J. Phys. Chem. 1995,99, 16636-16640
16636
Periodic Perturbations and Three-Branch Return Maps of an Oscillatory Chemical System Andrzej Lech Kawczydski Institute of Physical Chemistry, Polish Academy of Sciences, 01224 Warsaw, Poland Kedma Bar-Eli* Sackler Faculty of Exact Sciences, School of Chemistry, Tel-Aviv University, Ramat Aviv 69978, Israel Received: July 26, I995@
Previous results of a periodically perturbed Oregonator have shown synchronized periods and patterns composed of large and small oscillations. Next amplitude maps of the small oscillations show the existence of threebranch return maps. A family of maps consisting of three branches was constructed in order to mimic the behavior of the results of the differential equations. These little known maps show clearly the behavior of the perturbed system. The maps as well as the real system do not have any chaotic motions and mainly have periodic ones.
I. Introduction
11. Summary of Results for the Oregonator
Since oscillating chemical reactions have been discovered, there has been a growing interest in periodic perturbations of such systems. One can expect quasiperiodicity, resonances, modulation, and chaos in such modulated systems. These phenomena have been observed in some experimental systems.1-4 Zhabotinsky has investigated the behavior of the BelousovZhabotinsky (BZ)5-7 reaction under the influence of a periodic light modulation. Dulos2 has done similar experiments on the Briggs-Rauscher oscillator.8 Synchronization of the oscillations with the modulation period has been found in both cases. The peroxidase reaction has been perturbed by periodic changes of oxygen influx in CSTR experiments. Resonances and chaos have been observed by Lazar and Ross3 and by Forster et aL4 when the unperturbed system exhibits limit cycle oscillations. Theoretical models were also studied for periodic perturbations. Thus, Ito9 and Kai and Tomita'O 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 Ross," and specific models were examined by Rehmus et a1.I2. Bar-EliI3.l4 has investigated the peristaltic effect (which is equivalent to periodically perturbing the flow) on the BZ and other systems. Weiner et al.15,16have modulated the minimal bromate o ~ c i l l a t o r ~ ~by- *a~delayed signal from another, similar, oscillator. Kaw~zyriski*~~*~ has investigated periodic perturbations on an enzymatic model exhibiting an infinite number of the period adding bifurcations and again found chaos, quasiperiodicity, and resonances. The results of Markman and Bar-Eli25,26have shown the existence of large and small oscillations in a modulated system of the BZ type (the Oregonator model). They have found regularities in sequences of small- and large-amplitude oscillations. These findings prompt us to construct next amplitude plots (return maps) for small oscillations only. In some cases these maps have three branches. Very little is known about the properties of three-branch maps. Therefore in the present paper the properties of three-branch maps are studied and their relevance to the results of the perturbed Oregonator is discussed.
The model used here is the one used earlier by De Kepper and Bar-ElL2' This model is the same as the Oregonator model described by Field and NoyesZ8except that terms describing the flow of reactants to and from the vessel are included. The relevant equations, written in a dimensionless form are
'
@
Abstract published in Advance ACS Abstracts, October 15, 1995.
k = a(x
+ y - x y ) - bx2 - ko(t)x (3)
where x, y , and z are the dimensionless concentrations of bromous acid, bromide, and ceric ions, respectively, f is the stoichiometric factor, a, b, and c are appropriate combinations of the chemical rate constants, and ko(t) is the flow rate modulated according to ko(t)= ko(l
+
6
sin or)
(4)
with w = 2nIT,,,,where T,,, is the modulation period. 11.1. Autonomous Behavior. Of the few cases described earlier,25s26we are interested here in the case where a = 550, b = 0.05, c = 25, yo = 10.8, f = 0.533, and the variable parameters are b,6, and Tin. When E = 0, the system is autonomous with the following properties: (1) There is a region where three steady states exist. These are denoted as SS1 where the value of x is the greatest, SS2 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 are x(SS 1) % z(SS1) % 5000, y(SS1) 0.5; x, y , z RZ 1-10 for SS2 and SS3, with x(SS2) sz z(SS2) < x ( S S 3 ) % z(SS3) and y(SS2) > y(SS3). (2) SS2 exists in the range h ( A ) = 0.473 596 505 6 < k~ < 00 and is stable. (3) SS1 exists in the range 0 < ko < b ( B ) = 124.424 356 4 with Hopf bifurcation occurring at b(0 = 0.522 404 333 5. Mathematically, SS1 exists even at h < 0, but these values of flow do not have any physical meaning. Oscillations occur in the range ko < h ( A ) (even in the nonphysical range of b < 0). The oscillations end at the point b ( A ) with saddle-loop and with period going to infinity
0022-365419512099-16636$09.00/0 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99,No. 45,1995 16637
Oscillatory Chemical System 7.50
-5.50
-
-
7.50
-
-
-5.50
+c
+c
?
