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Stabilization of chaotic behavior in a two-phase autocatalytic reactor
Chaos, Solitons and Fractals 12 (2001) 903±918
www.elsevier.nl/locate/chaos
Stabilization of chaotic behavior in a two-phase autocatalytic reactor Abdelhamid Ajbar * Department of Chemical Engineering, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia Accepted 1 March 2000
Abstract The eect of feed forcing on the chaotic behavior found by Alhumaizi and Aris (Chaos, Solitons & Fractals 1994;4:1985) in the k1 k2 autocatalytic reaction of Gray and Scott A 2B ! 3B; B ! C, taking place in a two-phase reactor, is numerically investigated. It is shown that by periodic forcing of the feed ¯ow rate, the chaotic behavior of the reactor can be controlled for even small forcing amplitudes or frequencies. The behavior of the forced system alternates between chaotic and period behavior via period adding, quasiperiodic and intermittent bifurcations. The results of the stabilization of the chaotic behavior are summarized in an amplitude±frequency excitation diagram showing the dierent bifurcation mechanisms. A numerical investigation has also revealed that the reaction conversion can be improved by an appropriate selection of the forcing frequency. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Models of chemically reactive systems are known to exhibit a variety of non-linear phenomena ranging from steady-state multiplicity to complex quasi-periodic as well as strange chaotic and non-chaotic behavior. Bifurcation and chaotic behavior exhibited by a number of these models are not only of theoretical importance, but also these phenomena are rather important in a number of industrial chemical and biochemical processes [2]. Some of the important examples include reactors for the production of polymers and co-polymers either in liquid phase or in gas-phase catalytic processes [3,4], the industrial ¯uid catalytic cracking (FCC) units [5±8] for the cracking of gasoil to high octane number gasoline, the catalytic oxidation of CO in car exhaust systems [9], the catalyzed OXO reactions involving the sequential conversion of ole®ns to aldehydes and aldehydes to alcohol [10], the highly exothermic catalytic partial oxidation reactions (e.g., ethylene to ethylene oxide, o-xylene to phthalic anhydride) [11±13] as well as a number of biochemical systems such as substrate inhibited enzymatic reactions [14,15]. There are many practical reasons for interest in studying the chaotic behavior in chemically reactive systems. As pointed out by a number of authors [16,17], a process operating around oscillatory trajectories may be more advantageous in terms of mean productivity or selectivity as compared to operating the process at steady-states. On the other hand, large ¯uctuations or instabilities in chemical reactors can pose safety hazards. Knowledge of how to predict and stabilize this behavior is necessary for proper design and operation. One necessary requirement for any chemical reaction system to exhibit `exotic' behavior such as multiplicity or sustained oscillations is that it should have a Ôfeedback mechanismÕ. Some intermediate species
*
Fax: +966-1-467-8770. E-mail address: [email protected] (A. Ajbar).
0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 0 5 4 - 0
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or products of the process must be able to in¯uence the rate of earlier steps. This eect may be thermal as in the classical case of ®rst-order exothermic reaction where the temperature rise, consequent on heat release, in¯uences the reaction rate constant. The feedback may also be chemical as for example in non-monotonic kinetics or in autocatalytic reactions, where at least one product of reaction increases the reaction rate and thus its own rate of production. One such system is the cubic autocatalysis with catalyst decay k1
A 2B ! 3B; k2
B ! C;
with rate r1 k1 AB2 ;
with rate r2 k2 B
1 2
taking place in an isothermal continuous stirred tank reactor (CSTR). This system has been studied extensively by Gray and Scott and co-workers [18±22]. Using a two-dimensional continuous model, the authors showed that this reactive system can display a whole range of non-linear phenomena ranging from multiplicity to self-sustained periodic solutions. The considerable interest in studying autocatalytic reactions stem from the fact that these reaction models are at the core of some common reactions such as hydrogen and carbon monoxide oxidation [23], and oreganator reaction [24]. Moreover, the richness of the two-variable Gray±Scott model is another evidence that low-dimensional deterministic chemical systems can describe much of the complex behavior found in more complicated ones [25]. The original two-variable Gray±Scott model was further developed by a number of investigators. It was found for instance that complex periodic and aperiodic behavior can result when coupling the cubic autocatalytic reaction with self heating feedback element [21] or when the reaction is coupled with an equivalent isothermal feedback step [22]. In their contribution, Alumaizi and Aris [1] studied the autocatalytic reactions (1) and (2) in a two-phase reactor, consisting of two isothermal well-stirred tanks separated by a semipermeable membrane (Fig. 1). The autocatalytic reaction (1) occurs only in one tank, Ôthe reactorÕ to where the autocatalyst B is fed directly. The reactant A diuses through the membrane from the other compartment ÔreservoirÕ. This two-phase reactor allows the analysis of the reactive system using a three-variable model, with the concentration of the reactant in the reservoir providing the third variable in this extended Gray and Scott model. The authors showed that the modi®ed reactive system can exhibit complex chaotic and nonchaotic behavior. This also con®rms that chaotic chemical systems can be obtained when two or more gradientless system such as CSTR are connected to each other via heat or mass transfer. In this paper, we analyze the dynamics of the chaotic behavior found by the authors [1], when the reactive system is subjected to periodic perturbations in feed conditions, such as the feed ¯ow rate. Chaotic attractors show extreme sensitivity to initial conditions as nearby process trajectories diverges
Fig. 1. Schematic diagram of the two-phase autocatalytic reactor. Fig. 2. Poincare bifurcation diagram of the autonomous system (point (C) is the chosen attractor for forcing).
