J . Phys. Chem. 1993,97,7258-7263
7258
Deterministic Chaos Arising from Homoclinicity in the Chlorite-Thiourea Oscillator Christopher J. Doona, Ralph Blittersdorf, and Friedemann W. Schneider’ Institute of Physical Chemistry, University of Wiirzburg, 911 I Marcusstrasse, 0-8700 Wiirzburg, Germany Received: March 26, 1993
Analysis of aperiodic oscillations in the chlorite oxidation of thiourea in a continuous-flow stirred tank reactor in terms of time series, reconstructed attractors, Fourier spectra, Poincard sections, return maps, and generalized Renyi dimensions confirms the existence of deterministic chemical chaos in this system. The coincidence of an experimentally observed hysteresis and a discontinuous transition from a stationary state to aperiodic behavior signifies this as homoclinic chaos, which arises from the interaction of a local subcritical Hopf bifurcation and a global homoclinic bifurcation. Homoclinic chaos is characterized by a spiral-type strange attractor.
Introduction Nonlinear chemical systems can exhibit a wide assortment of exotic dynamical phenomena when maintained far from equilibrium using a continuous-flow stirred tank reactor (CSTR). The Belousov-Zhabotinskii reaction (BZ), the best known example of these systems, displays behavior ranging from periodicity! to quasiperiodicity,? intermittency,’ phase-locking,4 and chaos.5 Further insight into these behaviors has been gained by the design of other chemical systems that exhibit similarly intriguing dynamical phenomenaq6Consequently, entire families of oscillators have been designed based on the chemistry of halogens, chalcogens, and transition metals, with chlorite (ClO2-)-driven systems constituting the largest class of these oscillators. Although a variety of phenomena have been observed in the chlorite systems containing several chemical component^,^ the systems comprised of only chlorite as oxidant and a sulfur species as reductant are especially attractive because they exhibit a startling array of dynamical behavior, which belies their apparent simplicity as “two-component” systems. The reaction of chlorite with thiosulfate (Cl02- S Z O ~ ~ - the ) , * best studied from this subclass of oscillators, displays simple as well as complex oscillations, chemicalchaos arising from a period-doubling route, and chaos arising from a quasiperiodic route,q travelling waves and spatial patterns,Ioand extreme sensitivity to fluctuationsand mixing.” The chemical mechanism of this reaction12 has been studied at high pH and T = 90 OC; however, the mechanism at oscillatory conditions is not well-understood. The systems composed of chlorite and thiocyanate (C102- + SCN-)13 or of chlorite and thiourea (C102-+ CS(NH2)2)l4or thiourea derivatives comprise the remaining members of this subclass. Both the thiocyanateand the thiourea systems exhibit a remarkablevariety of dynamical behavior, including aperiodicity.I5 The chloritethiourea reaction,14studied both in batch and in flow conditions over a wide range of reactant concentrations and their relative ratios, solution pH, and flow rates, displays bistability between two stationary states, simple oscillation, complex multipeaked oscillation, birhythmicity (bistability between two different oscillatory states), and aperiodicity. Deterministicchaos in chemical systems is a phenomenon that requires verification by precise experimentation and thorough analysis to distinguishthe observed aperiodicity from experimental artifacts.5c Thus, experimentsshould establish one of the known routes to chaos,l6 and the data should be analyzed using the tools of dynamical systems theory.” Rigorous application of these procedures to the BZ has reaffirmed “low-flow rate” deterministic chaos and the absence of determinismin the aperiodicity observed in the “high-flow rate” region.18 Herein we investigate aperiodic oscillations in the C102--CS(NH2)2 reaction carried out in a
+
0022-3654/93/2097-7258%04.00/0
CSTR in order to distinguishthis behavior as truly deterministic chaos. At previously unexplored conditions, homoclinic chaos is found in the chlorite-thiourea system in a scenario resembling that observed experimentallyin the homogeneous BZ,’9 as well as in the electrodissolution of copper in sulfuric acid.20 Data are characterized by their Fourier spectra, phase portraits of the reconstructed attractors (reconstructedby the method of singular value decomposition21), Poincard sections, return maps, and calculated generalized Renyi dimensions.22 These results demonstrate and confirm the existenceof deterministic chemical choas in this fascinating system.
