5305 J. Phys. Chem. 1984, 88, 5305-5308 Dynamical Behavior of Coupled Chemical Oscillators: Chlorite-Thiosulf ate- I odide-I odine‘ Jerzy Maselko2 and Irving R. Epstein* Department of Chemistry, Brandeis University, Waltham, Massachusetts 02254 (Received: January 30, 1984) Bifurcation (phase) diagrams have been determined for systems consisting of C102-, S2O3’-, and I- or CIOz-, S20;-, I-, and I2 at pH 4 in a stirred tank reactor. From an analysis of the experimental behavior, topological models (phase portraits) are proposed which account for the observed dynamics. These systems display the features of the component C102--I- and C10<-S2032- subsystems as well as new modes of behavior such as tristability and birhythmicity. Introduction The investigation of chemical systems which possess multiple attractors-bi~tability,~tristability,“ birhythmi~ity,~ and other combinations-has become one of the most active areas in chemical dynamics. Progress in the mathematical analysis of dynamical systems possessing multiple attractors has paralleled experimental studies of reactions having two or more stable steady states or limit cycles. One method which has recently been exploited to construct systems which display complex multiple-attractor dynamics is to couple a pair of chemical oscillators which share a common specie^.^,^ Several phenomena of interest including birhythmicity: compound oscillation; and a Lorentz-type chaotic attractor’ have been found in a system resulting from the coupling of the Br03--Iand C102--I- oscillators. Here we report on the results of joining together the C102--S2032- and CIOz--I- reactions. The effects of introducing I2 into this composite system are also discussed. Experimental Section The reactions were carried out at 25 OC in a 27.5-mL glass stirred tank reactor (CSTR) which has been described el~ewhere.~ All chemicals were the highest grade commercially available and were used without further purification. Sodium chlorite and sodium thiosulfate solutions were prepared in M NaOH and stored in dark bottles. “Triiodide” solutions were prepared by adding an equimolar amount of solid I2 to a previously prepared iodide solution. All experiments were carried out a t pH 4 maintained with a phthalate buffer. Potential measurements were made using a Pt redox electrode vs. Hg/Hg2S0, reference electrode. In most experiments the flow rate and two of the input concentrations were fixed and the third input concentration was varied by substituting different reservoirs of the component whose effect was being probed. Some experiments were also conducted in which all reservoir concentrations were fixed and the flow rate was varied. In all cases the parameter was varied in both directions in order to reveal any hysteresis effects. After each parameter variation, sufficient time was allowed for the system to reach a stable state whether stationary, periodic, or chaotic. The C102--S2032--I-System. The chlorite-thiosulfate-iodide reaction was looked at from two points of view. First, we started from the pure C102--I- system8 and made stepwise increases in (1) Part 22 in the series ‘Systematic Design of Chemical Oscillators”. Part 21: Maselko, J.; Alamgir, M.; Epstein, I. R., submitted for publication in Phys. Reu. A . (2) Permanent address: Technical University of Wroclaw, Institute of Inorganic Chemistry and Metallurgy of Rare Elements, Wroclaw, Poland. (3) Epstein, I. R.; Dateo, C. E.; De Kepper, P.; Kustin, K. In “Nonlinear Phenomena in Chemical Dynamics”; Vidal, C., Pacault, A., Eds.; SpringerVerlag: West Berlin, 1982; pp 188-191. (4) Orban, M.; Dateo, C.; De Kepper, P.; Epstein, I. R. J . Am. Chem. SOC. 1982, 104, 5911-5918. (5) Alamgir, M.; Epstein, I. R. J. Am. Chem. SOC.1983, 105, 2500-2502. (6) Maselko, J. Chem. Phys. 1983, 78, 381-389. (7) De Kepper, P.;Epstein, I. R.; Kustin, K. J . Am. Chem. SOC.1981, 103, 6121-6127. 0022-3654/84/2088-5305$01.50/0 the SzO?- input in order to study the transition between the single and the coupled oscillators. In another set of experiments, the starting point was taken as the pure C10<-S20?- oscillator9 and I- was added in increasing amounts. The behavior observed is summarized in the bifurcation diagrams of Figures 1 and 2. It is useful to attempt to understand the dynamical behavior of the system in terms of the topological models or phase portraits presented in Figure 3. We have divided Figure 1 into four different regions according to the types of transitions observed for a fixed [S2032-]oas [C102-], is increased. These transitions are shown schematically in the diagrams at the right of Figure 1. In the ~ O W - [ S ~ O region ~ ~ -A, ] ~the behavior is essentially that of the Cl0a-I- subsystem, Le., transition from one steady state to another with hysteresis. This sequence is shown in panel A of Figure 3, where the numbers 1 and 2 between successive frames correspond to the bifurcation lines 1 and 2 in the phase diagram of Figure 1. As [S2032-]ois increased into region B, the behavior becomes more