J . Chern. SOC., Faraday Trans. I , 1988, 84(6), 1751-1771 Cooperative Effects in Heterogeneous Catalysis Part 2.-Analysis and Modelling of the Temperature Dependence of the Oscillating Catalytic Oxidation of CO on a Palladium A120,-supported Catalyst Peter J. Plath,* Karin Moller and Nils I. Jaeger Institut fur Angewandte und Physikalische Chemie, Universitat Bremen, Bibliothekstrasse, N W 2, 0-2800 Bremen 33, Federal Republic of Germany The temperature dependence of the dynamics of the heterogeneous oxidation of CO on palladium supported by an amorphous A1,0, carrier has been studied. The complex structure of the observed time series has been investigated by fast-Fourier-transform analysis. The resulting spectra are characterized by at most three frequencies, the ratio of which is nearly an integer, and by frequency locking.For lower temperatures sub-harmonics of these frequencies become important. The fractal character of the time series could be explained qualitatively by a one-dimensional automaton rep- resenting a concentration wave travelling on a fractal network of Pd particles in the catalyst bed. The complex phenomenology of the dynamics of CO oxidation on palladium embedded in a zeolite matrix has been presented in detail in ref. (1). The interpretation of the observed oscillations was based on the assumption of phase transitions between a catalytically active Pd-metal phase and an inactive palladium oxide phase. The idea of phase transitions involving the catalyst has been proposed by Wagner2 and was used in early models put forward to account for observed reaction rate oscillation^.^-^ Phase transitions have since been observed in situ on supported Pd catalyst' and on single-crystal surfaces of Pt.' A mathematical model of ideal storage' involving the phase transition of the bulk of catalytically active crystalli tes showed important dynamic properties observed in experimental systems without considering the necessary coupling of active palladium particles.This model represents a fully synchronized catalyst and might be valid only under special experimental conditions. Strong coupling between the catalytically active parts of the catalyst is needed to fulfil the conditions of this model, For instance, thermal coupling of neighbouring catalytic areas undergoing phase transitions has been used successfully to simulate the dynamics of the oxidation of methanol on a supported Pd ~atalyst.~-ll However, the phenomenology of catalytic CO oxidation is of much higher complexity than that of methanol ~ x i d a t i o n ' * ~ * ~ ~ , l3 because of the smaller coupling between the Pd particles within the catalyst.Again using the idea of chemical stores undergoing phase transitions, the temporal pattern of this reaction could be described qualitatively by a linear deterministic cellular automaton. 14* l5 This automaton models the temporal development of the number of catalytic centres, which are blocked for the conversion of CO at a given time. Even though this model creates a discrete temporal pattern, which is very similar to the experimental time series, a detailed analysis of these time series has not been made.It is our aim now to analyse the time series of catalytic CO oxidation by Fourier analysis in order to obtain a deeper insight into this reaction. 17511152 Oscillating Catalytic Oxidation of CO Theoretical Background The short temporal decrease of the conversion requires the synchronization of very large numbers of Pd crystallites. Synergistic effects concerning the chemical behaviour of Pd particles located far from each other on a microscopic scale should therefore be taken into account. Synchronized areas may be formed by agglomerates of zeolite crystals or even by single crystals,' or by regions of sufficiently closed packed amorphous material in the case of Pd/A1,0, catalyst.Consider the synchronized domains as new active units, where all Pd crystals undergo phase transitions between catalytic active Pd and inactive PdO at the same time. We now have a set of more-or-less randomly distributed reactive units among which coupling should occur, mainly via diffusion of CO and 0,. This leads to a system of loosely coupled reactive units, each of which can be regarded as a bistable oscillator with respect to the Pd/PdO phase transition. Such a system may form an excitable medium of discrete nature and the number of leading centres working as oscillators will depend on the amount and packing of the catalyst. The chemical excitation, which starts from such an unstable leading centre, will spread out with time and might dominate macroscopically large parts of the catalyst.It has to be emphasized that it is the chemical excitation which is spreading out and not the concentration of