11724
J. Phys. Chem. 1993,97, 11724-1 1730
Periodic Perturbation of a Drifting Heterogeneous Catalytic System M. A. Uauw, K. Koblitz, N. I. Jaeger, and P. J. Platb’ Institute for Applied and Physical Chemistry, University of Bremen, FB 2, P.O. Box 330 440, 0-28334 Bremen. FRG Received: June 30, 1993; In Final Form: August 31, 1993’
The investigation of the dynamic behavior of chemical systems by experiments can be seriously impeded by a drift of variables. Sometimes, this drift is taken advantage of by considering it as an autonomous change of a bifurcation parameter. As an example, this approach is applied to the dynamic behavior of the oscillatory oxidation of formic acid over Pd/A1203. The system is driven with a periodic variation of the reactor temperature with constant frequency and an amplitude of just 0.25 K. With the catalyst temperature changing autonomously, a drift from quasi-periodicity through various entrainment bands is observed. It is shown through a simulation with the simple circle map that this experiment is essentially equivalent to a classical entrainment experiment. A phase multistability is observed within the 1:l entrainment band which is reminiscent of the behavior of an extended sine map.
Introduction In systems far from equilibrium, phenomena like oscillatory behavior, pattern formation, and chaos can be observed. The analysisof the dynamic behavior of experimentalsystems is often impeded or even made impossible by intrinsic features of experimental setups. One difficulty is the coarse adjustment ofparameters. There are interesting phenomena that occur in very narrow regions in parameter space. For their experimentalobservation, a very fine variation of a bifurcation parameter may be required. This variation cannot be arbitrarily fine in real experiments,however, and consequently it is sometimes impossible to observe all phenomena in an experimenteven though the region in parameter space may be known. Drvt is another problem. Many experimental systems are subject to very slow internal changes. In heterogeneous catalysis, aging of the catalyst can occur due to poisoning, sintering, or redispersion of the active metal phase, destruction of the support, etc. As long as the time scale of the aging process is significantly slower than that of the dynamic behavior, a stationary situation can be assumed for short parts of the time series. This allows the application of standard methods for the analysis of the dynamic behavior. Particularly at normal and high pressures, however, heterogeneous catalytic systems may exhibit a fairly strong drift. Since the determination of entrainment diagrams as well as of bifurcation diagrams requires that the system remains constant throughout the whole experiment, these experiments cannot be carried out with quickly drifting systems. It has repeatedly been attempted to utilize the unstationary aspects of systems to investigate their dynamic behavior. The idea is that a slowly changingvariable is considered as an internal bifurcation parameter. If all external parameters are kept constant while the behavior of the system is being monitored, the drift may simulatea very slow change of a bifurcation parameter. The advantage is that this change may be on a much finer scale than could be achieved by an external control. For the CO oxidationon Pt/AlzOp, Kapi6ka and Marekfound a drift through a period doubling sequence.’ Another example is the electrodissolution of copper where the working electrode was slowly covered by a film of CuCl.* Since the time scale of the oscillations was short with regard to the time scale of the formation of the film, short fragments of the time series could be regarded as stationary, which allowed Fourier analysis and the construction of attractors and Poincare maps. .Abstract published in Advance ACS Abstracts, October 1, 1993.
0022-3654 f 93 f 2097- 1 1724$04.00/0
However, one should be aware of the limitations of this approach. Baesens reported for the logistic map that changing the bifurcation parameter very slowly (“sweeping through the bifurcation diagram”), the bifurcation behavior depended on the direction and the velocity of the drifte3Another problem is that, by definition, the drift is not a controlledvariationof a parameter. This makes it impossible to plan or reproduce the experiments. In this paper, the influence of drift on an entrainment experiment will bediscussed. In a typical entrainmentexperiment, an oscillatory system with constant autonomous frequency is periodically perturbed by an external drivingforce the frequency and amplitude of which can be changed. Depending on the ratio of the two frequencies, Sa, quasiperiodicor periodic response can be observed. At larger amplitudes of the driver, chaos can be found. While it has been possible to obtain detailed entrainment diagrams for oxidation reactions on single crystal surfaces under ultrahigh vacuum (UHV) conditions? this is difficult for experimentsunder normal pressure condition^.^ The mentioned drift is one difficulty. If, however, the autonomous frequency of the system is slowly changing while the external driver is kept constant, then Sa changes, hence yielding a slow sweep through an entrainment diagram. This will be demonstrated for the heterogeneous catalytic oxidation of formic acid over Pd/A1203 that has been reported to exhibit oscillatory behavior.6 The presented experiment will be compared with classical entrainment experimentsrepresented by the well-known circle map as well as with an extended circle map and it will be discussed whether they can be considered to be equivalent.
