J. Phys. Chem. 1994, 98, 12248-12254
12248
Periodic Perturbations of an Oscillatory Chemical System Gregory Markman Department of Mathematics, University of Rostov, Engelsa 105 Rostov on Don, Russia
Kedma Bar-Eli* Sackler Faculty of Exact Sciences, School of Chemistry, Tel-Aviv University, Ramat Aviv 69978, Israel Received: March 24, 1994; In Final Form: June 28, 1994@
The Oregonator model in a continuously stirred tank reactor has been subjected to periodic modulation of the input and output flows of materials. Two cases has been investigated. In both cases the system has three steady states (one of which is always a saddle). In the first case, two steady states are stable. Transitions from one steady state to the other via the perturbations are investigated. In the second case oscillations occur, but they end via saddle-loop bifurcation. Small harmonic oscillations occur if the modulation spans only the stable steady state. When modulation period and amplitude are such that the modulated flow stays partly in the stable region and partly in the unstable region, the oscillations become synchronized with the modulation period. There is a range of modulation amplitude (or period) in which there is a rational ratio between the oscillations period and that of the modulation. Between any two such steps of synchronization there is always another step in which the period is the sum of the two periods, and the pattern is a combination of the patterns on its two sides. As the period increases its step size decreases. A simple power relationship exists between the step size and its period. These synchronized oscillations are always periodic, and no period doubling or chaotic oscillations have been observed.
Introduction
The Model
The Belousov-Zhabotinsky (BZ)1-3reaction has been subject to many investigations. Many of the experimental results has been explained via the Field-Koros-Noyes (FKN) mechan i ~ m .A~ simplified version of the above mechanism which includes flows to and from the system, and thus particularly useful for investigating continuously stirred tank reactors (CSTR), has been used by De Kepper and Bar-Eli5 and by Showalter et De Kepper and Bar-Eli5 compared the simplified and complete versions as to the behavior regarding the various control parameters. Zhabotinsky7has investigated the behavior of the BZ reaction under the influence of a periodic light modulation. Dulos8 has done similar experiments on the Briggs-Rauscher os~illator.~ Synchronization of the oscillations with the modulation period has been found in both cases. Ito’O and Kai and Tomitall studied the effect of small periodic perturbations on the limit cycle of the Brusselator model near its Hopf bifurcation point. The effect of small sinusoidal perturbations on a general limit cycle has been taken by Rehmus and Ross12 and specific models were examined by Rehmus et al.13 Bar-Eli14.15has investigated the peristaltic effect (which is equivalent to periodically perturbing the flow) on the BZ and other systems. Weiner et has modulated the minimal bromate o ~ c i l l a t o r ’ ~by - ~ ~a delayed signal from another, similar, oscillator. More recently Forster et aLZ4perturbed periodically the oxygen flow in peroxidaseoxidase system. All these results have prompted us to investigate more extensively the influence of modulation of a parameter of the oscillating system on the system characteristics. The simplest parameter to use is of course the flow to and from the system. This flow can be very conveniently controlled, and thus the results reported here can be easily compared to experiments.
The model used here is the one used earlier by De Kepper and Bar-Ek5 This model is the same as the Oregonator model described by Field and NoyesZ5except that terms describing the flow of reactants to and from the vessel are included. The relevant equations are
al.16917
@
Abstract published in Advance ACS Abstracts, October 15, 1994.
