J . Phys. Chem. 1988,92, 6963-6966
6963
Shil’nikov Chaos during Copper Electrodissolution M. R. Bassett and J. L. Hudson* Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, Virginia 29901 (Received: December 18, 1987)
Experiments performed on the electrodissolution of a copper rotating disk in a H2S04/NaC1solution are presented. Time series, attractors, next-maximum maps, and return maps are used as evidence for the existence of Shil’nikov chaos. Also, the development of the chaos is compared to the behavior of a set of model equations as a bifurcation parameter is varied.
Introduction In a recent publication,’ Argoul et al. have presented experimental evidence for the existence of Shil’nikov (homoclinic) chaos in a chemical system. They analyzed data obtained with the Belousov-Zhabotinskii reaction in a continuous, stirred reactor using, inter alia, one-dimensional maps and symbolic dynamics. In state space, trajectories obtained from the experimental data appear to approach close to a saddle focus and to satisfy the conditions of Shil’nikov2 for the existence of homoclinic chaos. The presence of a saddle focus was one of the key features of the original models presented by R o ~ s l e r for ~ , ~chaos in chemical systems. The work of Argoul et al. appears to be the first careful analysis of experimental results from a chemical reaction showing the existence of homoclinic chaos. In this paper we present experimental evidence showing Shil’nikov chaos in another chemical system, viz., the electrodissolution of copper. We see that many of the features seen in the stirred reactor occur also in a completely different system, one involving the electrodissolution of a surface. Furthermore, the electrochemical system has the advantage compared to the Belousov-Zhabotinskii reaction in that the oscillations in the former are considerably faster so that it is possible to accumulate more data over the time interval in which the time series is stationary; this aids in the construction of Poincare maps, return maps, next-maximum maps, and attractors used to compare the experimental results to theory. Oscillatory behavior in electrochemical systems has been known to exist for over 150 yearss and has been the focus of many experimental studied.+I0 However, it was not until very recently that the methods of nonlinear dynamics have been applied to electrochemical systems, e.g., to the galvanostatic dissolution of nickel,” the potentiostatic dissolution of iron,’* and the potentiostatic dissolution of copper.I3J4 Experiments The experiments were carried out using a copper rotating disk electrode, which was a copper rod 8.26 mm in diameter embedded in a 2-cm Teflon cylinder so that only one end of the copper rod was exposed. The solution which was 1 N in HzSO4 and 0.1 M in NaCl and was held at 25 OC. The cylinder was rotated at 200 or 1000 rpm. Experiments were performed potentiostatically at E = 350 or 400 mV (versus a saturated calomel reference electrode). At the beginning of an experiment the potential was changed in a stepwise manner from its rest potential to the working potential. The current in milliamps was then measured as a function of time at 120 Hz with the aid of a voltmeter and a PDP laboratory computer. More details on the experiments can be found in a previous p~blication.’~ Experimental Results During the course of an experiment a film builds up 071 the copper surface. The experiments are thus being carried out under conditions in which there is a slow variation of a parameter. After the step change of disk potential, there i s an initial nonoscillatory
* To whom correspondence
current. The system eventually makes a rapid transition to chaotic behavior. We shall discuss this transition further below. The oscillations endure for some time, after which the film becomes so thick that the oscillations die out. The data presented in Figures 1 and 2 were taken from a single experiment. During the 3000 s of oscillatory behavior in this experiment there were two types of chaos observed having small and large amplitudes, respectively. There were three intervals of large-amplitude chaos and two interspersed intervals of small-amplitude chaos. The large-amplitude intervals lasted 270, 730, and 1200 s, while the two small-amplitude intervals lasted 150 and 650 s. It is not known why the system alternated between the large- and small-amplitude chaos. It should be noted that although the chaotic behavior is transitory, it is stable for long enough periods of time, or is changing so slowly over that period of time, so that attractors, Poincare sections, next-maximum maps, and return maps can be generated. In other words, over the short period of time from which the attractors are made the behavior can be assumed to be stable and thus all nearby trajectories in phase space converge to this attractor. We also note that the system, when perturbed intentionally, always returned to the state in which it existed before the perturbation in a short period of time. For example, we disturbed the system by opening the electrical circuit momentarily; upon reclosing the circuit, the current goes through a transient of less than a few seconds before returning to the original state. We observed Shil’nikov chaos in the potential range 320 2 E 5 650 mV and at disk rotation rates of 200-1000 rpm. However, at higher potentials and/or rotation rates only the larger amplitude chaos was obtained. We will now show and analyze representative examples of the two types of chaos. The time series and attractor for the small-amplitude chaos are shown in Figure 1, a and b, respectively. Figure l a was made with 20.83 s of data taken from the second interval of small-amplitude chaos which lasted 650 s. The attractor, shown in Figure lb, was constructed with 41.67 s of data, the middle half of which is shown in Figure la. The attractor was constructed by using ~