Experimental Section Nonisothermal experiments were carried out at atmospheric pressure in a flow apparatus that has been described in detail elsewhere.’ Nitrogen (99.996%, Messer-Griesheim)as the carrier gas was led through a saturator filled with liquid formic acid (>98%%,Riedel-deHaen). The temperature of the saturator was kept constant at 293.2 f 0.1 K. Thereafter the gas stream passed a condenser which was held at a lower temperature (288.0 & 0.1 K). This led to full saturation at the temperatureof the condenser which allowed the control of the concentration of formic acid in the gas phase. The gas stream was then mixed with oxygen (99.99596, Messer-Griesheim), preheated, and led into the reactor designed as a flow calorimeter.* All gas flows were controlled by thermal gas flow controllers (HI-TEC, accuracy 1%). The catalyst was a laboratory synthesized monodispersed Pd catalyst (0.1 wt % Pd on A1203) with an average Pd particle size Q 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97,No. 45, 1993 11725
Periodic Perturbation of a Drifting System 0.25 K -1 I
20 min
4
AT
6
I
I
0 0
40
WI '
1
6
I
1
2
3
4
5 6 7 Time t [h]
8
9
1
I 0 1 1 1 2
I
f 30 hHz1 ' t Ihl
20
,--A
8
9
10 0
0
e
9
10
11
12
't [hl
Figure 1. Time series ( c ~ t1.6 vol %, CO, 49.2 vol %, TR = 419 K,u = 102 mL/min). Inset: The reactor temperature as the driving function. Thestructurewithin theoscillations isan artifact due to the finite resolution of the graphics.
of 1.5 nm and a grain size of 5100 ym. A 20-mg portion of the catalyst was evenly distributed on a sample holder of silver (8.4 mm diameter, weight 74 mg, fixed bed). A NiCr-Ni thermocouple soldered onto the sample holder served to monitor its temperature Tc. The difference AT between TCand the reactor temperature TRwas taken as a measure for the reaction rate. The data were collected with a sampling rate of 5 Hz and a resolution of 12 bit, stored and processed on an Atari Mega ST. The formic acid inlet concentration was 1.6 vol %; the flow rate was 102 mL/min. The temperature of the reactor was held at 419 f 0.13 K.
Experimental Oscillations. The temperature of the reactor was modulated roughly sinusoidally with an amplitude of 0.25 K and a period of approximately 110 s (Figure 1, inset). After a step change of the oxygen concentration from 32.6 to 49.2 vol %, modulated harmonic oscillations appeared (Figure 1). The amplitudes of the oscillations and their beat period increased. After 3.5 h, the modulation vanished and the oscillationsremained constant for about 4 h. The period was 107.5 s; the amplitude was 2.5 K. Then, the oscillations started to be modulated again. This time, the beat period decreased from 50 min to 23 min in the next 4 h. During this experiment, the average temperature differencedecreased slowly. In Figure 2a, the time series is shown again. The oscillationscan no longer be seen here because of the compressed time axis. The modulation of the envelope of the oscillationscan be distinguished even more clearly than in Figure 1. The dotted line representsthe average temperature (arithmetic mean over one period) as a function of time. It decreases from 3.8 to 3.0 K. From the measured temperature difference, the conversion was estimated at 20-30%. In order to check for transport limitations, the dimensionless Thiele modulus 4 was calculated which should be 4 > 10 in the purely diffusion-controlledregime
1
2
3
4
5 6 7 Time f [h]
8
9
1 0 1 1 1 2
Figure 2. (a) Time series from Figure 1 with compressed time axis. The decrease of the average temperature is depicted by the dotted line. (b) Spectrogram (rectangular windows, 256 data points, 43 min length, sampling rate 0.1 Hz). The power spectra are plotted vertically, dark areas correspondingto a high spectral densityof the frequency component. The system is characterized by two frequencies fd and & and their harmonics, respectively. At the beginning, fd = 9 X l t 3 Hz and & = 13 X lW3 Hz. Whilefd remains constant,f, decreases.