0022-3654/94/2098- 12248$04.50/0
k = k,Ay - k n
+ k,Ax - 2k4x2 - ko(t)x
(1)
+ k f z + ko(t)(yo- y)
(2)
j , = -k,Ay - k n Z
= k&
- k5z - ko(t)Z
(3)
The rate of flow term is given by k(t),and it is the reciprocal of the residence time of the reactants in the CSTR vessel. In our case the inflow consists only of the species yo, Le., bromide ions, while all the species, namely, bromous acid (x), bromide (y), and ceric ions ( 2 ) are included in the outflow. In the present work we shall modulate the system by modulating the flow rate in the following manner:
ko(t) = ko(l
+
E
sin ut)
(4)
This is similar to the modulation which has been used earlier14J5in the investigation of the effect of the peristaltic Pump. The system can be nondimensionalized using the following transformation of the variables:
x = k,Ax’/k, t = t’/(k,A)
y = kgy’lk, z = k,k3AAz’/(k2k5) yo = (k3A)y’dk2 w = w’(k,A) ko = k’,(k,A)
using the following constants:
a = k,/k,
b = 2k4/k,
0 1994 American Chemical Society
c = k,/(k,A)
J. Phys. Chem., Vol. 98, No. 47, 1994 12249
Periodic Perturbations of an Oscillatory Chemical System TABLE 1: Bifurcation Points (Values of ko) for the Various Cases ~
~~~~~
CASE 1
~
ss2
case YO f A“ Bb C‘ 1 4.0 0.5045 1.774 430 068 222.389 593 5 2 10.8 0.533 0.473 596 505 6 124.424 356 4 0.552 404 333 5 A: lower saddle-node SS2 and SS3. B: upper saddle-node SS1 and SS3. Hopf bifurcation on SS1. TABLE 2: Range of Stable Steady States and/or Oscillations case ss1 s s 1 + ss2 ss2
a(x
Y = -y
- xy
+ k,(f)OiO - Y )
z = c(x - z ) - k&)z
osc
(6) (7)
These equations can be solved as they are, using eq 4 for ko(t), or two more dimensions can be added, by adding two more equations to describe the time variation of the flow. These equations are U = V
(8)
2 V=--Ou
(9)
The solution of these equations with the initial conditions u(0) = 0 and v(0) = w = 2n1Tin (with Ti, being the period of the modulating force) gives u = sin wt and thus eq 4 becomes
ko(t)= ko(l
+
CASE 2
C=.52240
ss1
B =124.42
> k-0
+ y - xy) - bx2 - ko(t)x +fz
A -1 .?744
A47360
and omitting the primes, the following equations are obtained:
k
B= 222.38
ss1
EU)
Using the set (5-7) with eq 4 is thus completely equivalent to the set (5-7) with eqs 8 and 9. The larger set is more convenient to use, since one does not have to compute the values of sin Lot for large values of t. Autonomous Equations
The modulation of the system has been investigated in two cases. In the two cases the values of the rate constants namely a-c are set to be a = 550, b = 0.05, and c = 25, while the other parameters vary. When the system is unperturbed, i.e., when the value of E is taken to be zero, the system becomes an autonomous one as dealt earlier by De Kepper and Bar-ElL5 In both cases described, there is a region where three steady states exist. These are denoted as SS1 where the value of x is the greatest, S S 2 where the value of x is the smallest, and SS3 in the middle which behaves always as a saddle with one positive eigenvalue. Typical values arex(SS1) % z(SS1) % 5000 y(SS1) % 0.5, x, y, z % 1-10 for SS2 and SS3, with x ( S S 2 ) x z(ss2) < x(ss3) % z(ss3)y ( s s 2 > y(ss3). For each case the values of yo andfare fixed while the value of b-the flow rate-is used as a bifurcation parameter. The description of the 2 cases, together with the flow values for the critical points A-C is given in Table 1. Table 2 gives the range of existence of the stable steady states, oscillations, and combinations of them. Figure 1 shows schematically the above regions and points A-C for quick reference. Typically in the oscillations, which occur in the indicated range, namely, up to point A, the system spends most of the time at low values of x near SS2 and only comparatively short times at the peak values near x = lo4. In most of the oscillation
Figure 1. Schematic plot of the stable and oscillatory steady states. S S 3 (not shown) connects points A and B in all cases. This figure is a pictorial representation of Tables 1 and 2. TABLE 3: Values of ko and Ti,Where Transition to SS1 Occur@
ko
b u n
ko(A) - h
2.0 3.2 3.4 3.5
1.0 1.6 1.7 1.75
0.77443007 0.17443007 0.07443007 0.024 430 07
aE
mn
Tin tdZn AtJTin 0.895 0.96379 0.42758 6.325 0.8250 0.1500 14.565 0.79727 0.09454 43.465 0.77662 0.053249
= 0.5 initial point
Atc
Ako(Atc)*
0.381 0.95 1.3769 2.314
0.1124 0.157 0.141 0.131
near SS2.
