Dynamical Behavior of Coupled Chemical Oscillators: C hlorit e

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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
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20’l
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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.
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