and 4 < 0.3 in the purely kinetic regime. Assuming Knudsen diffusion with DK x 1W2cm2.min-l, the Thiele modulus did not exceed 4 = 3 even for the largest catalyst particles. It was less than unity for the upper 5% of the catalyst bed. About half of the catalyst bed was in the transition region (4 < 10) rather than in the purely diffusion-controlled regime. Fourier Analysis. In Figure 2b, the spectrogram is depicted. Short parts of the time series were Fourier transformed (rectangular window, 256 data points, 43 min length, sampling rate 0.1 Hz). The power spectra are plotted vertically, dark areas corresponding to a high spectral density of the frequency component. During the first 40 min, e.g., the system is characterized by two frequencies, atfd = 9 X l e 3 Hz and atf, = 13 X le3Hz and their harmonics, respectively. While fd remains constant,& decreases. This is accompaniedby an increase in its intensity and the appearance of higher harmonics. After 4 h,fs has reachedfd. Another 4 h later, the frequencyf, decreases further whilefd still remains constant. This is more obvious for the harmonics than for the principal frequency itself. The splitting into two frequencies can be seen best in the last segment. Within the segments, the frequencyf, changed (e.8. by about 1.7 mHz for the first segment). This led to some broadening of the peak in the corresponding power spectra. This effect could not be avoided by analyzing shorter segments as the decrease of the drift-induced broadening would have been compensated for by a loss of frequency resolution. However, the main results of the analysis were not affected by this broadening. Poincarb Sections of Short Segments. The time series of the interval 2.01-3.12 h (length 67 min) is depicted in Figure 3a. In Figure 3b, the reconstruction of the trajectory in a threedimensional phase space with cylindrical coordinates is shown. AT(t) is plotted vs AT(t 20 s) in the x-y plane which rotates around the y-axis with the frequency of the driver (9.3 X le3 Hz). The attractor is a torus. In Figure 3c, the PoincarC section with thex-y plane is depicted. The points lie on a one-dimensional
+
11726 The Journal of Physical Chemistry, Vol. 97, No. 45, 1993
Liauw et al.
AT
i
,
,
200 s ,-
'
Tim6 f
)
)
,
,
,
,
,
91
/ f
/
, Time t
Bi e
im t
'
Figure3. Interval 2.01-3.12 h: Superharmonicquasi-periodicbehavior. (a) AT vs time. (b) Trajectory reconstructed in a three-dimensional phase space with cylindrical coordinates. AT(f + 20 s) is plotted vs A T ( f )in the x-y plane which rotatesaround the y-axis with the frequency of the driver (9.3 X lP3Hz). The attractor is a torus. (c) PoincarC section. The average of the time series is denoted by 0 , the reference line for the calculation of 0 is indicated by the solid line. (d) Return map. The pointslie on a one-dimensionalinvertiblecurvewhich is characteristic for quasi-periodicbehavior. (e) Plot of the phase angleeversus time (elf plot).
Figure4. Interval 5.W5.55 h: 1:l entrainment. The attractor is a limit cycle (b). This stable periodic orbit is also represented by the single points in the PoincarC section (c) and in the return map (d).
closed curve of triangular shape. In order to calculate the phase between the driving force and the response of the system, every intersection point was connected with the average of AT(r) in this segment (marked by the open circle). The phase angle was then the angle 8 between this line and the reference line (solid line). In Figure 3d, the corresponding return map is depicted (@,,+I vs 8"). The points lie on a one-dimensional invertible curve which is characteristic for quasi-periodic behavior. Obviously, this curve does not change significantly within this interval, confirming that the chosen part of the time series was short compared to the drift velocity. Furthermore, the curve is considerably wrinkled. This indicates that in this particular system, the nonlinearity introduced by a driving amplitude of only 0.25 K cannot be neglected. In Figure 3e, the 8 values are plotted vs time ( 8 / t plot). This is another method to show that the intersection points in the PoincarC section wander around the smaller diameter of the torus. The same analysis yields a different picture for the modelocked interval 5.00-5.55 h (33 min, Figure 4). It should be noted that an amplitude of 0.25 K is sufficient for entrainment. Theattractorisessentiallya limit cycle (Figure4b) with a Poincart section that consists of a single point (Figure 4c). This is also reflected by a point attractor on the diagonal in the return map (Figure 4). 8 values are approximately constant with time (Figure 4e). After reentering the quasi-periodic region (7.78-8.89 h, 67 min), the attractor reconstruction in phase space yields another torus (Figure 5b) which has a one-dimensional PoincarC section (Figure 5c) and a one-dimensional invertible return map (Figure Sd). In the return map, the points lie above the main diagonal
Time I
400s
I
I
/..-.
ei
,...'
e)
......