(1) Argoul, E ; Arneodo, A.; Richetti, P. Phys. k t r . A 1987, 120, 269-275. (2) Shil’nikov, L.P. SOC.Marh. Dokl. 1965, 6, 163-166. (3) Rossler, 0.E. Z.Naturforsch. 1976, 31A, 259-264. (4) Rossler, 0.E. 2.Naturforsch. 1976, 31A, 1664-1670. (5) Fechner, G.Th. J . Chem. P h p . 1828,53, 129. (6) Wojtowicz, J. In Modern Aspects of Electrochemistry; Bockris, J. O., Conway, B., Eds.; Plenum: New York, 1972;Vol. 8. (7) Cooper, J.; Muller, R.; Tobias, C. J. Electrochem. SOC.1980, 128, 1733-1744. (8) Lee, H. P.; Nobe, K.; Pearlstein, A. J . Electrochem. SOC.1985, 132, 1031-1037. (9) Jaeger, N.I.; Plath, P. J.; Quyen, N. Q.In Temporal Order, Rensing, Jaeger, N. I., Eds.; Springer-Verlag: Berlin, 1985. (10)Tsitsopoulos, L. T.; Tsotsis, T. T.; Webster, I. A. Sur/. Sci. 1987, 191, 225-238 (11) Lev, 0.; Wolffberg, A.; Sheintuch, M.; Pismen, L. M. Chem. Eng. Sci. 1988, 43, 1339-1353. (12) Diem, C . B.; Hudson, J. L. AIChE J. 1987, 33, 218-224. (13) Bassett, M. R.;Hudson, J. L. Chem. Eng. Commun. 1987, 60, ~~
145-1 .- 59
should be addressed.
0022-365418812092-6963$01.50/0
(14) Albahadily, F.N.;Schell, M. J . Chem. Phys. 1988,88, 4312-4319.
0 1988 American Chemical Society
6964
Bassett and Hudson
The Journal of Physical Chemistry, Vol. 92, No. 24, 1988
25 d
8
4
2
n v
+
U
I
it-5
s+
TIME I
2 2
1
30 30
N
I
’22
(
/
, 25
31 e
’
‘
/’
e
32 ’-l ,
Figure 1. Small-amplitude chaos ( E = 350 mV (SCE)); rotation rate = 200 rpm; all current values are in milliamps): (a) Time series. (b) Attractor; the two projections are made at angles differing by 6’. The right picture is for the left eye and the left for the right. The attractor was constructed by using the time delay method with the delays being 37 and 67 where T = 1/120 s. (c) A single loop around the attractor. (d) Return map made from Poincare section taken at I ( t - 3 7 ) = 26.5. Cut was taken as trajectories left the area of the saddle focus. (e) Next-maximum map.
the method of time with delays of 37 and 67; throughout this paper T = 1/120 s. (The attractor was made by plotting every current measurement of the time series with that current as one coordinate, the current at time 37 before that as the second coordinate, and the current a t time 37 before that as the third coordinate.) A portion of the trajectory making a single loop around the attractor is shown in Figure IC. There appears to be a saddle focus at the point S. Trajectories spiral out from that point, make a large circuit around the attractor, and are then reinjected to the vicinity of S. Since the point of reinjection varies slightly each time, the number of small-amplitude oscillations around the saddle focus can also vary. The facts that the trajectories are reinjected in a nonoscillatory manner near the supposed saddle focus and leave the area of the supposed saddle focus by oscillating away from it indicate that the saddle focus has eigenvalues X and p f iw where X is negative and p is positive. This then is a “spiral” type strange attractor as described by Rossler.” Shil’nikov chaos, which exhibits the “spiral” type attractor, has an interesting feature; the return map of a Poincare section taken transverse to the trajectories as they leave or return to the area of the saddle focus is multibranched. ~
~
(15) Takens, F. Lect. Nofes Math. 1981, 898, 366-381. (16) Packard, N. H.; Crutchfield, J. P.; Farmer, J. D.; Shaw, R. S. Phys. Reu. Left. 1980, 45, 712-716. (17) Rossler, 0 . E. Bull. Math. Biol. 1911, 39, 275-289.
That is, it consists of a set of 1-D curves, each of which is associated with some nonnegative integer. This integer represents the number of small-amplitude oscillations a trajectory completes around the saddle focus between two successive crossings of the Poincare plane. (For a discussion of Poincare sections and return maps see, for example, ref 18.) A Poincare section of the attractor was made at Z(t-37) = 26.5. The Poincare section is not shown. A return map made with this Poincare section is shown in Figure Id. (This map was made by using the same 41.67 s of data used to construct Figure :b.) Each point (1 17 total points) on the return map is represented by some integer 0 to 4. Within experimental error all the zeros and positive integers fall on approximately 1-D curves. This then yields a multibranched, 1-D return map which is characteristic of systems exhibiting Shil’nikov chaos. A next-maximum map made from the data shown in Figure l b (again using 41.67 s) by plotting the value of the maximum of a current oscillation versus the value of the previous maximum is shown in Figure le. This map, which is also approximately 1-D, will be used to compare the small- and large-amplitude chaotic behaviors. A time series and attractor for a portion of the larger amplitude chaos are shown in Figure 2a, b. A trajectory which travels once (18) Eckman, J. P.; Ruelle, D. Rev. Mod. Phys. 1985, 57, 617-656.