Experimental Section Fresh stock solutions of sodium chlorite (Aldrich) were prepared daily by dissolution in distilled and deionized water. Before filling the stock solution to volume, 10-3M NaOH was added to stabilize the chlorite. Stock solutions of CS(NH2)2 and potassium hydrogen phthalate ( C ~ H S K O ~both ) , purchased from Merck, were stable over longer times and were therefore prepared as necessary. The inflow solutions were freshly prepared immediately prior to carrying out the experiments. To generate pH = 2.5 of the reaction mixture inside the CSTR, sulfuric acid (5.5 X 1V M) was added to the phthalate buffer input solution. A cylindrical CSTR of 3.8-mL volume and with baffles was used. The stirring rate was controlled by a magnetic bar stirred at 850 rpm. A piston-driven high-precision pump directed the flow of solutions into the CSTR at an electronically controlled rate.23 The syringescontaining the inflow solutions,the tubing connecting the syringes to the CSTR, and the CSTR were thermallyregulated by a circulating water bath. Oscillations were monitored with a Pt redoxelectrode containing a Ag/AgCl/KCl internal reference (Ingold). Data were collected on both a chart recorder and on a Cetera PC using an AID converter and analyzed according to previously described methods.24 ReSults
The various behaviors originally observed in the chloritethiourea system14 were found at T = 25 OC and over a wide range of different concentrations, solution pH, and flow rates. Presently, various regions of phase space have been probed in order to optimizeconditions in such a way that the chaos and its associated route could be observed by varying a single bifurcationparameter while all other variables were held constant. Beginning the examinationof dynamical behavior at [Cl02-]0 = 2.96 X 10-3 M, [CS(NH2)2]0= 1.2OX l0-3M (thesymbol [ lodenotesthereactor concentration), and the flow rate (expressed as inverse residence times) ko = 1.04 X 10-3 s-1, the system was monitored as the temperaturewas increased over the range of T = 1 7 4 5 OC (Figure la-d). With the corresponding increases in temperature, the 0 1993 American Chemical Society
Homoclinicity in the Chlorite-Thiourea Oscillator a
The Journal of Physical Chemistry, Vol. 97, No. 28, I993 7259 c
1700
-
1600
'ii
'?
'I
1200
i
9
1200
3
3 P 8
700 800 200
0
5000
10000
15m
0
20000
3600
Time [SI
d
1600
-
1200
9
.E: L
2 P
$
800
400
1500
E
Y
5
1200
8 900
1
i
0
7200
3600
0
10800
800
'I
1600
2400
Time [SI
Time [SI
Figure 1. Dynamical behavior in response to temperature changes at [ClOz-jo= 2.96 s-1 at T = (a) 17, (b) 30, (c) 40, and (d) 45 OC.
a
7200
Time [SI
X
lo-' M, [CS(NHz)z]o= 1.20 X lo-' M,
= 1.04 X 10-3
0.016
0.012
D
9
$
0.008
3
0.004
.s
0
0.1
0
0.3
0.2
0.4
Omega [rad/sI
d
2nl
e
P o i ncoro'
1-0 Hop
. ..
:.s..!
C
f r o m NNR
Diqi-spectrum
b
I IIn1
x
-2
0
2
4
6
9
Figure 2. Analysis of chaos in Figure IC in terms of (a) Fourier spectra, (b) reconstructed attractor, (c) Poincare section, (d) 1-D map, and (e) Renyi dimensions.
time series show periodicity followed by increasing complexity with the emergence of small- and medium-sized peaks, aperiodicity, and another periodic state. In Figure 2a-e appears the Fourier spectra, attractor, Poincart section, return map constructed from the plot of Xn+l vs X,, of the Poincare section, and
the calculated Renyi dimensions for the aperiodic time series (Figure IC). The power spectrum shows broad band noise, the phase portrait shows a spiral-type"strange" attractor, thePoincart section is a continuous curve, the 1-D map shows a single maximum, and the Hausdorff dimension is greater than two (Do
Doona et al.