complicated. Now, with increasing [C102-]o,the system jumps from the lower (Le., low potential) steady state to an oscillatory state on curve 2. A further increase in the chlorite input causes a transition from the oscillatory state to the upper steady state on curve 5. If we now reverse the sequence of operations and lower [C102-],, the system jumps from the upper to the lower state on curve 1. By starting from an intermediate [ClOp], value and the oscillatory state, we may observe the transition to the lower state on curve 3. The schematic diagram shows how the above behavior implies the existence of tristability among the two steady states and oscillatory state in region I11 between curves 2 and 3. A phase portrait which accounts for this behavior is shown in panel b of Figure 3. The model suggests that the transition from the oscillatory to the steady state which occurs on curve 3 takes place via the formation of a homoclinic orbit formed as the limit cycle and the saddle point collide in Figure 3 Bd. The approach to such an orbit should be characterized by the period of oscillation diverging to infinity.1° Figure 4 shows that this divergence is in fact observed experimentally. At still higher [SZO3*-],(region C of Figure l ) , a different sequence of states is observed as [ClO,-], is increased. Starting from the initial low-potential steady state at low [C102-]o, we observe a transition to oscillation as curve 2 is crossed, followed by a jump to the upper steady state on curve 5 . If [C1OZ-], is now lowered, the reverse transitions to the oscillatory and lower steady states occur on curves 4 and 3, respectively. Again, there is a region of tristability. The corresponding phase portrait is shown in Figure 3C. When the input thiosulfate concentration is very high (region D of Figure l), the only transition observed on increasing [CIOJo ~ ~~~ (8) Dateo, C. E.; Orban, M.; De Kepper, P.; Epstein, I. R. J . Am. Chem. SOC.1982, 104, 504-509. (9) Orban, M.; De Kepper, P.; Epstein, I. R. J . Phys. Chem. 1982, 86, 431-433. (10) Maselko, J. Chem. Phys. 1982, 67, 17-26. 0 1984 American Chemical Society 5306 The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 Maselko and Epstein IS 7 '. fl\ ob' 2 lI' 3 IOe [ClO& 5 4 6 ,M Figure 1. Bifurcation (phase) diagram for the C102--S2032--I- system M and reciprocal residence time ko = 5 X with [I-], = 6.25 X s-l. Numbers in transition scheme at right correspond to numbered bifurcation curves in diagram at left. Roman numerals denote behavior observed in each region: I, low-potential steady-state SSI, 11, bistability between SSI and oscillatory state 0; 111, tristability among SSI, 0, and high-potential steady-state SSII; IV, bistability between SSI and SSII; V, SSII; VI, bistability between SSII and 0. '.' f-. 6- 5- Figure 3. Phase portraits corresponding to behavior shown in Figures 1 and 2 . Numbers between panels correspond to numbered bifurcation curves of those figures. t t l 4 5 3 2 2- I- .. _ - _ __ ___ F I 2 3 IO2 [ C l O i l , 4 , 5 ,M Figure 2. Bifurcation diagram for the C~OC-S~O~~--Isystem with [S20?-],, = 6.25 X lo-' M and ko = 5 X lo-' s-l. Bifurcation curves and regions numbered as in Figure 1 (see text). Also, in regions numbered VI1 and VIII, only oscillatory state 0 is stable. from zero is from the lower to the upper steady state at curve 2. When [CIOz-]ois decreased, transitions take place to the oscillatory state on curve 4 and to the lower steady state on curve 3. A transition from oscillation to the upper steady state may be observed on curve 5 if one increases [CIOz-]o starting from an oscillatory state between curves 3 and 5. This phenomenon of being able to generate a particular transition only if one starts from a certain parameter range and state is characteristic of a tristable system. The phase portrait in this region appears in Figure 3D. The phase diagram describing the transition from the chlorite-thiosulfate subsystem to the full coupled oscillator is presented in Figure 2. At low [I-lO,we observe a region of oscillations characteristic of the chlorite-thiosulfate system under these conditions. In an idlermediate range of [I-lO(region E), a new I O' '310 315 410 10' CC1O~l0,M Figure 4. Dependence of oscillation period T o n [C102-], in neighborM, [SZOp2-],= 6 X hood of curve 3 in Figure 1 ([I-], = 6.25 X M, ko = 5 X lo-' s-'). Divergence of Timplies formation of a homoclinic orbit. sequence is formed. When [ClOcl0is increased, the system jumps from the low-potential state to oscillation on curve 2 and then to the upper state on curve 5 . The reverse transitions occur along curves 4 and 3. We thus have two independent hysteresis loops. The phase portrait corresponding to this behavior is shown in Figure 3E. Only at high [I-lOin region C do we observe tristability. The behavior here is essentially identical with that in region C of Figure 1. The C10z--S2032--13-System. Since I2 is produced in the reaction of chlorite with iodide, it would seem unlikely that adding iodine to the C10z--Sz032--I- system would produce any new behavior. However, we find that if the iodide flow is replaced by an equimolar solution of I2 and I-, referred to for simplicity as 13-,then the phase diagram becomes even more complex owing The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 5307 Chlorite-Thiosulfate-Iodide-Iodine - /[' 4 1 , 5 IO 12 Figure 7. Bifurcation diagram for the C102--S20,2--I< system with [S2O?