7
- 1
W
m
m
1.50
1 ,
1.50
Figure 1. Next amplitude map of the small x oscillations of the perturbed Oregonator for the case 6 = 0.5529, Ti,= 1, and ko = 0.475. The pattem is LS(LS2)8,and the period is 35.
as l l a , where A = h(A) - h. During these autonomous oscillations the system spends most of the time near SS2 with short spikes to large x (%lo4) values. 11.2. Modulation. When the system is modulated such that h(min) -= b(A) < b(max), i.e. the modulation “straddles” the point h(A), the resulting oscillations are (for details see Markman and Bar-Eli25-26)as follows: (1) A combination of large and small oscillations. (2) The oscillations period To,, is always an integer multiple of the modulation period Tin;Le., P = T o u t / ~isn an integer. The resulting periods are always synchronizedz9with the modulating periods. (3) Between any two periods there is always a period which is the sum of the two; Le., P12 = P I f P2. (4) When the amplitude 6 is changed, a sequence of patterns LS” with n = 0, 1, 2, ... is observed in successive intervals of 6 . These intervals are separated by subintervals in which some combinations of patterns i.e. LSnLSn+Iexist. (5) The oscillations are arranged in steps, Le., there is a range of the parameter (6 or Tin)where the same period and pattem exists. (6) These results are of the same nature whether the modulation period T,,, is kept constant and the amplitude 6 is changing or the converse is true, Le. the amplitude is constant and the period is varied. Also the particular value of the flow is not important as long as the modulation “straddles” the point A.
111. Return Maps for the Perturbed Oregonator The appearance of the sequence of patterns LS” with n going to infinity together with the addition of periods as shown above, Le. P12 = P I P2, and also the pattern addition, as for example the pattern LS3LS4appearing between the patterns LS3 and LS4, prompt us to construct next amplitude maps for the small oscillations only. Next amplitude maps for the variable x , Sn+l vs S, were constructed. Two types of next amplitude maps are obtained. Figure 1 shows a map composed of two branches. Although we see the isolated points, the two branches are clearly distinctive: one is nearly vertical while the other is nearly horizontal. Figure 2 shows another case where, in addition to the two branches shown in Figure 1, another branch appears at the lower right hand side of the figure.
+
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s (n) 5’50
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Figure 2. Next amplitude map for 6 = 0.222; the other parameters are as in Figure 1. Three branches are clearly seen.
Another example which illustrates the formation of the third branch is given by the following set of values of x belonging to the pattern (LS4LS3)3LS4 (appearing at ko = 0.475, Tin = 1, E = 0.221) and having the time period P = 45 = 3(7 6) 6: [(1.99, 2.95, 3.70, 4.49;) (2.60, 3.44, 4.14;)] [(2.41, 3.31, 4.00, 5.29;) (2.73, 3.54, 4.25;)] [(2.52, 3.39,4.08, 5.87;) (2.82, 3.60,4.33;)] (2.56, 3.41,4.11, 6.38;). Note that the last “group” should have been LS3 (to conform with the period 6), but being LS4 it gives a single point on the third branch of the map. Note also that the number of points appearing on the map is 25 = 3(4 3) 4 and not 3(4 3) 3 = 24 as expected. The maps obtained have the following properties: (a) Each branch increases monotonically with x. (b) The maps are invertible, which means that their reciprocals are single-valued functions. According to our knowledge, there is no mathematical theory for such maps. Since we have found maps with three branches and since the mathematics of these maps is unknown, we are prompted to construct such maps and to investigate their properties in relation to the results quoted above.