7260 The Journal of Physical Chemistry, Vol. 97, No. 28, 1993
1
400
I
I
0
I200
2400
3600
0
2500
Time [SI
C
'$
7500
Time [SI
4
1000
5000
0.04
0.03 800
n 2
0.02
I
3 8
600 0.01
400
0
700
1400
2100
0.1
0
Time [SI
0.03
'I
e
aB
0.3
0.4
Omega [rads]
S
0.2
0.02
1
700
,
i
0.01
a"
I 0
2000
0
0
4000
0.1
Time [SI
0.2
0.3
0.4
0.3
0.4
Omega [rad/s]
0.03 m
'I a
0.02
700
'n
B
a"
400
'
0
I 2000
4000
0
0.1
0.2
Omega [rad/sl Time [SI F i p e 3. Response of dynamical behavior to flow rate changes at [ClOz-]o = 2.96 X l t 3 M, [CS(NH2)2]0 = 1.22 X l t 3 M, T = 40 OC, and ko = (a) 5.59 X 10-3, (b) 1.72 X l t 3 , (c) 7.31 X lo-", (d) 6.40 X l w , and (e) 5.13 X lo-' s-l. The Fourier spectra corresponding to c,e, and g are shown
in d,f, and h, respectively.
= 2.3), all of which are characteristic of deterministic chaos. Examining the system under the same external constraints with T = 40 O C and varying only the flow rate reveals similar interesting behavior. At the highest flow rate setting of our pump (ko = 2.05 X 10-2 d),the system shows large relaxation oscillations (data not shown) similar to those seen in Figure la. As the flow rate is decreased, small-amplitude oscillations and medium-sized spikes emerge between the large-amplitudebursts.
Illustrated in Figure 3a-c,e,g is the increasing complexity that develops in the oscillations with decreasing flow rate, which parallels the response to increasing temperature (Figure 1a-d). The aperiodic time series observed in Figure 3b is also analyzed by the tools of dynamicalsystems theory (Figure 4a-f). A portion of the "strange" attractor, shown in Figure 4b, magnifies the spiralling behavior occurring at the center of the attractor in Figure 4a. The evidence for deterministicchaotic motion is seen
The Journal of Physical Chemistry, Vol. 97, No. 28, 1993 7261
Homoclinicity in the Chlorite-Thiourea Oscillator
I
C 0.006
11
' : i l r
d
f
* *
f
I
0
0
0.2
0.1
Omega
f
0lqi
0.3
0.4
z
[rads] f r o m NNR
-spectrum
2.6
-6.1
-2
0
2
L
6
4
Figure 4. Analysis of chaos from Figure 3b,d. For Figure 3d, (a) the entire attractor, (b) spiralling behavior, (c) Fourier spectra, (d) PoincarC section, (e) 1-D map, and (f) Renyi dimensions.
in the broad banded power spectrum, continuousPoincart section, single-extremum 1-D map, and the Hausdorff dimension (DO= 2.3, Figure 4c-f, respectively). Decreasing the flow rate below the aperiodic regime reveals further interestingbehavior. Observed first is twepeaked complex oscillations (Figure 3c) followed by a period-2 oscillatory state (Figure 3e), which undergoes a reverse period4oubling bifurcation with a further decreasein flow rate to yield a simple periodic state (Figure 3g). This bifurcation is determined from the halving of the fundamental frequency observed in the corresponding Fourier spectra (Figure 3d,f,h). The system settles into a stationary state at ko = 3.63 X 10-4 s-l. Searching for hysteresis by now raising the flow rate shows the steady-state persists until reaching the critical value of ko = 1.08 X 1V s-1, at which the system undergoes a sudden discontinuous transition and exhibits aperiodic oscillations. The transition point of this system is sensitive to the magnitude of the increments in which the flow rate was increased; with larger increments the transition to complex behavior occurs at lower flow rates. Despite changing the conditionsof the system, similar trends in dynamical behavior are observed. Homoclinic behavior is seen at higher flow rates (Figure sa), multipeaked complex oscillations at intermediate flow rates (Figure 5b,c), and simple periodicity at lower flow rates (Figure 5d).
