501
J. Phys. Chem. 1980, 84, 501-506
Transitions from Bistability to Limit Cycle Oscillations. Theoretical Analysis and Experimental Evidence in an Open Chemical System J. Boissonade and P. De Kepper” Centre de Recherche Paul Pascal (C.N.R.S.), Domaine Universitaire, 33405 Talence Cbdex, France (Received February 15, 1979; Revised Manuscript Received July 16, 1979)
Some general relationships between bistability and limit cycle oscillations may be summarized in a typical two-dimensional cross-shaped phase diagram whose validity limits are specified. A simple model is presented and its linear and nonlinear stability is discussed. A formal kinetic scheme and two experimental examp1.e~ in Briggs-Rauscher oscillating reaction are produced as illustrations in chemical systems, and experimental evidence for a subcritical Hopf bifurcation is exhibited.
Introduction Experimental investigations of chemical oscillating reactions in a continuously stirred tank reactor (CSTR) have revealed nonequilibrium phenomena such as oscillations, bistability, cornplex oscillations, and quasi-chaotic behavior, which can be summarized in phase diagrams of the reaction~.,l-~ In fact, close relationships between some kind of oscillatory phenomena and bistability have been pointed out by several ;luthors.6-8 Our purpose is to clarify these relationships by giving them a general representation in the phase diagram and discussing a simple analytical mathematical model. A simple chemical-type scheme and two experimental examples are presented as an illustration. Theoretical Section Generul Analysis. Let us consider a system, driven by a constraint parameter that presents the classical configuratioin of two stationary stable steady states8 for C1 < < Cz (a.ccording to Figure l),with relaxation times T ~ R characterizing ithe approach of these states; in the following, index i will refer to the associated branch (1or 2). Let us also consider a negative feedback t i (on branch i) exerted on lo,where tois the externally controlled value of the constraint, and characterized by a time ~ ~ ( of5 establish) ment of the feedback effects so that, at each time, the system does not appear as being driven by lobut by an effective const:raint to- E ~ . If we suppose that T~ >> 7iR7the system relaxes quickly to a quasi-stationary state on branch i that should correspond to the instantaneous value Cl’). On brunch 2 it will be stable for - tZ< C2 (or io> C;) and unstable for - c2 > Cz (or Eo > C;).
T R (TR will be further quantitatively explicit for this example), y relaxes slowly to x via eq 111. Then €(A’) may be regarded as the implicit function E(?) introduced previously. We shall now build the phase diagram (k,X) a t a given po by analytically discussing the linear stability of the stationary state solutions xo and yo of eq I1 and I11 and numerically discussing the nonlinear behavior around these solutions. Linear Stability Analysis. For stationary state equations we have yo = xo where xo is a solution of the equation xO3- (po - k)xo + X = 0. There are three solutions for k
-
Figure 3. (k,h) phase diagram with p0 = 3 and 1 / r = 1. Full curves correspond to linear stability analysis. Dashed curves correspond to nonlinear stability analysis. Letters A-M correspond to topological configurations in Figure 4.
< p o and a single one for k given by equation W’
> pD. The normal modes are
+ ( 3 x 2 - po + 1 / 7 ) ~+ -71[ ( 3 ~ 2- po) + k] = 0
(IV)
implying the following criteria for the stability of x:
+k >0
3~2-po
3~2-pyO+
; 1 >0
We summarize the main results by setting X J k ) = 2[ .,]’I2 A,,(k) =
[‘
3312
po -
!I1’ [
3k - 21, - 7
whose graphs are tangent for k = 117. We have three roots for -A1, < X < X1, and k < po but since the middle one is always unstable (saddle point) the discussion of the stability is limited to the other two. Case 1 / 7 < po (Figure 3 ) . For k < l / r the two roots are stable, so the system is bistable for -Xlc(h) < X < +A&). For 1 / lX2,(k)1, For k > 1/3[2po (1/7)] there are no stable roots for -A&) < X < +A&), so the system undergoes limit cycle oscillations; for 1 / 3 [ 2 p ~+ ( 1 / ~ ) 1 / we ~ get the “linear” cross-shaped phase diagram shown as a solid line on Figure 3 as ex-
+
Oscillating Chemical
g
The Journal of Physical Chemistry, Vol. 84, No. 5, 7980 503
Systems
a
E
I
I
l-
Y
2
4
(r0
Figure 6. Limit (k,X)phase diagram for kLg = 3 and 1 / =~ 0. Letters correspond to topological configurations in Figure 4.