the reactants. The nature of this excitation is assumed to be the phase transition of the reactive units from the chemically active Pd to the inactive PdO state. The functions of the time dependence of the phase transition itself cannot be specified at the present time. Furthermore, we do not have any detailed knowledge about the spatial distribution of the reactive units within the catalyst. The reactive units may form a fractal or a regular lattice and the excitation may travel as a spherical wave along the fractal arrangement of the reactive units, or the excitation itself may form a fractal in the latter case. An approach to understand and describe the dynamics of this process is suggested by the self-similar pattern of the time series of the conversion (see fig.1). This temporal behaviour could be described by a model, having the intrinsic feature of self-similarity . To simplify our system, first we restrict ourselves to one dimension, and secondly we take a discrete arrangement of cells as a valuable model, since our process acts on a set of distinguishable reactive units. The state of each cell now represents the phase of the reactive units. Thirdly, we introduce a discrete time to model the discrete character of the events, i.e. the abrupt decrease of the conversion. These events are correlated to the phase transition of the reactive units, and we have to introduce a transition rule for the states of the cells.The state of a cell at time (n + 1) depends on its state at time n and also on the state of some neighbouring cells at time n. With this set of assumptions a one- dimensional cellular automaton can be established, the temporal sequence of states of which represents a discrete version of the Sierpinsky gasket, a typical example for self- similar s t r u ~ t u r e s . ' ~ ~ ~ ~ In the framework of this elementary model we now have to explain the periodic behaviour of our real system. Since we have a recurrence of self-similar intervals we can simply model this by a reinjection process of the bounded automaton, starting the same process again from the same initial cell of the automaton. In chemical reality this means that after a given period of time our system reaches its boundary, i.e.under the constraints of the reaction the maximum number of reactive units become inactive. After reactivation, anywhere in the excitable medium one unit becomes active and may develop itself into a leading centre. It is not necessary that the same cell always becomes the leading cell, but in the mathematical model we can identify any of the possible leading centres with the starting cell of the automaton. With this model in mind, the reconstruction of the time series in three-dimensional space and their Fourier spectra will be presented and discussed.P. J . Plath, K . Miiller and N . I. Jaeger 1753 Experimental The reactions were carried out in a differential flow reactor' (volume 16 cm3) at different temperatures To between 434 and 507 K with a CO concentration of 0.37 YO by volume in the feed.The gas mixture was prepared from synthetic air (impurities < 0.1 ppm) and a certified gas mixture containing CO and N, (Messer Griesheim). A molecular sieve was used for additional purification of the gas mixture. Measurements were carried out with 50 mg of the Pd/AI,O, catalyst with 0.5 YO Pd by weight (Heraeus). The average size of the Pd crystallites was less than 2 nm. The CO or CO, concentration in the outlet of the reactor was continuously recorded by i.r. spectroscopy (URAS, Hartmann and Braun). The analogue signal was converted into digital data, stored and evaluated off-line on a PDP-11 computer (Digital Equipment). Prior to starting a catalytic reaction the catalyst was heated in synthetic air up to 600 K (heating rate 5 K min-') and kept at this temperature for 12 h.The catalytic experiments were conducted thereafter at constant CO concentration (0.37 vol YO) and with stepwise variation of the temperature (AT= 2-5 K). The system was kept under constant conditions for at least 22 h after each temperature variation. For further experimental details see ref. (1). Results Fig. 1 shows characteristic patterns of the time series for various temperatures of the reactor. The time series represents sudden breaks in the rate of conversion. With decreasing temperature an increase in frequency as well as in temporal deactivation can be observed. Multistability can be observed at lower temperatures, and eventually the reaction is quenched.In this region between 433443 K [see fig. 1 (B)] small changes in temperature force the system into an inactive state for a finite time, where the duration of the suppression depends on the specific reactor temperature. The exponential increase of the ignition times