Discussion Analysis of the aperiodicity seen in Figures I C and 3b reveals deterministic chaos as characterized in terms of the reconstructed attractor, Fourier spectra, Poincart section, 1-D map, and Renyi dimensions. The abrupt transition to nonperiodic behavior is associated with hysteresis, and the origin of the chaos therefore appears to be homoclini~ity,2~ probably of the Shil'nikov type.16 As discussed in its experimental observation in the BZ,19 homoclinic chaos shows some special properties, whose applications to the C102--CS(NH2)2 system are described below. Whether temperature or flow rate serves as the bifurcation parameter, similar trends in qualitative behavior are observed. Inspecting the time series provides some clues to the occurrence of homoclinicity. Time series displaying sequences of smaller oscillations, both irregular in the number of peaks and whose successive amplitudes grow in an exponential fashion, followed by large relaxation oscillationsare an indication of homoclinicity. This behavior is better resolved with flow rate acting as the control parameter. As many as nine smaller cycles can be distinguished between bursts (Figure 3b). The spiralling centers of the reconstructed attractors (Figures 2a, 4a,b) also signal homoclinicity. Beginning in the general vicinity of a saddle focus (Figure 4b), the trajectory traces orbits that spiral away from the saddle focus (the number of spirals around the focus corresponds to the
Doona et al.
7262 The Journal of Physical Chemistry, Vol. 97, No. 28, 1993 a
C
1700
d i
57
‘I
9 3 I
1100
B
1800
1100
I
B
&
5001 0
,
,
800
1600
,{
400
2400
0
Time [SI
b
1000
1500
Time [SI
d
1700
1800
-
,B
b
500
t i
1200
B
B
1200
I
I
B
&
700
600 0
900
1800
2700
0
2000
4000
Time [SI Time [SI Figure 5. Dynamical behavior in the chloritbthiourea system in other regions of phase space at T = 45 ‘C and all other conditions as in Figure 1: k0 = (a) 4.73 X 10-3 s-l and (b) 2.21 X l p 3and 2.67 X 10-3s-I (arrow indicates thechange inflow rate). With [C102-]0 = 3.55 X l t 3 M,[CS(NH2)2]0 = 1.52 X 10-3 M,pH = 2.5, T = 25 OC, and k,~,= (c) 4.12 X 10-3 s-l and (d) 1.33 X le3s-l.
Figure 6. Spiralling trajectories and aperiodicity arising from a pleated slow manifold effect” (see text).
number of small and medium peaks occurring between large oscillations)and departs on a large excursionaround the attractor, retuning to the saddle focus, where it is once more reinjected. “Spiral”-type attractors involving a reinjection process can arise from a pleated slow manifold effect as described in a model R B ~ s l e r .Thus, ~ ~ the trajectories leave the slow manifold, reverse direction by making a loop on the upper part of the manifold and then are reinjected in the region of the unstable (saddle) focus on the lower surface of the manifold (Figure 6). The spiralling out, large excursion, and reinjection process are evident in the reconstructed attractors. The interaction of a local subcritical Hopf bifurcation and a global homoclinic bifurcation are the source of homoclinic chaos in both the homogeneous BZ19and the electrochemicaloscillator.20 A homoclinic connection arises from the collision of a saddle point and a limit cycle, and homoclinic chaos is distinguished by the coincidence of a discontinuous transition from a stationary state to aperiodicity and a hysteresis phenomenon.25 Chaos in the C102--CS(NH2)2 oscillator arises from a similar homoclinic source. The system shows an analogous bifurcation structure with an abrupt transition from a stationary state to aperiodicity
occurring in the vicinity of a subcritical Hopf bifurcation. The bursting behavior seen in the time series and the spiralling strange attractor support this interpretation. The deterministic character of the homoclinic chaos in the chlorite-thiourea system is reinforced by the broad banded noise of the Fourier spectrum, the continuous curve of the Poincar6 section, a well-defined maximum appearing in the 1-D map, and the calculated Renyi dimensions above 2. A system of ordinary differential equations (ODE) that simulates homoclinic chaos was presented by Argoul et al.19 and adopted and modified by Hudson20 without any mechanistic studies. The next step would be to interpret the ODE in terms of a chemical model. Recent kinetics studies28 have greatly advanced the understanding of the mechanism of the complex oxidation of thiourea by chlorite and found an overall stoichiometry of
A mechanism comprised of 13 steps simulated the autocatalytic production of the intermediate C 1 0 ~in batch conditions at pH C 3 and in unbuffered solution in good agreement with experiment.28 Further developments are needed to construct a model to account for the homoclinic chaos observed in the C102--CS(NH2)2 system.