1I . a * *
The I
Figure 5. Interval 7.78-8.89 h: Subharmonic quasi-periodic behavior. The three humps in the return map (d) suggest that there are three stable periodic orbits cocxisting within the 1:l entrainment band.
due to the transition from the superharmonic to the subharmonic regime. There is a long residence time of the system in three constrained 8 regions. This can be seen by a clustering of points
The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 11727
Periodic Perturbation of a Drifting System
AT 400s Time t
b,
A
AT(t + r)
t Q
?
2
3
5
4
6
7
8
9
10
Time t [h]
Figure 7. t9/t plot for the entire time series. Between the regions of quasipcriodicity,the broad 1:l entrainment band can be distinguished. The arrows point to sudden changes of 0. The dashed lines depict the 4:3 and 5 4 entrainment band, respectively (see Figures 8 and 9 and text).
I
-!
I
'
Time t
-
'
f dT(t+r)
b)
1 T it
Figure 6. Interval 11.1-12.2 h. The 'ghosts of the fix points" from Figure 6 have almost disappeared.
in the Poincart section and in the return map as well as by a very slow phase shift in parts of the 8 / t plot (Figure 5e). This phenomenon, sometimesreferred to as the'ghost of the fix point", is well-known for the saddle-node bifurcation at the transition from the entrainment regime to the quasi-periodic regimeag After some time (11.1-12.2 h, 67 min), the clustering has decreased (Figure 6). The ghosts of the fixed points have almost disappeared. This is accompaniedby a decreaseof the modulation frequency. O/t Plot of the Entire Time Series. For the calculation of the 8 values, the center of the smaller diameter of the torus had to be determined. A problem arises as due to the slow drift of the average temperature, the attractor moves through phase space. This leads to a change of the center position as well, which makes it impossible to calculate 8. For the short parts of the time series analyzed above, this movement could be neglected. In order to calculate the 8 / t plot for the entire time series, however, the drift of the baseline was subtracted. The resulting 8 / t plot is depicted in Figure 7. The 1:l entrainment band can clearly be distinguished between the regions of quasi-periodicity. One interesting feature is that while in the beginning of the 1:1 entrainment band, the phase angle between driver and response was almost constant, the system sometimes seemed about to leave the entrainment region but autonomously returned to a mode-locked state. These phase jumps are marked with arrows. They strongly suggest that there is more than just one single limit cycle within the entrainment band. This will be discussed below. High-Order Eatrpinment. From the Fourier analysis it can be taken that the ratio Q =fi/fd slowly decreased from (13 mHz9 mHz) = 1.44 to (9 mHz:9 mHz) = 1. This means that there must have been time intervals where Q = 1.33 (= 4:3) and Q = 1.25 (= 94). respectively. If the amplitudeof the forcing function was high enough and if the drift was not too fast, entrainment should have occurred. The 8 / t plot is a useful representation to
"
i
I
J
find these narrow time intervals: It is known that within the 4:3 entrainment band, the Poincart section is characterized by three different intersection points whereas the 5:4 band will give four different intersection points. These regions are the arcas "4:3" and '54" in Figure 7. For these short segments, the attractor reconstruction yielded limit cycles of period 3 and 4, respectively (Figures 8 and 9). It is also confirmed by the PoincarC sections and the corresponding return maps that for approximately 20 min,the system drifted through these higher entrainment bands. Closer inspection of Figure 7 reveals that the points in these regions lie on U-shaped curves (dotted lines). This behavior can be found in simulations with circle maps where Q is changing. It is connected with an attractor changing so quickly that the system does not come arbitrarily close to it. Arrheairur-Type Behavior. From the e/r plot, the oscillation frequency of the system& can be obtained. To some extent, this
Liauw et al.
11728 The Journal of Physical Chemistry, Vol. 97, No.45, 1993
here to a single rate-determining step. It will only be stated here that the combination of steps of transport and reaction yields a global Arrhenius-type behavior.
2
AT
120 8
I
ATU
"3 .
Figure 9. Interval 1.06-1.39 h. A limit cycle of period four (b) and four groups of intersection points in the PoincarC section (c) can be observed as the system passes the 5:4 entrainment band.