range, the period is near 2. A plot of such oscillations are abundant in the literature, e.g., ref 25. When the flow approaches point A , at which the oscillations end (as can be seen from Tables 1 and 2 and Figure I), the time spent near SS2 as well as the period, increases to infinity as ll&, where A = ko - ko(A), since the oscillations approach saddle-loop b i f u r ~ a t i o n .It~ ~should ~ ~ ~ be noted that the oscillations exist also at negative ko, but since these flows do not have any physical meaning, these regions were not investigated. The behavior of the system has been investigated as a function of the three free parameters, namely, ko, E , and Ti,. We have chosen the flow, i.e., ko as the perturbed (amplitude and period) parameter, although any other parameter could have been used, since an experiment was in mind, and this parameter is most easily controlled by the experimentalist. Further, although peristaltic pumps are hardly used any more, this investigation shed more light on experiments conducted with such pumps, as a continuation to previous result^.'^,^^ Case 1
In this case the two steady states (SS1 and SS2) are always stable; thus, there is no autonomous limit cycle associated with it, and the system will be stable at either SS1 or SS2. When the modulation is activated, one obtains small harmonic oscillations either around SS1 or around SS2, depending, of course, on the initial conditions. The period of these harmonic oscillations is the same as that of the modulation. If the initial point is at S S 2 and the value of the flow rate ko is near point A, the saddle-node point, and E is such that during the oscillation the system spends some time below the point A where only SS1 exists, then the system may jump to SS1 and continue its harmonic oscillations around it. Just before the “jump” to the other steady state, the simple harmonic oscillations are slightly distorted and become more pointed at the top (high x’s) and more flat at the bottom (low x’s). This distortion becomes more pronounced as the “jump” point is approached. The occurrence of the “jump” to SS1 occurs only if Ti, is large enough, Le., the modulation is fairly slow. Table 3 shows the critical flow needed to transfer from the vicinity of S S 2 to the vicinity of SS1, as a function of the modulation period. It is seen that as komi, approaches b ( A ) , a longer Ti, is needed to enable the transfer from one steady state to the other.
Markman and Bar-Eli
12250 J. Phys. Chem., Vol. 98, No. 47, 1994
Similar phenomenon will occur when the system is near point B; in this case the system, initially at SS1, will under similar conditions go over to S S 2 . The transfer of a system from one steady state to the other has been studied extensively. Thus Bar-Eli and Geiseler26 perturbed a bistable bromate-cerium ions in sulfuric acid and measured the kinetics of the transition from one steady state to the other. The “successful” perturbation, Le., the one that can cause the transition, must be with appropriate amplitude and duration. When the perturbing amplitude becomes smaller, its duration must become longer. Showalter et al.27-29studied extensively, both theoretically and experimentally, the transitions of the bistable iodatearseneous system located near the saddle-node critical point. They perturbed the system beyond the critical point and found that as the perturbation amplitude becomes smaller, the transition to the other steady state takes longer. They also found the relationship, initially pointed out by Dewel et aL30s31namely, Akot2 = const, where A h = h(A) - ko(min) = M A ) - h ( 1 - E)is the deviation from the critical point (the hysteresis limit, i.e., point A) and t is the duration of the transfer to the available stable steady state. The system located near the hysteresis limit is critically slowed d o ~ n since ~ ~ it, must ~ ~change , ~ ~its kinetics near the saddle-node point due to one of the eigenvalues approaching zero. This change of oscillation type, Le., departure from harmonicity can be checked experimentally. The explanation of our results depicted in Table 3 seems obvious. As the system leaves the A-B region it starts moving toward SS1. If the time spent below point A is too short, the system will revert to S S 2 , since it does not have enough time to reach the separatrix and traverse it; as the flow increases, the system spends a smaller portion of the cycle below point A, and thus we need longer modulation times to compensate for in order to have enough time to traverse the separatrix. Proper calculation can be done as follows: the total time spent below point A is given by Atc = 2(tc1- tmin) where
and
tcl
is given by the equation
tmin the
time to reach the minimal ko(min) by t,JTin
= 314
A plot of Atc2 vs llAko gives a straight line with slope of 0.129 26 and intercept of 0.087. Thus, either from the linear plot or from the last column of Table 3 (average 0.135) we obtain that the minimum flow deviation (from point A) needed to transfer the system from SS2 to SS1 is inversely proportional to the square of the time spent below point A. This result is in complete agreement with the results cited above27-30in order for the transition to the upper, stable SS1 to occur, the system, during its modulated perturbation, must spend enough time below critical point A. This time depends on the modulation amplitude E, the modulation period Tin and the average flow ko. An experimental verification of this result will, therefore, complement the results obtained previously in refs 27-30 in a different way. Case 2 E
Change with E. The behavior of the system with change of has been done mainly under the constant flow and period
x * s tlme
e?s=.6
vo=ic.a k0=.4:5
A.