computer. Due to the symmetry of the diagram the following description is limited to X > 0. According to the general theorems of the nonlinear differential equation theory,l3?l4when k > 1 / the ~ points where 3x02 - po 1 / =~0 in eq IV, corresponding to +A,, and -Azc are bifurcation points for periodic solutions (Hopf ~ ~ for bifurcations).16J6 It can be shown a n a l y t i ~ a l l ythat k < 2p0 - 1 / in~our case, these bifurcations are subcritical and then give rise to unstable limit cycles surrounding the bifurcating solutions. The neighborhood of point P is characterized by complicated dynamical behavior due to the “interaction” between the two Hopf bifurcations along +Azc and -Azc. The different zones are indicated in Figure 3 by letters (A, B, ...) M), bounded by dashed curves and their associated topological configurations are sketched in Figure 4. This allows us to see how their portraits metamorphose from one type to another as a function of k and A. Some of these configurations have been already recognized by Uppal et al.17 using a first-order exothermic reaction in a CSTR as a model system. Let us now proceed to a brief discussion of the nature of the zones whose existence is not predicted by the linear stability analysis. Zone C is characterized by a simple limit cycle surrounding an unstable stationary state. The associated oscillations appear, for k < 2p0 - 1 / ~with , nul amplitude a t X = &Azc where the bifurcation is supercritical. On the other hand, for k < 2p0 - 1/7, where the bifurcation is subcritical, finite relaxation oscillations occur before the stable stationary solutions become locally unstable which implies bistability between a steady state and an oscillating one in zones B and H and tristability between two steady states and an oscillating one in zone L. In zone I an unstable limit cycle whose amplitude goes to zero on +Azc is surrounding the stable positive solution which implies excitability properties for the stable negative one. Zones F, G,J , and K have very special characteristics due to the “interaction” of the two Hopf bifurcations along +Azc and -A2,. Zones J and G present two similar configurations including a stable limit cycle, corresponding to the relaxation oscillation surrounding an unstable singularity and a negative stable stationary state whose uttractive domain is bounded by two close spirals coiling around the unstable singularity (a positive unstable stationary state for zone G and an unstable limit cycle surrounding a positive stable stationary state for zone J). Continuity between zones G and J is ensured along +Azc
+
Figure 4. Sketch of the different topological configuration in response space.
Figure 5. (k,X) phase diagram for k,, = 3 and 117 correspond to topological configurations in Figure 4.
> 3.
Letters
pected from the general analysis. The area of the instability zone decreases as a function of 7 and goes to zero for 7 < l / p o (Figure 5). This gives us a quantitative expression for the condition T >> TR by setting T R = l/po, and, as we shall show further, in the limit 1 / TR is not strictly ensured. A Chemical Scheme. These arguments could be applied to any nonlinear systems but in the following we shall limit ourselves to the case of chemical systems. We shall give first, as a kinetic example, the following simple chemical scheme:
-
kl
A-X
lo5.
0,05
-
0,04
A o steady state 1
A stcrdy state 2 oscillating state
c 0
‘I 1
[1210 (10-4 mol 1-1)
B+X-2X
(2)
+X
products
(3)
X’
(4)
but the general characteristics of the phase diagram are not m ~ d i f i e d . ~
(5)