1
For the observation of oscillations of a global variable like the average temperature, the synchronization of a large number of catalytic centers is required. Due to this lumping, the dynamic behavior could be described in terms of a periodically perturbed oscillator in the preceding section. The description of the oscillationswas further reduced to a singlevariable, Le. the phase difference between the oscillatory driving function and the oscillations of the system. The most simple approach to model the response of an oscillatorysystem toa periodicperturbation is theone-dimensional circle map. In this section, the validity of this approach will be checked where the ratio of the frequencies of system and driving function 52 in the circle map will be slowly varied. The essential features from the experiment that will have to be captured are the change from quasiperiodic behavior to 1:l entrainment and back to quasiperiodicity,the phenomenonof critical slowing down near the "ghost of the fix point" just outside the entrainment band, the appearance of the high-order entrainment regions, and the constant degree of wrinkledness of the corresponding return map. In the second part of this section it will be shown that the phenomenon of a phasejump requires the treatment of an extended sine map. Simple Circle Map. Circle maps are one-dimensional maps that describe the dependency of the phase angle e,+, between an oscillatory system and an external driver from the previous phase angle e,, the ratio of the two frequencies, 52 = fd/fi, and the amplitude K of the driver, normalized by 2r. Their general form is
=fl',) - Me,)L
(la)
with
fle,) = 0,
-4.71 1.95
'
I
1.96
'
I
1.97 1000/T
'
I
1.98
'
I
1.99
'
! I
2.00
[l/K]
Figure 10. Logarithm of the frequencyj, (E In(u0)) versus the inverse absolute temperature 1/T (Arrhenius plot).
-
will be related to the rates of the reactions. In order to check the assumptionthat the frequency obeysan Arrhenius lawfs e-EIRT, InG) is plotted versus the inverse absolute temperature 1 / T (Figure 10). Within experimental error, the points in this plot lie on a straight line. From the slope, an energy of 66 kJ/mol is calculated which is well within the range of activation energies for heterogeneouscatalytic oxidation reactions. It should be noted, however, that during one period of the oscillations, a large number of diffusion, adsorption, reaction, and desorption steps will occur. Moreover,the catalyst itself may undergo changes like oxidationreduction cycles or adsorbate-induced reconstructions. The sequence of all these processes will probably be quite complex. In the case of relaxation oscillations that are made up by processes with widely differing time scales, one of the processes may be much slower than all others. Then, the frequency of the oscillations will be determined by this slowest proccss, and an Arrhenius plot will yield its activation energy. The concept of activation energy holds because it is possible to isolate a single rate-determining step. As theoscillations are rather harmonic in the described system, the calculated "overall activation energy" will not be attributed
+ s2 + K / ~sin(2r0,) u
(Ib)
The expression He,)[stands for the smallest integer for which holds If(@,)1 Ifid,). By this notation, &,isconfined to the interval [0;1[. It should be stressed that the circle map is a strictly phenomenological description of the function @,+I =fie,) and not a model where the several terms can be attributed to parts of a particular system. The corresponding return maps are depicted in Figure 11 for different values of 52. At low 52 (52 = 0.92), the return map has no intersection points with the diagonal Bi+l = 0,. This leads to a constant phase shift (quasiperiodic behavior). At 52 = 0.936, a snp-bifurcation (saddle-node of periods) occurs which yields two intersection points (a stable and an unstable stationary state, S and U), indicating the transition to the mode-lacked state at the edge of an entrainment band. At further increase of 0, the intersection points move along the diagonal. This is reflected by a shift of the phase between driver and response. In a second snp-bifurcation at 52 FI 1.064, the stable stationary state is annihilated by a collision with the unstable stationary state and the system becomes quasiperiodic again. In Figure 12,the correspondingB/t plot of the 1:l entrainment band is shown where 52 is increased by 5.3 X 1 V after each iteration. This is close to a stationary experiment. The sharp borders of the entrainment band can be seen as well aa the clustering of points in the quasiperiodic region near the f i i point. The variation of 0 with 52 within the entrainment band shows inversion symmetry with respect to the point 52 = 1, B = 0.5. Decreasing 52 gives essentially the same picture. However, the sweep through this diagram leads to a small bend at the left edge of the entrainment band.
The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 11729
Periodic Perturbation of a Drifting System
0
e,
1
o
e,,;
0
o e, io el 1 Figure 11. Simple circle map: return maps for different n values, K = -0.4. S andUdenotethestableandunstablestationarypoint, respectively. 0
el
i
1 .o
Figure 13. Extended sine map: return maps for different 0 values, K1 = -0.2, K2 = -0.2. SIand U1 denote the first pair of stationary points, S2 and U2 the second
1.o
0.8
e
0.8
0.6
e
0.4
0.6 0.4
0.2 0.0 0.92
pair.