connected with decreasing the temperature of the reactor, To, is plotted in fig. 2. Aside from different pre-exponential factors, an apparent activation energy between 3 15 and 322 kJ mol-' can be estimated. Taking into account the time of low activity during multistable behaviour one can calculate an apparent activation energy of ca. 479 kJ mol-' even for the autonomous changes of the systems in the range of its multistable situation. A similar activation energy has been observed by Turner et al." obtained from measurements of the induction period of the oscillations. The order of magnitude of these activation energies points to processes which are not compatible with desorption, adsorption or surface reaction. For instance the chemisorption of CO and 0, is not achieved in this temperature range,18 the desorption of CO possesses an activation energy of 117-1 51 kJ mol-l, whereas for oxygen it is ca.180-251 kJm0l-l.l' For the surface diffusion of oxygen 63 kJ rnol-l2O and for CO oxidation on a PdIII surface following the Langmuir- Hinshelwood mechanism, an activation energy between 59-105 kJ mol-' 21 is required. A possible explanation could be given by the assumption that the diffusion of oxygen from the bulk to the surface of an oxidized and inactive palladium crystal requires this high activation energy.This would be in agreement with our previous interpretation' of the observed patterns involving phase transitions between Pd and PdO. The observed temperature dependence of the time needed for the reactivation leads to the conclusion that the PdO phase should be stabilized at lower temperature by shifting the thermodynamic equilibrium towards PdO. Comparable conclusions can be drawn by looking at the stability of surface PdO;,, therefore, one expects an increase of the number of palladium oxide particles for decreasing reaction temperatures. The assumption leads to a stronger decrease of the conversion of CO to CO, due to the more favourable condition to form PdO crystals. On the other hand, taking CO to reduce PdO1754 Oscillating Catalytic Oxidation of CO A 50 60 min t f B I I 60 min t Fig.1. Characteristic self-similar time series. Dependence of the conversion of CO into CO, on the temperatyre To of the reactor. [Pd/Al,O, catalyst (0.5 YO Pd by wt); 50 mg catalyst, Pd particle diameter 20 A, surface area of Pd particles 1.25 m2 (g catalyst)-', 0.37 YO CO by volume.] A (a) T = 223 "C, (b) T = 213 "C, (c) T = 203 "C, ( d ) T = 175 "C; B (0) T = 168 "C, (b) T = 164 "C.P. J . Plath, K. Moller and N . I. Jaeger 1755 I \ \. t I I 160 170 180 190 T0l"C Fig. 2. Ignition times of the reaction after lowering the temperature to the finite reaction temperature for two experiments (a) and (b) with the same amount of the catalyst (50 mg). The inset shows an Arrhenius plot. no formation of CO, could be observed for temperatures less than 403 K, which can be understood by the high activation energy of the reduction of PdO.Analysis of the Time Series To gain deeper insight into the nature of the process underlying the time series, we have investigated their pattern by reconstructing the trajectory of the system in a three- dimensional phase space using the method mentioned ea~1ier.l~' l5> 23 Fig. 3 shows the trajectories of our system at four different temperatures in the range 438-496 K. For the graphic representation we have chosen a time interval of 1 h from the corresponding time series, which are given in fig. 1. The trajectories form attractors, the shape of which reminds us of 'strange attractors' representing chaotic behaviour. 23-29 The characteristic feature of ' strange attractors ' is their sensitive dependence upon the initial conditions.For our experimental investigation this means that the sequence of the orbits of the attractor differs for each interval of the experimental time series, although the topology of its embedding remains constant with time. However, one has to be very careful in drawing conclusions from details of the geometrical shape of the attractors, since the shape depends upon the chosen time interval z used for the construction of the phase space. This can be understood by looking at the attractors resulting from the same time series at 496 K the sequences of the realizations of the two bundles of orbits differ [see fig. 3(A) and 41.At 476 K the system is represented by an attractor, the orbits of which differ extremely and cover a large area of the phase space, although one can recognize several bundles of orbits [see fig. 3(A)]. The appearance of multistability for lower temperatures leads to a complex1756 Oscillating Catalytic Oxidation of CO Z Z X ' Fig. 3. Representation of the time series of fig. 1 by the trajectories of the system in a three- dimensional