Experimental Section The Briggs-Rauscher (BR)18oscillating reaction which has been extensively s t ~ d i e din~our ~ ~laboratory ~ in a CSTR shows, among many other phenomena, clear experimental evidence of transitions of the previously discussed type in constraint space. In the following, we shall report two such examples. One deals with bistability between two steady states which will give rise to an oscillating state and the other refers to bistability between a steady state and an oscillating state which will undergo a transition to complex oscillations.1-5~19-21The experimental setup has been described elsewhere.22 The external parameter or constraints for this system are [KI03]0, [H20&, [CHz(COOH)~lo, [HC10410,[MnS0410,0 the residence time, and T the temperature. The quantities [Ail0 are the concentrations species Ai would attain if no chemical reaction took place in the tank. The responses measured are the optical density at 460 nm which is proportional to the iodine concentrationz3and the chemical potential E between a platinum rod electrode and a Hg-HgzS04 reference electrode. The first example summarized in Figure 8 represents part of the phase diagram in the plane ([Iz]o,[KI03]J,all other constraint values remaining constant. Note that in this experiment, which has been proposed elsewhere as an illustration of “inverted regulation” on the iodine,24there
xk3
k4
k5
B+X’-Y
Y
k, =
(1) kz
D
Figure 7. Chemical scheme. (Js,ks)linear phase diagram with k , = 2, k, = 2.5,k, = 5.5,k, = kE = 1, J A = 30, JD = 75,and
ks
X’
+ products
(6)
which is based on the autocatalytical step (2). In a flow reactor, JA,JB, and JD,the incoming fluxes of the respective species A, B, and D, and kE, the inverse of the resident time, are controlled externally. If one derives the kinetic equations under the assumptions of the law of mass action, it has been shown in ref 7 that steps 1-4 may give a bistability and that steps 4-6 may be handled as a feedback on the constraint parameter J B of the autocatalytic step, inducing oscillations of the studied type for suitable values of the amplitude feedback parameter, kg. We have numerically computed the normal modes associated to the stationary states of these equations and constructed the phase diagram in space (JB,k5),all other parameters being constant (Figure 7). We get the characteristic cross-shaped diagram with bistability and oscillations according to the predictions of linear analysis. Due to the high values of the feedback evolution time T = l / k 4 = 100 (with rR= 1) the time scale condition is fulfilled and the linear stability diagram leads to good predictions for this system’s evolution but deviations are observed for lower values of 7 (7 = 10). When one of the branches is an intrinsic oscillating state the oscillating region becomes a complex oscillation region
FI ure 8. BR. ([12]o,[KI03]o) phase diagram for [H20plo = 0.34 mol L- , [CHz(COOH)2]o= 0.0015 mol L-’, [HC10410 = 0.057 mol L-‘, [FJ~~ISO,]~ = 0.004 mol L-l, = 2.6 mn, and T = 20 OC.
7
Oscillating Chemical Systems
,
The Journal of Physical Chemistry, Vol. 84, No. 5, 1980 505 11~1(10-4
A
1-1)
C
0
.4,85
.3.25
'I[KIOrlo :0,035
-2.44
mal 1-l
[12lo: 4,7 10-5
.1.60
mol 1-l
[KI03lo 10,035 mol I-' 200
stater
#.....---
180
7
-1.30
=(A)
.0,97
[1210: 5.7 10-5 mol 1.1
[KlOslor 0,035 mol k1
.0,65
[1210: 5,7 1 0 - ~ mol I.'
8: 5.45 min
CK10~~0:O.OM mol Id [1210: 4.7 10.5 mol 1.1
O
[12]0:3,4
0
I
10-5 mol I-'
IO.
L
I
I
2
i
I
~ 1 2 1(10-4 mol 1-1)
Figure 9. BR. Projection in response space ([I2],€) of different steady states 1 and 2 and a limit cycle trajectory. Dots on this trajectory are time equidistant at 12 s. All these states differ only by the values of [I2lO and [K1O3],,. For all other constraints see Figure 8.
0'04
4.84
-
[K10310: 0,030 ml I-! d
e:
8 : 5.06 min
M
Figure 11. BR. Iodlne concentration evolution for different values of residence time 8: (A) 5.45 min; (B) 5.06 min; (C) 4.84 min, with [CH2(COOH)2]o= 0.022mol L-'. For all other constraints see Figure
[KIOsIor 0,030 mol 1.l Oscillattnq state (-*-*-)
160
5
ElmVl 240
*I 200
I
1
160
t
-
6 [12i i10 I Figure 12. BR. Projection of steady state 2 (A)in the response space ( [ I ] , E )oscillation state trajectory (-*-*-) with [KIO lo = 0.026mol L-q [IJO = 3.25 X lo4 mol L-', [H2Oplo= 0.33mol L- 9, [CH,(COOH)2]o = 0.0145mol L-I, [HC10410= 0.057 mol L-I, [MnSO,], = 0.004 mol L-I, 8 = 2.7 min, and T = 21 OC. 2
steady state
v I
I
orillalxq able11 complex acillatimr
Figure I O . BR. (1/8,[CH,(COOH),]o) phase diagram for [KI0310 0.047 mol L-I, [H20plo= 1.0 mol L- , [HCIO4I0 = 0.056 mol L- , [MnS04Jo= 0.004 mol L", and T = 24.5 OC.