0.2 0.96
1 .oo
1.04
1.08
R Figure 12. Simple circle map: O/t plot.
A higher drift velocity leads to an overshoot at entering the Amol'd tongue but does not affect the leaving of it. Also, the high-order entrainment bands change from gauche to U-shaped: the change of the underlying return map is so fast that the system response does not settle on the attractor. Extended Sine Map (ESM). It can be seen in the e / t plot in Figure 7 that the phase between driver and response is almost constant within the 1 :1 entrainment band until it suddenly changes to another value. This can be explained by a phase multistability as already been predicted by Rehmus et aI.loJ1and Vance12and found experimentally for the combustion of acetaldehyde by Hjelmfelt et aI.l3in the caseof a two-componentdriving functi0r1.I~ The corresponding map has the form
with
fle,) = [e, + s2 + K J ~ usin(2u0,) + K , / ~ usin(4~0,)l (2b) The return maps are depicted in Figure 13 for different values of Q. The transition from quasiperiodicity to entrainment at Q = 0.944 generates a pair of stationary points (stable SI and unstable Ul). At fl = 0.988, a second pair of stationary points appears (SZ, U2). There is no autonomous transition from one stable fix point to the other because they are separated by the unstable fix points. At Q = 1.012, SIdisappears by collision with U2 and the system will jump to the second fix point. At Q = 1.056, the bifurcation to quasiperiodicity takes place. In the region Q = 0.988-1.012, two stable stationary points coexist.
nn V."
0.92
0.96
1.60
n
1.04
1 .os
Figure 14. Extended sine map: O / f plots for increasing and decreasing
n superimposed. Note the region of bistability.
This is equivalent to a phase bistability: When Q is decreased from higher values, 0 = 0.6 at Q = 1. When fl is increased, 0 = 0.4 at Q = 1. A transition between the stationary statca will only occur if either a perturbation of sufficient amplitude shifts the system across the unstable stationary state or if Q is varied such that the stationary state disappears and only the other one is left. This is what can be observed in the 0 / t plot in Figure 14.
Discussion In the typical entrainment experiment, two parameters can be varied, viz. the amplitude of the driver ( K ) and the ratio D of the frequenciesof the driver and the autonomoussystem ( f d / f ) . These are the parameters that are normally varied while the frequency of the autonomous system is constant. This is in contrast to the presented experiment where the temperature of the system slowly decreased and with it fi. Advantage was taken of this drift by keeping the drivingfrequency constant, which results in a variation of fJfi like in the normal entrainment experiment. It will now be verified whether these two experiments are indeed equivalent. Like in a classical entrainment experiment, a transition from superharmonic quasiperiodic behavior to 1:1 entrainment and to subharmonicquasiperiodicitycould beobserved. While thiscould be taken from visual inspection of the time series only, it was by reconstruction of the trajectories in phase space and by PoincarC sections and return maps that even more detailed information on the system was gathered: The dynamic behavior just outside the
11730 The Journal of Physical Chemistry, Vol. 97, No. 45, 1993 entrainment band wascharacterized by the‘ghost of the fix point” and high beat periods. By means of a elf plot, even two narrow regions of high-order entrainment (5:4 and 4:3, respectively) could be detected. From the circle map it is known that these are exactlythe most pronounced entrainment bands within this region of $2. The phasespacereconstructionyielded limit cyclesof periods four and three, respectively. The shape of the O / t plot suggested that in this regime, the drift was too fast for the system to be arbitrarily near to the attractor at every time. It must be noted that it has not only been the frequency of the system that was changing but also the distance to the next bifurcation. The slow drift could have led the system beyond the bifurcation where the oscillations cease. Then, no entrainment would have been possible any longer, and the analogy with the classical entrainment experiment would have broken down. However, in this particular experiment, no such bifurcation has occurred, suggesting that the system has been sufficiently far away from the next bifurcation point. Another point to consider is that although the absolute value of the amplitude of the driver has remained constant, its relative impact on the system may have changed with the drift. This means that the way through the entrainment diagram must not have necessarily been parallel to the abscissa. However, the wrinkledness of the return maps (the degree of nonlinearity introduced) did not change significantly during the experiment. This indicates that the effective amplitude of the drive could be estimated to remain roughly constant and well within the interval 0
