 (1 M),but different bromate concentrations: (a) 0.093, (b) 0.1 18, and (c) 0.133 M. The stirring rate is 600 RPM. The transmission of light is measured at a wavelength of 344 nm. (The absorption is mainly due to Ce4+.)The transmission at the start of the oscillations ranges from about 30 to 80%. common initial concentrations of all reactants except for slightly different bromate concentrations. In each of the experiments there is a short (5-min) induction period after which simple oscillations start. These continue with almost constant amplitude for about 2 h, when a “period doubling” occurs in each case. In the experiment of Figure la, the twopeak oscillations last for a little less that 1 h and are followed by a ”reverse period doubling- back to simple oscillations. In the experiment of Figure lb, the complex oscillations last much longer, and the two-peak oscillations are interrupted by another period doubling bifurcation leading to four-peak oscillation. These return to simple oscillations through two reverse period doublings. In Figure ICthe oscillationsdevelopinto three-peakoscillations which change through complex transitionsincludingreverse period doublings into simple oscillations. The figure illustrates a systematic increase of complexity of the transient oscillations as the initial concentration of bromate is increased. Indeed, for initial bromate concentrationssomewhat lower than in Figure la, e.g. 0.066 M, we find simple damped oscillations which are similar to those of Figure la outside the region of period doubled oscillations. And, for initial bromate concentration larger than in Figure IC,the complex oscillations become even more irregular, and by small adjustments of the initial concentrations of some of the other species, we get highly irregular transient chaos. We exemplify transient chaos in Figure 2 by an experiment with initial conditions quite different from those of Figure 1 in order to exhibit also transient torus oscillations. Here the complex oscillations start after about 1.5 h and continue for about 7 h, after which they change to regular torus oscillations of gradually decreasing modulation (and amplitude). Note that theoscillations last more than 11 h.
Discussion Characterization of irregular oscillations in chemical systems is subject to a number of difficulties. In a CSTR, one difficulty
is to make sure that irregularities are not due to incompletemixing of the feed streams with the contents of the reactor. In closed systems, that problem is absent. Spatial inhomogeneities may nevertheless develop as a result of small local differences of reactionsrates and nonideal hydrodynamicmixing,’* and irregular oscillations may also be caused by fluctuations in an excitable system with a very small threshold? In experiments, these situations may manifest themselves in a dependence of the oscillations on the stirring rate.I0 Such dependence cannot be demonstrated for states on the chaotic attractor itself because genuine chaotic oscillations will change from one experiment to another, even with ideal mixing. A way of revealing insufficient stirring is to find recognizable patterns that are known to be characteristic of routes to chaos and demonstrate an independence of the stirring rate and cell geometry for these. In the present experiments,we checked that each of the regular (nonchaotic) complex patterns could be reproduced when the experiment was repeated with all conditions unchanged. In addition, the oscillations in Figure la,b were essentially exactly reproduced with double the stirring rate, 1200 RPM, as well as in a different reactor with different geometry and stirrer. Figure 3 shows the complex part of a time series from an experiment run at the same conditions as that of Figure l a except for the stirring rate of 1200 RPM. The cell is the same as for Figure la, so the curves are directly comparable. The “period doubling” starts at almost exactly the same time after the start of the experiment as that for the lower stirring rate. With increasing complexity, regular patterns can be increasinglydifficultto reproduce in detail. The highly irregular oscillationsconsideredto contain transient chaos can be reproduced in the sense that the regular parts beginning and ending the time series are essentially the same, whereas the irregular middle part differs from one run to another, no matter how carefully one tries to keep all conditions fixed. Such behavior is of course characteristic of chaotic oscillation^.^^ In summary, we report the evolution of fairly complex, reproducible patterns (Figure la,b), known to be possible precursors of chaos, close to conditions where highly irregular
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time I hours Figure 2. Transient chaotic oscillations ending with transient regular torus oscillations, measured as in Figure 1. Initial concentrations are [BrOo-] = 0.071 M, [MA] = 0.267 M, [Ce3+] = 0.001 33 M, and [H2S04] = 0.5 M (MA is malonic acid). The stirring rate is 600 RPM.
We conclude from these experiments in closed systems that complex and chaotic oscillations exist as transients phenomena, created by the intrinsic chemical dynamics and by implication that chaos observed in open systems may well result from the chemistry and not from incomplete mixing of reactants.
References and Notes Figure 3. Time evolution of a closed BZ system showing a section of period doubled oscillations. The conditions are exactly the same as for Figure l a except for the stirring rate which is 1200 RPM. The time series from the experiment appears the same as for the one recorded in Figure l a in all essential details (including the parts not shown here), and the forward and reverse period doublings occur at almost exactly the same times after the start of the experiment as in Figure la.
oscillations appear. This fact suggests that the irregular oscillations may indeed be viewed as transient chaos. Substantiation of this interpretation through calculation of classical indicators of chaos such as Liapunov exponents is surrounded by serious methodological problems. We have also observed, at quite different initial conditions, irregular oscillations that end in transient torus oscillations. The very long duration of the irregular oscillationsof Figure 2 suggests that the irregularity here is not due to transitions between a series of regular oscillations, thus supporting an assumption that they too represent transient chaos.
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