+ +
+ +
+ +
IV. Family of Three-Branch Maps A family of maps with similar features such as those shown in Figure 2 was constructed. The maps have three branches defined on three intervals, namely, kl 5 x < k2, k2 5 x < k3, and k3 5 x 5 k4, where kl < k2 < k3 < k4. All three branches are monotonically increasing. Let us denote the image of kl belonging to the first branch as 11,the image of k2 belonging to the second branch as 12, and the image of k3 belonging to the third branch as 13. In this way l l , 12, and 13 define the minimal values of the first, second, and third branches, respectively. The maximal values for the branches are denoted by ml, m2, and m3, respectively. Let us notice that the image ( I I ) of the left most point of the first branch kl is equal to the image (m2) of the right most point of the second branch k3 and that the image ( 1 2 ) of the left most point of the second branch k2 is equal to the image (m3) of the right most point of the third branch. We discuss a particular example which approximately describes the results for the perturbed Oregonator. Consider a family of 1D maps consisting of three hyperbolic curves:
Kawczynski and Bar-Eli
16638 J. Phys. Chem., Vol. 99, No. 45,1995
+-(k3 - c2)+ d,
(k2 - b2)
for k, Ix < k3
(6)
I
/
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no.60
i
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where
0 00 000
020
040
080
060 X
Figure 3. Three-branch map of eqs 5-7 with k = 0.31. Other constant parameters are given in the text. In this case period 3 is attracting (see Figure 6).
....
and
. : . :
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:
.*.; . .!:.
. .(
0.60
.. . .. . .. . . . ..... . . . . * .... . . . .. . . ... . .... .. . . . .. . . . . ...... . .
X
It easy to check that
0.19
0.39
0 29
0.49
k as described above. To approximate the behavior observed in the perturbed system, a bifurcation parameter is introduced which controls the position of the first and the second branches. This can be achieved if one assumes that k2, b2, and C I depend linearly on the bifurcation parameter k.
k2 = ko2
+k
b, = bo,
+k
+k
c , = co2
Increasing k causes the first branch to shift toward the diagonal and simultaneously the left most point of the second branch to be shifted to the right; Le., the interval of the second branch decreases when k increases. Assuming that 11 = m? = 0.1, 12 = m3 = 0.05, kl = 13 = 0, and k4 = ml = 1, one gets the transformation of the interval [0,1] into itself. The following values of the remaining parameters are assumed: bl = -0.5, b02 = 0.00, b3 = 0.0, coI = 0.11, c2 = 20.0, c3 = 2.0, kl = 0.0, ko2 = 0.01, and k3 = 0.9.
Figure 4. Histogram of the map for 0.19 < k < 0.49. The simple periods 2, 3, 4, etc. are clearly seen: some combined periods can be seen too.
Figure 3 shows the map with three branches for eqs 5-7 with k = 0.31.
V. Results A histogram showing the dependence of the attracting periodic orbits on the parameter k is shown in Figure 4. A more detailed figure showing an expansion of the above figure is given in Figure 5. One notes first the simplest sequence of the general form F,-lS (such as FS, F2S, etc.), which consists of n points, n 1 of which are on the first branch and 1 of which is on the second branch. (We use the symbols F, S, and T for points appearing on the first, second, and third branch, respectively). Between the intervals of k with the above periods and patterns, we notice subintervals with combined sequences such as FzSFS, (FzShFS, (F2S)3FS, and so on, appearing between the FS and the F2S with periods 3 2 = 5, 2 x 3 2 = 7, 3 x 3 -t 2 = 11, etc. All these orbits consist of fixed points belonging to the first and the second branch only. This situation resembles
+
+
J. Phys. Chem., Vol. 99, No. 45, 1995 16639
Oscillatory Chemical System
TABLE 1: Patterns for bo2 = 0.00, COI = 0.11, and k02 = 0.001 for Selected k"
k Figure 5. Detail of the histogram of Figure 4 for 0.21 < k < 0.28. Combined periods between 2 and 3 such as 5 etc. are seen.
well-known picture characteristic for the circle map. However, the Farey30 arithmetic which is characteristic of the circle map is not valid in our case when the third branch comes into play, as we shall see below. If we look at the sequence described above, the next item should have been (F2S)4FS with a period of 4 x 3 2 = 14. But this sequence is not found, and instead 3 = 15 of it the sequence ( F 2 S ) D F with period 4 x 3 appears. The point appearing on the third branch with a point on the first one following immediately after gives FTF instead of the expected FS and has caused the period to be greater by one than that expected for two-branch maps. Let us look at another example. Between the periodic orbit (F$3)3FS with 11 fixed points and the periodic orbit (FzS)2FS with 8 fixed points, the periodic orbit (F~S)~FS(FZS)~FS with 19 fixed points is expected from the circle map. Instead we get (FZS)~FS(F~S)~FTF with 20 fixed points. Again, the sequence FTF has replaced the sequence FS, and another point was added to the period. Table 1 summarizes the results described above in the region 0.2 < k < 0.3. The sequences which include the third branch and have one or more T's are marked with an *. Each pattem (period) occupies a certain interval of the parameter k, meaning that it appears (becomes attractive) and disappears (becomes repellent) at the ends of the interval. In order to explain these bifurcations, let us consider an example. In Figure 6 we see the third iterate of the map for the value of k = 0.31 when the F2S (period 3) is attracting. The iterate consists of seven parts; three of them cross the diagonal with a slope smaller than 1. Increasing k causes these parts to shift down and to the right. It is clear now that the interval of this period will start at the value of k when the upper right hand points of the three parts "touch" the diagonal (simultaneously) and will end when the curved parts become tangent to the diagonal, Le. when the stable and unstable points will coincide-saddle-node bifurcation. Finding the limits of these intervals seems to be extremly difficult, especially for the high iterates, and we will pospone it for another publication. One can assume that Farey type arithmetics combined with replacement of FS segments by FTF ones characterize the orbits which appear between the given orbits, but because there is no theory for three-branch maps the concrete formulas are difficult to predict.