is an inlet flow of iodifie [Iz], which must be distinguished from the response [I2]: the iodine concentration of the reacting mixture. Two steady states (state 1 and state 2) are observed in this space. Their stability frontiers are crossed lines dividing this plane in four regions: (a) a region of stability of steady state 1 characterized by high iodine coiicentr,ztion and high chemical potential E values; (b) a region of stability of steady state 2 which has low iodine concentration and low chemical potential E values; (c) a region of histability between states 1 and 2 (in this region steady state 1is reached from region a and steady state 2 from region b); (d) a region of sustained oscillations: neither state 1 nor 2 are stable in this region but they correspond to the pseudosteady states of the oscillatory phenomena. This can easily be seen in Figure 9 which represents projections, in the response space (&],E), of a limit cycle of region d and of steady states 1 and 2 for different values of the constraints corresponding, respectively, to region a and b. On the cyclic trajectory dots are time equidistant; this allows us to show that there are two parts on the trajectory with relatively low velocity (high density of dots) which correspond respectively in the response space ([ &],E)to evolutions in the quasi-stationary states 1 and 2.L9 The second example is depicted in Figure 10; it represents our now classical cross-shaped phase diagram in the
4
'
5mp~
constraints space (8,[CH2(COOH),],). In this experiment the four regions are as follows: (a) the region of stability of an iodine high concentration steady state I; (b) the region of simple sustained oscillations (state 11); (c) the region of bistability between states I and 11; (d) the region of complex oscillations. Figure 11B represents the typical evolution of the iodine concentration for these complex oscillations. When compared with the evolution of the neighboring steady state (Figure 11A) the simple sustained oscillations (Figure 1lC) one can readily see that this complex oscillation consists in a periodic transition from a pseudosteady state I to a pseudooscillatory state I1 and vice versa, just the kind of behavior our analysis predicts. We must add that, in both experiments, excitability phenomena are observed in region a and b. Furthermore, the oscillations in region d burst with full amplitude at the transition lines. As we noted above, this is generally characteristic of a Hopf subcritical bifurcation and should imply the existence of a bistability zone along these lines. This was not experimentally observed in the constraint domain reported in Figures 8 and 10. These regions might be beyond our experimental accuracy which is consistent respectively with the very stiff transitions between the two pseudosteady states of the limit cycle ([I,] is fairly constant and transitions are fast processes in Figure 9) and the absence of significant overshoots in complex oscillations in Figure 10. Thus we infer that the time scale of feedback evolution is much greater than the time scale of relaxation to each of the pseudostationary states. We have already emphasized that in these conditions the subcritical bifurcation region is very small. Yet in the experiments of Figure 9 when increasing [CH2(COOH),], the limit cycle projection exhibits smoother transitions between pseudosteady states (Figure 12) and we get a region of hysteresis between the oscillating state and steady state 2. Figure 13 illustrates this hysteresis as a
506
The Journal of Physical Chemistry, Vol. 84,
No. 5, 1980
Flgure 13. BR. Hysteresis loop in iodine concentration [I2] as a function of the iodine flow between steady state 2 (V)and an oscillating state. The vertical segments correspond to the amplitude of the
[&I0
oscillations.
function of [I2lO; a spontaneous transition from the oscillating state to steady state 2 is observed when increasing [I2lOabove 10 X mol L-l; the reverse transition is only mol L-l. observed by decreasing [IzlObelow 0.7 X Although this is the first experimental evidence for a subcritical Hopf bifurcation in this reacting system, topological situations corresponding to D, H, and I (Figure 5 ) have already been recognized by one of us as a least topological requisite to take account of a whole set of experimental features.lg A question arising is to establish the level of generality of our theoretical considerations with regard to systems with a level of dimensionality higher than two in the response variable space. Their successful application to the BR reaction have already demonstrated their utility. Due to the importance of such considerations for checking the validity of a mechanism, further analyses of these questions are in progress with the analytical model. The experimental research for these topological situations in the well-known Belo~sov-Zhabotinskii~~ reaction for which detailed mechanisms are available26-28is thus of great interest. Preliminary experimental investigations have exhibited a new bistabilitym between an oscillating state and a steady state in good analogy with the Hopf subcritical bifurcation in the BR reaction and very suggestive in connection with some complex oscillations often observed in the BZ reaction when performed in flow reactor^.^^-^^
References and Notes (1) A. M. Zhabotinskii, "Biological and Biochemical Osciihtws", Academic Press, New York, 1972, p 71.