J 19
1
I
,
,
‘
‘
20
23
22
23
24
1 ‘ 25
I‘ I
,
26 27
I
28
29 30
time Figure 2. Case 2: plot of x vs time for k~ = 0.475, T,, = 1, e = 0.6 with pattem LSLP and reduced period of 7.
conditions, namely, ko(A) < ko = 0.475 ko(C) and Tk = 1. Under these conditions the flow is partly above and partly below the critical point A. In all cases the resulting oscillations are composed of a series of large (x,, % lo4 marked L) and small (1.0 < ,x < 10.0, marked S)oscillations. Typical oscillations are shown in Figure 2 , where a time plot of x is given. The resulting oscillations have the pattern LSLP. The pattern is thus made of a large peak followed by a small one and then again a large one followed, this time by two small ones, and the above repeats itself indefinitely, with reduced period of P = T,,JTi, = 7. The period is thus seen to be s y n ~ h r o n i z e dwith ~ ~ the modulation period, Le., the latter is an exact multiple of the former. Note also the similarity to the autonomous oscillations, described earlier, the large oscillations look the same while small ones (in the vicinity of SS2) with different amplitudes, as described, appear. As E increases from 0, small oscillations ensue around SS2 (which is the only stable steady state) and the pattern obtained is S only. As E increases further, the pattern changes to LS“ with decreasing n as can be seen in Table 4 and Figure 3. For the sake of completeness, the table includes nonphysical values of E 1. It is seen from the table and the figure that the oscillations are arranged in steps, Le., there is a range of E in which the same period and pattern persists, thus making the plot of P = T,,JZn vs E (P being an integer) a step-like plot. Since P is an integer, it means that the oscillations are synchronized with the perturbation. When the scale of E is expanded, it is seen that between any two steps of certain pattern and period, say P I and P2, there exists another step with a pattern which is a combination of the two namely, P12 = PI f P2 with period which is the sum of the two. Thus between E = 0.551 34 (which is the end of period 4 step) and E = 0.617 77 (which is the beginning of period 3 step), there are steps with periods 7, 11, 15, etc., converging on period 4,on the low+ side, and periods 7, 10, 13, etc., converging on period 3, on the high+ side. Period 7 with pattern LSLP is that shown in Figure 2. The middle part of Table 4 and Figure 4 show the “two horned” set of combination periods obtained in this way. Similar results are obtained between any two periods.
’
J. Phys. Chem., Vol. 98, No. 47, I994 12251
Periodic Perturbations of an Oscillatory Chemical System TABLE 4: Pattern and Reduced Period as a Function of (ko = 0.475, Ti. = 1) pattern
E
combination
G
0
;I
T,,JTi,
~
0+-0.08957 0.09129-0.09130 0.09256-0.09260 0.10658-0.10690 0.145378-0.14820 0.19453-0.20991 0.22808-0.25520 0.28197-0.33820 0.38394-0.55134 0.67177-1.5741 2.173-
s
1 LSS2 2+53x1 LS39 2+40x 1 LS’’ 2+16x1 L5” 2+8x1 LS4 2f5x1 LS3 2+4x1 L g...L p 2+3x1 2+2x1 LS...L g LS 2+1x1 L 2fOx1 Detail: Pattems and Periods between 3 and 4 (Ls)2(Lw 3+4x4 0.55585-0.557 (Ls)2(Ls-)2 3+3x4 0.55830-0.561 (Ls)2 LS(LP)Z 3 2 x 4 0.5635-0.571 (LS)‘... LSLP 4+3 0.57802-0.61629 0.625-0.63947 (LSyLs2 4+2x3 (Ln3L9 4+3x3 0.6505-0.65057 0.6551-0.65667 (Lq4LS2 4+4x3 (Lq5m 4+5x3 0.6601-0.66059 0.6631-0.66312 (La6LP 4+6x3 Detail: between 3 and 2 (above E =- 1) 1.6 LSL2S 3+5 1.7-1.85 L2S 3+2 1.9 L2SL3S 7+5 1.95-2 L3S 2+5
m...