phase-space constructed by the variables x = x(t), y = x(t - z) and z = x(t - 22) and a generic time interval z = 3 s: A (a) T = 223 "C, (b) T = 203 "C; B (a) T = 168 "C, (b) T = 164 "C.P. J. Plath, K. Moller and N. I. Jaeger 1757 X Z Fig. 4. The projection of the trajectories of the system at T = 223 "C into the (a) two-dimensional and (b) three-dimensional phase-space; z = 8 s.attractor shape [fig. 3(B)], again characterized by a high sensitivity of the system to the initial conditions. For example, there are some orbits which pass the middle state without touching, while others stay at this state before reaching the lower state of conversion. The appearance of bundles of orbits gives rise to the question whether or not hidden periodicities can be observed, which might be due to dominating leading centres in the excitable two-dimensional catalyst bed. In order to study this problem the power ~ p e c t r a ~ ' ? ~ ~ obtained by Fourier analysis (see fig. 5 and 6) of the time series were analysed. Decreasing the temperature from 606 K the first small-amplitude oscillation occurs at 529 K.The birth of very small oscillations at this critical temperature can be characterized by a weak excitation as it is typical for a Hopf bifurcation. The oscillations show a simple periodic pattern [fig. 5(a)-(c)]. At 496 K one can observe oscillations almost of period two [fig. 5(e)]. Thereafter a more or less periodic pattern develops into oscillations with a fractal structure in time [fig. 5(g)-(i)]. One can follow this looking at the Fourier spectra (see fig. 6). They have been taken by analysing an interval of 1024 s and 150 given harmonics. The fundamental frequencies X (i = 1, 2, 3) can be approximated easily by counting the number of given harmonics within the interval up to the frequency of interest and multiplying this number with the factor 0.058, giving the number of oscillations per minute. For higher temperatures between 508 and 501 K a periodic pattern can be recognized within the Fourier spectra.The fundamental frequencies could be interpreted in a first attempt as a main mode followed by several harmonics since they are close to multiples of this lowest frequency. But in the linear representation of a Fourier spectrum harmonics of the fundamental frequency of a circle should not be observable unless patterns occur, which are non-sinusoidal to such a degree that many higher harmonics with strong amplitudes are required. Otherwise, they become visible only in the logarithmic plot. In our case, the occurrence of 'harmonics' already in the linear plot indicates resonance with other fundamental closely neighbouring frequencies.For example, the number of harmonics visible in the linear plot can grow up to six in the case1758 Oscillating Catalytic Oxidation of CO J A CI YP . J . Plath, K. Moller and N . I . Jaeger 1759 i--1760 A h Y Oscillating Catalytic Oxidation of CO i - c YP . J . Plath, K . Moller and N. I. Jaeger 1761 of T = 507. If one compares a linear spectrum of these time series with the linear representation of the squared amplitudes of a sine function one can recognize that for instance the spectrum for 507 K [see fig. 6(a)] is not only constructed by one frequency, f,, and its harmonics but by at least three frequencies f1(6), f2( 13) and f3(19) and their linear combinations Ifi + mf, + nfk. The simple structure of the spectrum is due to the fact that in the limit of the accuracy of the numerical analysis f, = f3/3 and f 3 - f, = f,, f2 = f,f3, 2f3 = 3f2; 4f1 +f3 = 5f1 +.f2.There is a locking of frequencies. At 505 K the lowest frequencyf1(9) grows in comparison to 507 K. The spectrum can again be formed by three frequencies, f1(9), f2(17) and f3(26). At 501 K the spectrum becomes very simple and it seems to be harmonic; however, the amplitudes of the higher harmonics are too large to be formed by only one frequency, f1(12). For this reason a second and third frequency have to be taken into account again. With decreasing temperature the fundamental frequencies are shifted to higher values. At 498 K bifurcation occurs, as observed by a splitting of the lowest fundamental frequency into two frequencies close together :fl, ,( 13) = 0.7685 min-l andf,, ,( 17) = 0.9087 min-l.One can see even a subharmonic, if,. There is a very strong periodicity of period length two at T = 496 K, as seen from the graph of the attractor. In the linear plot of the Fourier spectrum the period which is expected because of the pitchfork bifurcation according to the Feigenbaum scenario, cannot be detected.28 For this purpose one needs to look at the logarithmic power spectrum (see fig. 7). Besides the three fundamentals fl( 14), f2(29) and f3(44) one can detect the subharmonics of fl, :fl and ifl, and the