+
k
period
0.216 0.2216 0.2226 0.224 0.2257 0.2262 0.22734 0.22776 0.229 0.2322 0.2329 0.234 0.2362 0.244 0.2.53
2 11 = (3 4 x 2) 21* 9 = (3 3 x 2) 27* 17* 25* 32* 7 = (3 2 x 2) 28* 20* 13* 18* 5 = (3 2) 13 = (3 2 2 x 3 2) 8 = (2 x 3 2) 20* 11 = (3 x 3 2) 15* 18* 21* 24* 27* 33* 3
0.258 0.261 0.263 0.2656 0.267 0.268 0.2687 0.2691 0.2695 0.272
*
pattem FS F2S(FS)4 F2S(FS)3F2S(FS)3FTF F2S(FS)3
+
+
F~S(FS)~FZS(FS)~FTFF~S(FS)~FTF F&FS)zF2S(FS)zFTF
F~S(FS)~F?S(FS)ZFTFF~S(FS)FTF F2S(FS)2F2S(FS)2F2S(FS)2FTFF2S(FS)FTF
+
FzS(FS)z
F2S(FS)SS(FS)2F2S(FS)FTFF2S(FS)FTF F*S(FS)FzS(FS)2F2S(FS)FTF F2S(FS)F?S(FS)FTF
FzS(FS)F2S(FS)FzS(FS)FTF
+
F2S(FS) F~S(FS)(F~S)Z(FS)
+ +
+
+ +
(F*S)?(FS) (F~S)~(FS)(FZS)~FTF (F2S)dFS) (F2S)aFTF (F2ShFTF (FzS)6FTF (F2ShFTF (FzS)sFTF (F2S)ioFTF FzS
a indicates sequences which include the third branch and have one or more T's.
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Figure 6. Third iterate of the map shown in Figure 3 for k = 0.31. The three intersections with the diagonal-attractive period 3-are seen.
VI. Discussion The three-branch maps described above show quite good qualitative agreement with the results of the perturbed Oregonator described earlier by Markman and Bar-Eli.25s26The timeperiod-adding property of the perturbed Oregonator seems to operate in all cases; however, the pattem combination seems to fail sometimes (as shown in the example given in section 111) when the next amplitude plots give a point on the third branch. The results given in Table 1 show exactly the same phenomena; Le., pattem adding is valid until the third branch is involved. (Note that in the case of maps, when time does not appear, the pattem and the period are the same.) It is interesting to calculate the Lyapounov exponents and see how they behave, in particular in the regions of long periods, and see whether chaotic trajectories exist. Figure 7 shows the Lyapounov exponents in the same region of k as the periodic
16640 J. Phys. Chem., Vol. 99, No. 45, 1995
Kawczyiiski and Bar-Eli direct experimental test in the BZ reaction and in principle in other oscillating systems too. The behavior of the amplitude of the small oscillations can be examined and compared to the theoretical predictions given here.