Conclusions The experimental evidence and its analysis presented herein support the existence of deterministic chaos arising from homoclinic conditions in the oscillatory oxidation of thiourea by chlorite. These findings are consistent with the experimental observations of Shil’nikov-type homoclinic chaos in other syst e m ~ . It ~ is ~ of . ~interest ~ to probe other regions of phase space or to extend to this system physical methods comparable to those employed with the BZ reaction (e.g. the mutual coupling of
Homoclinicity in the Chlorite-Thiourea Oscillator oscillators,23~29 delayed feedback,30 and chaos controP) to uncover further nonlinear behavior in this system.
Acknowledgment. We gratefully acknowledge the National Science Foundation-Division of International ProgramsResearch at Foreign Centers of Excellence for providing a postdoctoral research fellowship (C.J.D.) and financial support from the Stiftung Volkswagenwerk and the Fonds der Chemischen Industrie. We thank Dr. A. Miinster for helpful discussions and Th. Kruel for implementing the computer programs used in the above analysis. References and Notes (1) (a) Belousov, E. P. Sbornik Referatov po Radiatsionnoi Meditsine; Medgiz: Moscow, 1958; p 145. (b) Zhabotinskii, A. M. Eiojzika 1964, 9, 306. (2) (a) Argoul, F.; Arneodo, A.; Richetti, P.; Roux, J. C. J. Chem. Phys. 1987, 86, 3325. (b) Richetti, P.; Roux, J. C.; Argoul, F.; Ameodo, F. J . Chem. Phys. 1987, 86, 3339. (3) Pomeau, Y.; Manneville, P. Commun. Math. Phys. 1980, 74, 189. (4) Maselko, J.; Swinney, H. L. J . Chem. Phys. 1986,85, 6430. ( 5 ) (a) Schmitz, R. A.; Graziani, K. R.; Hudson, J. L. J. Chem. Phys. 1977,67,3040. (b) Roux, J. C.; Simoyi, R. H.; Swinney, H. L. Physica 1983, 8D, 257. (c) Turner, J.; Roux, J. C.; McCormick, W. D.; Swinney, H. L. Phys. Lett. A 1981,85,9. (d) Coffman, K. C.;McCormick, W. D.;Noszticzius, 2.;Simoyi, R. H.; Swinney, H. L. J. Chem.Phys. 1987,86,119. (e)Schneider, F. W.; Mlinster, A. F. J . Phys. Chem. 1991, 95, 2130. (6) Epstein, I. R.; Orbin, M. In Oscillations and Travelling Waves in Chemical Systems; Field, R. J., Burger, M., Eds.; Wiley-Interscience: New York, 1985; p 257. (7) Dateo, C.; Orbin, M.; De Kepper, P.; Epstein, I. R. J. Chem. Soc. .. 1982,104, 504. (8) (a) Orbin, M.; De Kepper, P.; Epstein, I. R. J . Phys. Chem. 1982, 86. 431. Ib) Orbin. M.: Emtein. I. R. J. Phvs. Chem. 1982. 86, 3907. ’ (9) Maklko, J.; Epheh, I. R.J. Chem.-Phys. 1984, 80;3175. (10) (a) Nagypil, I.; Bawa, Gy.; Epstein, I. R. J. Am. Chem. Soc. 1986, 108,3635. (b) Szirovicza, L; Nagypil, I.; Boga, E. J . Am. Chem. Soc. 1989, 111, 2842.