ultrasubharmonics ifi. The ultrasubharmonic :fl closely resembles the subharmonic $f2. Furthermore, two linear combinations are located close tof2(29), 2f1 = f(28) and 2f2 = JT30), which are locked by $,(29).Looking at the higher harmonics off, in the logarithmic power spectrum one can estimate f, more exactly to be fl( 14, 2). The third frequency, f3(48), and its harmonics can be neglected with respect to the outstanding structure of the time series, which is obviously controlled by f, and f, and their sub- and ultrasub-harmonics. At 492 K two strong frequencies, f1(15) and f2(33), and their linear combination dominate the Fourier spectrum. Upon lowering the temperature this behaviour is continued at 498 K and then develops into a spectrum of only one main frequencyf'(24) and its subharmonics and harmonics at 486 K, while the pattern of the time series begins to show a fractal structure with small repeating areas (see fig.5). The existence of subharmonic frequencies if,, ifl and the ultrasubharmonic frequency ifl as well as the harmonic frequency 2f1 can easily be seen in fig. 6 at 486 K. The fractal structure developed at 476 K can mainly be described by the two frequencies of the doubletf,, ,(22), fi, ,(24), their subharmonics and linear combinations (see fig. 7). The higher frequency, &(56), loses its influence41 the structure of the spectrum and the time series. The shape of the spectrum becomes much simpler again for some special low temperatures, showing only one frequency and its subharmonics and ultrasubharmonics. Fig. 8 shows two examples for highly developed fractal patterns at low temperatures ( T = 468 K and T = 461 K), which differ greatly in the complexity of their spectra. To obtain better resolution we have taken 300 harmonics for the Fourier analysis in these cases.The number of the frequency has now to be multiplied by the factor y1z = 1.47 x lo-, to obtain the frequency in cycles min-'. As was the case for 476 K, the subharmonics of the old fundamentals now become the main frequencies of these spectra. For instance, the 468 K spectrum is determined by three new frequencies g0(26), g,(49) and g,( 59), which derive from the frequencies g( 1 1) = ifp, ,( 1 l), f,, ,(22) and f,, ,(24) in the spectrum at 476 K. In the linear spectrum at 468 K one can recognize the subharmonics f g0(9), f go( 13), 5 g,( 16), + g2( 19), the ultrasubharmonics f go( 17), 8 g,(32), f g,(40) and several linear combinations : (g2 -gl) ( lo), (go + g, - 8,) (36), (go + 8,) (7% (g2 - 5 8,) (43).Again the higher frequencies1762 L-4 x c w- 1" Oscillating Catalytic Oxidation of CO I ir 7 IN L O lamod h 0 v 1: m %.- =-I? 0 i, iamod A aJ Y m w iamod h Q v - x Y N x ? 4 ? m 1 i m cv x 0, 4 9 : i m -0 samod &iamod0 0.5 1.1 1.6 2 . 2 2 . 7 3.3 3 . 8 4 . 4 cycles min-' fl 2 j fl I 0 0.5 1.1 1 g = 2f1,l f l , l ( h ) P 4 i 1.6 2.2 2 . 7 3.3 3.8 4*4 cycles min-'1764 Oscillating Catalytic Oxidation of CO h 0 v 0 N II N, c \I, * m II N (Y x 00 * II N m I . k N N N c x m N Y, h( -. Y,- * N *-. m b, CDP. J. Plath, K. Moller and N . I. Jaeger 1765 can be understood by frequency locking, e.g. (2 g, - 2 g, +go + g,) (65) = f g,(65) ; Lowering the temperature the new frequencies, g, increase and lose their influence on the spectrum, whereas their own subharmonics h and linear combinations of them now become important.At 461 K the spectrum becomes very simple again, because of a strong frequency locking between the subharmonics and their differences. On the one hand it seems as if only one frequency i(7) and its harmonics will rule the whole spectrum, but on the other hand one can derive this spectrum from that at 468 K: the former frequencies g0(26), g1(49) and g,(59) are shifted and split now, yielding g0,,(36), go, ,(43), gl, ,(56), gl, ,(64), g2, ,(77) and g2, ,(86). 'I'heir diff~~ences ( g2,, -gl, (2% (g,, -gl, 1) (2 l), (gl, , -go, ,) (21) are very close to the subharmonic h2(22) = + go, ,. The difference (go, , -go, ,) (7) just equals i(7) the subharmonic h,(22) and + hl( M), whereas h,( 14) = 5 go, ,(43).h,(29) = f go, , can also be detected easily. Furthermore, g0,,(43) can be understood as a subharmonic of g,,,(86). It seems that the temperature dependence of the time series can be expressed by the following scenario : first, the system can be characterized by three fundamental frequencies, the ratio of which resembles a harmonic one. Lowering the temperature increases the amplitudes of these frequencies (see fig. 9), while subharmonics and ultrasubharmonics become more and more important and locking of frequencies occurs. The higher frequencies then lose their influence on the spectrum. Decreasing the temperature causes bifurcation mainly of the lowest fundamental frequency. The spectra become very