-l0 = 5 X lo4 M and ko = 4.8 X s-'. Bifurcation curves and regions numbered as in Figure 5. Also: VII, 011 only; VIII, birhythmicity between 01 and 011; IX, bistability between SSI and 011. A . ".. 1/2/3 8 io3[CIO,-I,, M .. *..I 6 IO 20 I5 IO4 tClO&, 600\ 25 M Figure 5. Bifurcation diagram for the C10y-S2032--I,-system with [I3-],, = 1.25 x lo-' M and ko = 4.8 X lo-) s-I: I, low-potential steady-stateSSI; 11, bistability between SSI and oscillation 01characteristic of pure C102--13- subsystem; 111, bistability between SSI and high-potential steady-state SSII; IV, tristability among SSI, SSII, and oscillation 011 characteristic of coupled system; V, bistability between SSI and 011; VI, SSII. 400 E 6 300 B 200 ..5 100 0 Time H 10 min Figure 8. Birhythmicity, Le., two different stable oscillatory states for the same set of parameters. At points indicated by arrows, flow rate is first decreased and then increased to values shown ([CIO,-], = 6.3 X lo4 M, [s20?-], = 5 x 10-4 M, [I,-], = 1.5 x 10-4 M). Figure 6. Phase portraits corresponding to behavior shown in Figures 5 and 7. to the presence of a second oscillatory state. It is, of course, possible that the behavior observed for our C10;-S2032--13systems is accessible for a C101-S2032--I- composition not investigated in these experiments. The phase diagram obtained by starting from the C102--13subsystem and adding increasing amounts of SZO3'- is shown in Figure 5 . The pure subsystem shows an oscillatory state for intermediate values of [C102-],. As we increase the chlorite input, the system jumps from the lower to the upper state on curve 3. On decreasing [C102-]o,we observe successive transitions to the oscillatory and lower states on curves 2 and 1, respectively. The topological model shown in Figure 6A indicates that the bifurcation at curve 1 occurs via formation of a homoclinic orbit. When [S2032-]o is sufficiently high, as in region B of Figure 5 , a second oscillatory state appears at relatively high [C1O2-],. The sequence of transitions observed as [C102-], is increased is equivalent to that of region A of Figure 5 followed by that of region B of Figure 1. Similarly, the phase portrait (Figure 6B) may be constructed by combining the models of Figures 6A and 3B. At still higher [S,032-]o, the tristable region between curves 4 and 5 vanishes, and we have the sequence of transitions shown in region C of Figure 5. As Figure 6C shows, the bifurcation on curve 6 is apparently of the saddle-node type. Comparing Figures 6Be and 6Ce suggests that on increasing [S2032-] there is a smooth transition from a homoclinic to a saddle-node bifurcation. The transitions postulated in Figure 6 between one type of bifurcation and another (cf. Figure 6Be,Ce) as an input parameter, in this case [S2032-]ois changed, are of considerable interest. Unfortunately, at present no general theory is available to describe this type of behavior, but perhaps such experimental data will provide an incentive for its development. The existence of two different types of oscillation in the C102--S2032--13- system suggests that birhythmicity may be possible in this reaction. In Figure 7, we see that this is indeed the case for [I3-lOvalues somewhat lower than in Figure 5 . As is increased, the birhythmicity found between curves 2 and 8 in region D of Figure 7 gives way to a separation of the oscillatory states and a bifurcation sequence resembling that found in region B of Figure 5 . The phase portrait in Figure 6D shows the generation of the two concentric stable limit cycles separated by an unstable limit cycle. This behavior, which is consistent with the topological models for the system at other concentrations, differs from that of the only other birthythmic system for which a phase portrait analysis has been carried out. In the C102--Br03--I- ~ y s t e m , ~ the two stable limit cycles are not concentric but are separated by a saddle point. Since as a function of the input concentrations the range of birhythmicity is relatively narrow, more detailed studies of this behavior were performed by fixing the concentrations and varying the flow rate. Figure 8 shows the two different oscillatory wave forms which can occur at reciprocal residence time: ko = 1.25 5308 The Journal of Physical Chemistry, Vol. 88, No. 22, 1984 ..... 20’l ;i.. . . 