The Journal of Physical Chemistry, Vol. 97, No. 28, 1993 7263 (11) Nagypil, I.; Epstein, I. R. J. Phys. Chem. 1986, 90, 6285. (12) (a) Nagyp61, I.; Epstein, I. R.; Kwtin, K. Int. J . Chem. Kinet. 1986, 18, 345. (b) NagypB1, I.; Epstein, I. R. J. Phys. Chem. 1986,90, 6285. (13) Alamgir, M.; Epstein, I. R. J. Phys. Chem. 1985,89, 361 1. (14) Alamgir, M.; Epstein, I. R. Int. J. Chem. Kinet. 1985, 17, 429. (15) Doona, C. J.; Doumbouya, S. I.; Schneider, F. W. Manuscript in preparation. (16) Swinney, H. L.; Roux, J. C. In Nonequilibrium Dynamicsin Chemical Systems; Vidal, C., Pacault, A., Eds.;Springer: Berlin, 1984, p 124. (17) Argoul, F.; Arneodo, A.; Richetti, P.; Roux, J. C.; Swinney, H. L. Acc. Chem. Res. 1987, 20, 436. (18) Blittersdorf, R.; Mdnster, A.; Schneider, F. W. J . Phys. Chem. 1992, 96, 5893. (19) Argoul, F.; Amtodo, A.; Richetti, P. Phys. Lett. A 1987, 120, 269. (20) Bassett, M. R.; Hudson, J. L. J. Phys. Chem. 1988,92, 6963. (21) Broomhead, D. S.; King, G. P. Physica 1983, ZOD, 217. (22) (a) Renyi, A. Probability Theory;North Holland Amsterdam, 1970. (b) Van der Water, W.;Schram, P. Phys. Rev. Lett. 1988, 37, 3118. (c) Badii, R.; Politi, A. Phys. Rev. Lett. 1984, 52, 1661. (23) Doumbouya, S. I.; Mdnster, A. F.; Doona, C. J.; Schneider, F. W. J . Phys. Chem. 1993,97, 1025. (24) Swinney, H. L. Physica 1983, 7 0 , 3 (reprinted in ref 25, p 332). (25) Thompson, J. M. T.; Stewart, H. B. Nonlinear Dynamics and Chaos; Wiley: Chichester, U.K., 1986. (26) (a) Shil’nikov, L. P. Soc. Math. Dokl. 1965,6, 163. (b) Shil’nikov, L. P. Math. USSR Sbornik 1970, 10, 91. (27) (a) RBssler, 0. E. 2.Naturforsch. 1976,31a, 259. (b) RBssler, 0. E. Z . Naturforsch. 1976,31a, 1168. (c) Rksler, 0. E. Ann. N.Y. Acad. Sci. 1979, 316, 376. (28) Epstein, I. R.; Kustin, K.; Simoyi, R. H. J. Phys. Chem. 1992, 96, 5852. (29) (a) Schneider, F. W.; Hauser, M. J. B.; Reising, J. Eer. Eurnsen-Ges. Phys. Chem. 1993,97,55. (b) Weiner, J.;Holz, R.; Schneider, F. W.J. Phys. Chem. 1992, 96, 8915. (30) (a) Weiner, J.; Schneider, F. W.; Bar-Eli, K. J. Phys. Chem. 1989, 93, 2705. (b) Chevalier, T.; Freund, A,; Ross, J. J. Phys. Chem. 1991, 95, 308. (c) Roesky, P.; Doumbouya, S.; Schneider, F. W.J. Phys. Chem. 1993, 97, 398. (31) Petrov, V.; Gispir, V.; Masere, J.; Showalter, K. Nature 1993,361, 240.