simple, essentially structured by only one frequency and its subharmonics and ultrasubharmonics, if the fractal character of the time series is fully developed [see fig.5(h) and fig. 6(h)]. Highly chaotic patterns arise [see fig. 6 0 and (i)] if additional bifurcations occur,31 creating a tremendous number of new linear combinations. The chaos in our system seems to be deterministic and temperature is an order parameter. There are small areas of this parameter Ton which strong periodic behaviour occurs, which can essentially be described by one fundamental frequency and its subharmonics or harmonics. There are two tendencies which lead to chaos: (a) lowering the temperature the fundamental frequencies increase, giving rise to bifurcation and (b) subharmonics and ultrasubharmonics occur on this line, taking over the leading role in the spectra by frequency locking.The spectra become simple even for highly developed fractal patterns of the time series if all main frequencies, the fundamental as well as their subharmonics and ultrasubharmonics, are well locked. In this case all frequencies occurring in the spectra seem to be harmonics of the lowest frequency, the amplitude of which becomes the highest. The time series correlated to such spectra show a clear fractal character, which can be modelled assuming an excitation wave travelling across a fractal framework to a Sierpinsky gasket.15 (2g,+;gz-g1)(88) zs gg,(88,5); ~g,(73)+(g,+g,)(75)(148) 3 g1. The Model To model this behaviour by using a one-dimensional cellular automaton as described e a ~ l i e r ~ ~ ' ~ ~ a re-injection procedure has to be established in order to obtain a periodic pattern similar to the experimental results.This can be achieved by giving the automaton a finite length. The automaton consists of a finite number of cells arranged linearly. The state of each of these cells can be one or zero, corresponding to the producing and non- producing state of the catalytic units or the Pd and PdO phases, respectively. The whole catalyst is represented by such a column of cells. The amplitude of the break-downs of the CO, production, which is assumed to be proportional to the amount of catalytic units in the PdO state, is now correlated to the number of cells of the zero-state. The expansion of the non-active PdO phase is simulated by a propagation of state-changes of the cells (1 -+ 0) in one direction only, starting from a cell whose state is always zero.If1766 Oscillating Catalytic Oxidation of CO A 0 Y JaMOd2 56 3 8 4 t l s 0 12 8 1 1 I I I I I I I t 1 I I I I I ( b ) I hl h2 h3 c 0) a II. d. 111 I,.I,II. 1.1 . I . , , I I I L . l * . ~ , 1 I I I . 1 1 2.70 4.05 1.35 $ cycles min-' 0 P Ir B cu op w 2 Fig. 8. Time series and logarithmic plots of the Fourier analysis (300 harmonics): (a) T = 468 K, go = 26, g, = 49, g 2 = 59; = 43, gl,l = 56, g1,2 = 64, g2,I = 77, g2,2 = 86, h, = 14, h2 = 22, h, = 29, i = 7. (b) T = 461 K, go,1 = 36, g0.21768 Oscillating Catalytic Oxidation of CO 50 40 rz 01 VI 8 2 30 - I c 2 2 20 m c( h 10 0 A \ 2 00 210 220 2 30 T/"C Fig.9. Fundamental frequencies of the power spectra taken as a function of the reactor temperature T. Fig. 10. Time development of one-dimensional automatons. Fundamental repetition units characterized by different AF. (a) Y not defined, u = 8, A F = 0.25, K(u) = 2; (6) r not defined, u = 4 , A F = 0 . 5 , K ( u ) = ~ ; (c) r = 2 , u = 2 , A F = 1 . 0 , K ( u ) = ~ ; ( d ) r = 3 , u = 2 , A F = 1 . 5 , K(u) = 3 ; (e) r = 4, u = 2, AF = 2.0, K(u) = 4.P. J. Pluth, K. Mdler and N . I. Jaeger 1769 Fig. 11. Variation of the length 1 of the one-dimensional automaton creating smallest new sub- harmonics $ of the fundamental frequency F,: K(u) = 2, u = 2 and F, = + in all cases. (a) E = 8, p = 8, Fp = i; (b) E = 10, p = 16, Fp = &; (c) I = 12, p = 16, Fp = &; ( d ) E = 24, p = 32, Fp = &.one cell that is non-catalytic (blocked) touches this boundary of the limited automaton at time n, the time development of the automaton will be stopped. The remaining K(n) blocked cells would produce K(n + 1) blocked cells at time (n+ 1) without reaching a boundary. However, beyond a boundary line no more blocked cells can be produced. The automaton is now re-injected into the starting position, but the point of departure only contains B(n + 1) < K(n + 1) blocked cells because of the boundary, which cuts off the rest K(n + 1) - B(n + 1). The ,automaton now develops starting with the B(n + 1) blocked cells remaining in the position in which they have been produced at time n. In this manner the automaton becomes periodic at time p , whereas the unrestricted automaton can never achieve periodicity apart from its dilatational symmetry.At time p the number K(p) of blocked cells is equal to the length 1 of the