References and Notes
\ .0 _ i
-0 40
i:
\
\
- 0 60
0 21
0 25
0.23
0 27
k Figure 7. Lyapounov exponents for the same region of k as in Figure 5. Note the resemblance between the region of periodicity and the magnitude of the exponent.
points in the histogram given in Figure 5 . There is a large drop in the exponent at every periodic region-the shorter the period the lower the exponent. The exponent is close to zero at some small intervals, indicating that in all probability the most complex trajectories can be quasiperiodic and not chaotic. In the perturbed Oregonator a scaling law between the period and the range of its existence of the form PRa = b (where P is the period, R is the range, and a and b are constants which depend on the patterns) has been f o ~ n d . It~ is~ interesting ,~~ to find whether a similar scaling law exists in the case of the threebranch map. We postpone this to a subsequent publication. In the above results we have changed only the value of k, thus controlling the positions of the first and second branches only. Further work needs to be done changing k3 and thus controlling the third branch too. In the perturbed Oregonator the position and size of the third branch depends heavily on the parameters E and Ti,, i.e. the amplitude and period of the modulation. These changes, reflected in the maps by k2 and k3, should be investigated. Further, different maps, not necessarily hyperbolas, can be taken into account. Last but not least, all the described results are amenable to
(1) Zhabotinsky, A. M. Concentrations Auto-Oscillations: Nauka: Moscow, 1974 (in Russian). (2) Dulos, E. In Nonlinear Phenomena in Chemicd Dynamics; Vidal, C., Pacault, A,, Eds.: Springer-Verlag: Berlin. 1981; p 140. (3) Lazar, J. G.; Ross, J. J . Chem. Phys. 1990, 92, 3759. (4) Forster, A,: Hauk, T.: Schneider, F. W. J . Phys. Chem. 1994, 98, 184. (5) Belousov, B. P. Sb. Rex Radiat. Med. 1958, 145. (6) Zhabotinsky, A. M. Dokl. Akad. Nauk USSR 1969, 157, 392. (7) Zhabotinsky, A. M. Biofiziko 1964, 9, 306. (8) Briggs, T. S.; Rauscher, W. C. J . Chem. Educ. 1973, 50, 7. (9) Ito, A. Prog. Theor. Phys. 1979, 61, 45. (10) Kai, T.; Tomita, K. Prog. Theor. Phys. 1979. 61, 54. (11) Rehmus, P.: Ross, J. J . Chem. Phys. 1983, 78, 3747. (12) Rehmus, P.; Zimmermann, E. C.; Ross, J.; Frisch, H. L. J . Chem. Phys. 1983, 78, 7241. (13) Bar-Eli, K. J . Phys. Chem. 1985, 89, 2852. (14) Bar-Eli, K. In Temporal Order; Rensing, L., Jaeger, N. I., Eds.; Springer Verlag: Berlin, 1985; p 126. (15) Weiner, J.; Schneider, F. W.; Bar-Eli, K. J . Phys. Chem. 1989, 93, 2704. (16) Weiner. J.; Holz, R.; Schneider, F. W.; Bar-Eli, K. J . Phys. Chem. 1992, 96, 8915. (17) Bar-Eli, K. J . Phys. Chem. 1985, 89, 2855. (18) Bar-Eli, K.: Geiseler, W. J . Phys. Chem. 1983, 87, 3769. (19) Bar-Eli, K.: Field, R. J. J . Phys. Chem. 1990, 94, 3660. (20) Dateo, C. E.: Orban, M.; De Kepper, P.; Epstein, I. R. J . Am. Chem. Soc. 1982, 104, 504. (21) Geiseler, W. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 721. (22) Geiseler, W. J . Phys. Chem. 1982, 86, 4394. (23) Kawczyhski, A. L. Pol. J . Chem. 1995, 69, 296. (24) Kawczyhski, A. L. In Chaos: The Interplay Between Stochastical and Deterministic Behauiour; Garbaczewski, P., Wolf, M., Weron, A,, Eds.; Lecture Notes in Physics Vol. 457; Springer-Verlag: Berlin, 1995: p 451. (25) Markman, G.; Bar-Eli, K. J . Phys. Chem. 1994, 98, 12248. (26) Markman, G.; Bar-Eli, K. In Far from Equilibrium Dynamics of Chemical Sysrems; Gorecki, J., Cukrowski, A. C., Kawczynski, A. L.. Nawakowski, B., Eds.; World Scientific: Singapore, 1994, p 11. (27) De Kepper, P.: Bar-Eli, K. J . Phys. Chem. 1983, 87, 480. (28) Field, R. J.; Noyes, R. M. J . Chem. Phys. 1974, 60, 1877. (29) Minorsky, N. Nonlinear Oscillators; D. Van Nostrand Company Inc.: Princeton, NJ, 1962; p 438 and references therein. (30) (a) Farey, J. Philos. Mag. 1816, 47. 385. (b) An Introduction to the Theorj of Numbers: Hardy, G. H.. Wright, E. M., Eds.: Clarendon: Oxford. 1979. JP952127U