Boissonade and De Kepper (2) A. PacauR, P. Hanusse, P. De Kepper, C. Vidai, and J. Boissonade, Acc. Chem. Res., 9, 438 (1976). (3) P. De Kepper, A. Rossi, and A. Pacault, C. R . Acad. Sci. Paris, Ser. C, 283, 371 (1976). 141 . . K. R. Graziani. J. L. Hudson. and R. A. Schmitz. Chem. €no. J.. 12, 9 (1976); R. A. Schmitz, K. R. Grazlani, and J. L. Hudson, j . Chem. Phys., 67, 3040 (1977). (5) M. Marek and E. Svobodova, Blophys. Chem., 3, 263 (1975). (6) B. F. Gray and L. J. Aarons, Faraday Symp., Chem. Soc., 9, 129 11974). (7) J. Boissonade, J. Chim. Phys., 73, 540 (1976). (8) U. F. Franck, "Bdogical and BiophysicalOscilbtws", Academic Press, New York, 1972, p 7. U. F. Franck, Angew. Chem., Int. Edit. Engl., 17, 1 (1978). (9) A. Nitzan, P. Ortoieva, J. Deutch, and J. Ross, J . Chem. Phys., 61, 1056 (1974). (10) W. Horsthemke and R. Lefever, Fhys. Left. A, 64, 19 (1977); I.Arnold, W. Horsthemke, and R. Lefever, Z.Phys. 6 , 29, 367 (1978). (11) P. De Kepper and W. Horsthemke, C. R . Acad. Scl. Paris, 287, 251 (1978). (12) 0. E. Rossler, Bull. Math, Biol., 39, 275 (1977). (13) D. H. Sattinger, "Topics in Stability and Bifurcation Theory", in "Lecture Notes in Mathematics", Vol. 309, Springer-Verhg, West Berlin, 1973. (14) G. Nlcolis and I.Prigogine, "Self-Organization in Non Equllibrium Systems", Wiiey-Interscience, New York, 1977. 0. Nicolis, T. Erneux, and M. Herschkowitz-Kaufman, Adv. Chem. Phys., 38, 263 (1978). (15) E. Hopf, 6 e r . Math-Phys. Klasse Sachs, Akad. Wlss. Leipzig, 94, 3 (1942). (16) J. Marsden and M. McCraken, "The Hopf Bifurcation", in "Lecture Notes in Mathematics", Springer-Verlag, West Berlin, 1977. (17) A. Uppal, W. H. Ray, and A. B. Poore, Chem. Engl. Sci., 29, 967 (1974). (18) T. S. Briggs and W. C. Rauscher, J. Chem. Educ., 50, 496 (1973). 119) P. De KeDDer. Thesis. Bordeaux. 1978. (20j P. De Kepper; A. Pacault, and A. Rossi, C. R. Acad. Sci. Paris, Ser. C. 282, 199 (1976). (21) K. Showalter, R. M. Noyes, and K. Bar-Eli, J . Chem. Phys., 69, 2514 (1978). (22) A. Pacault, P. De Kepper, P. Hanusse, and A. Rossi, C. R . Acad. Sci. Paris, Ser. C, 281, 215 (1975). (23) J. C.Roux, S.Sanchez, and C. VMal, C. R. Acad. Sci. Paris, Ser. 6 , 283, 451 (1976). (24) P. De Kepper and A. PacauR, C. R. Acad. Sci. Paris, Ser. C, 266, 437 (1978). (25) B. P. Belousov, "Ref. Radiats Med", 1956, Medgiz, Moscow, 1959, p 145; A. M. Zhabotinskli, DoklAkad. Nauk SSSR, 157,392 (1964). (26) J. J. Tyson, "The Belousov-Zhabotinsky Reaction", in "Lecture Notes in Blomathernatics", No. 10, Springer-Verlag, West Berlin, 1976. (27) R. J. Field, E. Koros, and R. M. Noyes, J. Am. Chem. Soc., 94, 8649 (1972); D. Edelson, R. J. Field, and R. M. Noyes, Int. J. Chem. Kinet., 7, 417 (1975). (28) C. Herbo, G. Schmitz, and M. Van Glabbehe, Can. J. Chem., 54, 2628 (1976). (29) P. De Kepper and J. Boissonade, "Kinetics of Physicochemical Oscillations", Meeting Aachen, Sept 1979. (30) J. S. Turner, Phys. Lett. A, 56, 155 (1976). (31) J. J. Tyson, J . Chem. Phys., 66, 905 (1977). (32) R. M. Noyes, In "Far from Equilibrium, Synergetics", Vol. 3, A. Pacault and C. Vidal, Ed., Springer-Verlag, West Berlin, 1979. (33) M. L. Smoes, J . Chem. Phys., 71, 4669 (1979). (34) J. Boissonade, Thesis, 1980.