restricted automaton. It has to be mentioned that p and not n, the point of re-injection, determines the period length. As we have seen from the Fourier analysis of the experiments, not only one frequency governs the system, but in most cases two or three frequencies and their linear1770 Oscillating Catalytic Oxidation of CO combinations. The ratio of such basic frequencies is nearly an integer. Furthermore, the experiments have shown that the frequencies depend on the temperature of the reactor: the higher the temperature the lower the frequencies. With respect to our model, frequency has to be defined by the number K(u) of blocked cells at the end of the smallest repetition unit.u is the number of time steps, for which all blocked cells are adjacent, forming a blocked row of cells of length K(u) # 1. The structure of this fundamental repetition unit is just the structure, which creates the dilatational symmetry of the automaton. Now a frequency can be defined as F = l / u correlated with the amplitude A = K(u). Fig. 10 shows some examples for different frequencies F and different products AF. In the experiments AF and Fare small for high temperatures. This can be achieved in the model either by using a low value for A = K(u) and a high value for u or by using very small values for r, where r(r 2 u) is the number of cells at time (n+ 1) following an isolated cell at time n [ref.(1 5)]. For very high temperatures no cells can be blocked, even though a blocked cell might exist at the beginning (stationary point), whether F or AF go to zero. The limit for low temperatures implies the existence of a very high frequency, the upper limit for which is F = i. The simulation of the experiment by the proposed model results in a diminished possibility for the formation of blocked cells with increasing temperature. By lowering the temperature more catalytic units can be blocked within a repetition unit, but beside the fundamental frequency, subharmonics can be observed in the experiments. The amplitudes correlated to these subharmonics determine the structure of the time series increasingly with lower temperature.This means that lowering the temperature increases not only r but also the length 1 of the automaton, whereas F tends to its upper limit. These subharmonics, 4, of the fundamental frequency, F,, = l/uo, can be defined by the time ui (i = 1, 2, ...) for which all blocked cells of the automaton are adjacent, forming a row of length K(u,): 4 = l/ui. The smallest subharmonic is determined by Fp = l/up = l/p, where up = p is the repetition time of the automaton. Increasing 1 with decreasing temperature the smallest new subharmonics are born at distinct values of I (see fig. 11). Now, if r becomes a discrete function of I, as for example in ref. (1 5), different K(u) and therefore different fundamental frequencies, F, will occur, However, because of the structure of the automaton the development of the subharmonics of all fundamental frequencies are of great importance with respect to the structure of the time series.This corresponds to the fact that the subharmonics of all fundamental frequencies in the experimental time series are of great importance for their global structure. Indeed, these subharmonics enable the determination of the rules governing the cellular automaton. Conclusion Fourier analysis of the time series leads to a deeper understanding of the dynamics of CO oxidation catalysed by Pd crystals supported on an amorphous A1,0, carrier. The resulting spectra are essentially structured by frequency locking. The temperature dependence of the temporal structure of the process can be modelled qualitatively by a one-dimensional automaton, as introduced in ref.(15) and (16). To understand why it is possible to describe the complex dynamics of this catalytic system simply in this manner one has to reflect on the fact that the cooperative effect is arranged by an excitation wave travelling along the two-dimensional network of Pd particles in the catalyst supported on the silver sieve or ceramic support. Assuming that this network is a fractal one because of the amorphous character of the A1,0, carrier and the random distribution of the pellets in the catalyst bed, the number of Pd particles touched by a cyclic wave at time n should be correlated to the state of the one-dimensional automaton at a corresponding time interval n.P. J. Plath, K. Moller and N .I. Jaeger 1771 Although the experimental facts can be described qualitatively by this model, this should not be overemphasized. For instance, there is a stochastic nature within the experimental results, which the complex but nevertheless strong periodicity of this model does not reflect. 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