17.5 e Maselko and Epstein + the occurrence of L nS oscillations for the corresponding [email protected]) of n. At low [I3-], the composition of the input flow and the resulting bifurcation sequence are the same as those of Figure 2F, with the two steady states and the L oscillation state occurring in the absence of hysteresis. As [I3-lOis increased, the S oscillation and complex L + nS states begin to appear. At values of [I3-], = 5 X M,the bifurcation sequence becomes particularly complicated. On increasing [C102-],, we observe successively steady-state I and S oscillations, a return to steady-state I, a sequence of L nS states with n decreasing from an initial value of 6, L oscillations, and finally steady-state 11. Traversing the sequence in reverse, we find hysteresis in the region marked H between the L + nS states with n = 4-6 and steady-state I. Such behavior has not been observed in the C1O2--S2O32- subsystem?,’ As [I3-], is increased further, the oscillatory regions narrow and ultimately disappear, the L region at about [I,-], = 1.35 X low5M and the L nS region at about 1.75 X M. In neither case do we observe bistability between two steady states in the region immediately above where oscillations cease as is found in systems characterized by the “cross-shaped phase diagram”.12 + “ ‘\i ’ + .. 8 0 0 . */. Figure 9. Bifurcation diagram for the C102--S2032--I< system in parameter range with complex oscillation: I, low-potential steady state; 11, high-potentialsteady state; L, large-amplitude oscillation; S , small-amplitude oscillations; arabic numbers n (1, 2, 4-6, 8-16) indicate region of L + nS complex oscillations; H, hysteresis between I and 4-6 ([S2032-]o= 1.25 X 10” M, ko = 5 X lo-) s-l). X s-l. Note that the lower frequency oscillations extend above and below the high-frequency oscillations in amplitude, as would be expected for the concentric limit cycles shown in Figure 6De. Thus far, the periodic behavior we have discussed has consisted of simple, single-peak oscillations. Previous studies” of the chlorite-thiosulfate system have revealed the existence of complex, multipeak oscillations as well. As a parameter (e.g., ko or [Sz032-]o) is varied, transitions take place from simple largeamplitude (L) oscillations through a series of complex periodic oscillations (L nS) consisting of one large-amplitude and n small-amplitude peaks per cycle, finally reaching a state of pure small-amplitude oscillation S. Values of n as large as 32 have been observed, and no hysteresis has been found in any transitions involving complex oscillatory states of the C10+Sz032-system. Figure 9 illustrates how addition of I< to the chlorite-thiosulfate reaction affects the conditions under which the various states occur. The regions marked I, 11, L, and S represent areas in which simple steady-state or periodic behavior is found. Regions designated by a single arabic number n or by a range of such numbers indicate + (11) Orban, M.;Epstein, I. R. J . Phys. Chem. 1982, 86, 3907-3910. Discussion The present study has provided further examples of the wealth of complex dynamical behavior which can result when chemical oscillators are coupled. It also shows that reasonable hypotheses may be constructed about the phase portraits even of systems which display highly complicated behavior. This analysis is aided immensely when information is available on the dependence of oscillation amplitudes and periods near bifurcation points.I0 It should be pointed out, however, that many aspects of the topological models presented here are hypotheses only, since it was not possible in all cases to obtain sufficient data to decide unambiguously among possible alternative bifurcation schemes. The phase diagrams given here represent, of necessity, only a few two-dimensional sections in a multidimensional space. We have limited the discussion to the stationary and periodic states of the system. A detailed exploration of the effects of additional species on the periodic-chaotic sequence in the C10z-S2032system” would certainly be of interest. The results obtained point to the need for further developments in bifurcation theory. While the theory of bifurcation as a single parameter is varied is relatively well developed,I3better descriptions are required of how one type of bifurcation transforms into another as a second parameter is varied, as in Figure 6. Finally, it may well prove possible from experiments like those presented here to describe and analyze these systems at the dynamical level of phase diagrams and phase portraits. However, from the point of view of the chemist, a true understanding of such reactions will require the development of a kinetic mechanism. Such studies, starting with the component CIOz--I- and C1O2-:S2O32- subsystems, are now well under way in these laboratories. Acknowledgment. This work was supported by the National Science Foundation under Grant 8204085. Discussions with Mohamed Alamgir, Kenneth Kustin, and Robert Olsen are gratefully acknowledged. (12) De Kepper, P.;Boissonade, J. In “Oscillations and Traveling Waves in Chemical Systems”; Field, R. J., Burger, M., Eds.; Wiley: New York, in press. (1 3) Guckenheimer, J.; Holmes, P. “Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields”; Springer-Verlag: New York, 1983.