+
1 55 42 18 10 7 6 5 4 3 2 19 15 11 7 10 13 16 19 22
8 5 12 7
In this way very long periods with complicated patterns may result, but careful examination shows that no chaotic or even quasiperiodic oscillations occur. Thus the totality of steps forms a Cantor set, and Figures 3 and 4 are self-similar as will be discussed below. In fact all the data shown in the upper part of Table 4 can be viewed as combination patterns between period 2 with pattern L and period 1 with pattern S, with one S added in each step. One expects to find patterns such as LnS appearing on the other side. Indeed the pattern LzS with period 5 appears between 1.7 < E < 1.85 and L3S with period 8 between 1.95 < E < 2 are shown in the lower part of Table 4. Figure 3 is therefore the “left (physical) horn” appearing between periods 1 and 2. As E increases, within the step, the maxima of the small oscillations increase too. In some cases (period 4 and period 5 ) , the pattern may change due to the appearance of a new small peak toward the end of the step. The appearance of the “new” peak is the result of a shoulder, which existed at smaller E ’ S , turning into a maximum. The total time involved does not change, and therefore this change in pattern does not change the period. This is the reason that period 7 (which is made up from 3 and 4) has a pattern of ( L q 2 on one side and LSLP on the other, as shown in Table 4. The results obtained in this, as well as those in the next, section are of a different nature than those obtained by earlier authors. The usual behavior of period doubling, quasiperiodicity and eventually chaotic oscillations are not observed. Instead of the familiar period doubling, a period “adding” with P12 = P1 Pz is seen. Similar, albeit somewhat different, period “adding” phenomena have been described by Aicardi and K a w c z y n ~ k i . ~In~ particular, the next amplitude plots (which will be dealt with in a future publication), in our case, are of totally different nature than those described earlier. Change of Modulation Period. When the amplitude of the modulation is kept constant and its period is being changed, Le., when, as in our case E = 0.6, very similar results to the previous ones are obtained. The results are summarized in Table 5 and in Figures 5 and 6 which show the behavior of the system
+
:ii
* .
.. .
.
I
.
0
900
0 40
0 20
060
083
130
€3S
Figure 3. Case 2: plot of period vs E for Q ! = 0.475,TI, = 1. Pattems are LS”.
:i $0
p e r ~ dvs r O = 475 y s = 12.8
65.l
steps
-..
j:
;etweer
. . .
255
eps
058
065
3 orid
4
1
052
054
060
068
eos
Figure 4. Case 2: plot of period vs E for steps with periods between 3 and 4 (expansion of Figure 3).
as Ti, is changed. The main finding is, again, that the period, or rather the reduced period P = T,,,/T.,, and the pattern are step function of the variable-in this case-the modulation period; moreover, between two periods (and patterns) there exist another period which is the sum of the two and a pattern which is a combination of the two. Thus a step of period 5 with pattern LSL2S is a combination of period 3 (L2S)and period 2 (LS) and so on. The “two horned” sets of steps appear in both Figure 5 and in the expanded Figure 6; each step is a combination of its flanking steps. Note that if Ti,, is increased along the step, the pattern may change from L to LS, Le., a small oscillation is added, due to a shoulder tuning into a peak without changing the period, similarily to what has been observed above. This may cause the pattern to change a little from one end of the step to the other, as shown in Table 5. When the modulation period is very large, the reduced period is exactly 1. The pattern is changing from LnS (with decreasing n ) to L as Ti, is reduced. This behavior is easily explained: the system stays a long time below point h ( A ) at which time it performs as many large oscillations as time permits with a period of m2. During the other half cycle the system is above ko(A), making a small change near the SS, thus allowing for one S type oscillation. Thus a very large step of period 1 with varying pattern is obtained. As the modulation period is further reduced, a series of LnS with periods of n 1. This series of steps converges at the
+
12252 J. Phys. Chem., Vol. 98, No. 47, 1994
Markman and Bar-Eli
; 11 c.
TABLE 5: Reduced Period and Patterns Dependence on Ti, (E = 0.6, ko = 0.475) TI, P = ToUt/T,, combination pattern 20 10-8 7-5 4.5-2.4108 2.344-2.3224 2.3206-2.2846 2.2813-2.2131 2.2063-2.0427 2.0202- 1.381 1.2689-1.033 0.9753-0.8568 0.8184-0.7477 0.7 187-0.67 16 0.6484-0.6146 0.5953-0.5699 0.162-0.05
Large TIn 1 1 1 1 1
5
6 7 8 1
L.S
Reduced
LS
1
\
+
Y0='38 eDS= 5
'