Shil'nikov Chaos during Copper Electrodissolution 50
1
I
The Journal of Physical Chemistry, Vol. 92, No. 24, 1988 6965
24 I
I
(I
9 I1
"
0
0 0
z
n
+
,
,o
W
1 0
Y
'
I
1 1
20
I
+IO
s--+
i
0
o
TIME
36 36
0
24
i
20 50
Figure 2. Large-amplitude chaos (same experiment as Figure 1; all current values are in milliamps): (a) Time series. (b) Attractor (spline fit). (c) A single loop around the attractor (spline fit). (d) Return map made from Poincare section taken at Z(r-27) = 28. Cut was taken as trajectories left the area of the saddle focus. (e) Next-maximum map.
around the attractor is shown in Figure 2c. In all cases, the results are qualitatively similar to those seen for the lower amplitude chaos. The differences are that in the larger amplitude case the trajectories can make more small-amplitude oscillations around the saddle focus and can make a larger excursion from the area of the saddle focus. A Poincare section (not shown) was made at Z(t-27) = 28. The return map, shown in Figure 2d, was generated in the same manner as Figure Id. In Figure 2d there is a larger range of integers than in Figure Id because of the greater number of small-amplitude oscillations a trajectory can make around the saddle focus between successive crossings of the Poincare plane. Figure 2d is not as nice as Figure Id; that is, the integers in Figure 2d cannot be approximated with some set of lines as was the case in Figure Id. This is due, at least to some extent, to experimental error, especially for the branches associated with the larger integers. It is known that the branches associated with the larger integers get increasingly closer. Therefore, even small amounts of experimental noise and error can cause points associated with certain integers to become mixed rather than falling on some set of lines. Also, the smallest amplitude oscillations around the saddle focus, for the larger amplitude chaos, are often difficult to distinguish from experimental noise. This could also give rise to the mixing of integers in the return map. Still, Figure 2d is qualitatively similar to maps generated from Shil'nikov chaos in that the smaller integers are on the right side of the map and the larger integers are on the left.
The next-maximum map shown in Figure 2e is similar to that in Figure l e except that it is steeper, narrower, and has a longer "tail" on the right side of the map. These features arise from the fact that the trajectories can make larger excursions from the area of the saddle focus. Also,there is a greater concentration of points along the 45' line in the lower left corner of Figure 2e than in Figure le. This is a result of the increased number of smallamplitude oscillations around the saddle focus in the larger amplitude chaos. Figure 2a represents 41.67 s of data taken from the last interval of large-amplitude chaos. Figure 2b is 16.67 s (approximately the middle 40% of the data shown in Figure 2a) of spline fit data. Figure 2d (which contains 131 points) was constructed from 125 s of data, the middle third of which is shown in Figure 2a. Figure 2e was made from 62.5 s of data, the first 41.67 s of which is shown in Figure 2a. Other experiments at these conditions were performed, and the oscillations observed in all cases were qualitatively similar to those shown in Figures 1 and 2. However, the duration of the oscillatory period changed from one experiment to the next due most likely in part to the complex nature of the development of the electrode surface and the growth of the surface film. Also, a different electrode was used for each experiment. In the three other experiments performed at the conditions of Figures 1 and 2, the oscillations lasted anywhere from 1200 to 3500 s. One experiment had three intervals of large-amplitude and two intervals of small-amplitude chaos, the second had two intervals of large-
6966 The Journal of Physical Chemistry, Vol. 92, No. 24, I988
amplitude and one interval of small-amplitudechaos, and the third showed only large-amplitude chaos. Again, it is important to remember that for both the large- and small-amplitude chaos the time series, attractors, and maps observed in all the experiments closely resembled those presented in Figures 1 and 2. We now compare the features of the experimental results to those of a mathematical model previously considered by Argoul et al. They treated the following three-variable ODE system:
Bassett and Hudson -3c
X
dX/dt = Y dY/dt = Z dZ/dr =
-2 - 1.3Y-
pX
+ X 2 - 1.425P + 0.2XZ - O.OlX2Z
(1)
The system has steady states at the origin and (p,O,O). As the bifurcation parameter p is increased past 1.3, the origin undergoes a subcritical Hopf bifurcation. When this occurs, the behavior makes an abrupt transition from steady state to Shil'nikov chaos. The origin undergoes a homoclinic bifurcation as p is increased past 1.505. The experimental results resemble very closely simulations made using eq 1. For example, p = 1.33 (small-amplitude chaos) yields a time series and maps very similar to those seen in Figure la,d,e, while p = 1.455 (large-amplitude chaos) yields results resembling those seen in Figure 2a,d,e. The eigenvalues of the steady state at the origin are X = 1.01, p f iw = 6.4 X f 1.140i and X = -1.06, p f iw = 3 X f 1.167i for p = 1.33 and 1.445, respectively. Note that Ip/Xl