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exponentially. This characteristic, known as the Ôbutter¯y eectÕ [26], is a formidable obstacle to an adequate control of chaotic behavior. The application of periodic parameter perturbations to stabilize unstable orbits of a chaotic attractor is one of the ideas that have been developed, in recent years, for means to control chaotic behavior [27,28]. While the eects associated with these perturbations are generally dicult to predict, nevertheless this technique is easy to implement. Furthermore in some cases [29±32] dramatic changes in the fundamental characteristics of the chaotic attractor occur at very small values of the forcing amplitude or frequency, which are almost impossible to avoid in practice. It should be noted that the eect of feed disturbances on the behavior of chemically reactive systems having periodic and quasi-periodic attractors has been investigated by a number of authors [33±40]. It was clearly demonstrated that for certain values of amplitude and frequency of the forcing variable, the periodic and quasi-periodic attractors of the autonomous system can turn into chaotic attractors. In this paper the opposite problem is tackled, starting from a chaotic attractor of the autonomous system, it is shown that a variety of periodic and aperiodic behavior can be obtained by appropriate forcing of the feed conditions.
2. Process model and presentation techniques The autonomous model of the two-phase reactive system is described by the following three ordinary dierential equations. The detailed derivation of the model was carried out by Alhumaizi and Aris [1] dx z ÿ x x ÿ ÿ xy 2 ; ds k h
3
dy b ÿ y xy 2 ÿ ky; ds h
4
q
dz zÿx r ÿ 1 ÿ z: ds k h
5
The model state variables are, respectively, the dimensionless concentrations of reactant A, intermediate product B and reactant A in the reservoir x
A ; A0
y
B ; A0
z
A : A0
6
The rest of the model parameters are the dimensionless concentration of the feed of B, b B0 =A0 , the ratio of the tank volumes q v=V , the ratio of the feed ¯ow rates r q=Q, the dimensionless decay rate constant k, the dimensionless mass transfer resistance k and the reactor residence time h. In this paper, we study the eects of periodically forcing the ratio r of feed ¯ow rates. The forcing takes the following simple form: r r0 Am sin ws;
7
where r0 is the baseline value of r and Am and w are the dimensionless forcing amplitude and frequency, respectively. Chaotic behavior in the autonomous model was found for a wide range of model parameters. For example, for the following values of parameters: b
4 ; 15
k 225;
r0 2;
q 20;
k
4 : 450
8
Fig. 2 shows the Poincare bifurcation diagram for h 2 3038; 3050. It can be seen that the autonomous system goes through a series of period doubling that culminates into banded and then fully developed chaos for h 2 3044:95; 3049:55. In this paper, we study the forcing of the chaotic attractor at h 3049:00 (point C) that lies in the chaotic strip of the autonomous system depicted in Fig. 2. Figs. 3(a) and (b) show the phase plane and the
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Fig. 3. Dynamic characteristics of the chosen forced attractor: (a) phase plane; (b) two-dimensional Poincare map.
two-dimensional Poincare map for this point. The maximum Lyapunov exponent is 0.122 further con®rming the chaotic nature of this attractor. The investigation of the behavior of the forced system is suitably carried out using a stroboscopic Poincare map [41]. Instead of following the whole trajectory, stroboscopic points are taken for every forcing period at discrete time intervals. Then transient motions appear on the map as scattered dots while the emergence of a periodic attractor of order n would be seen as jumps between n ®xed points. The stroboscopic map eectively reduces the dimension of attractors in the phase space by one-dimension. It is evident that the simplicity of the presentation is lost when the strobing period, i.e., forcing period is not exactly determined. High accuracy is then required for the determination of the forcing frequency wf 0:001563709 chosen to be the frequency of period-1 attractor of the autonomous system at h 3040 (Fig. 2). A shooting method [42] was used to compute this forcing frequency accurately. 3. Results and discussion In the ®rst part of the investigation we study the eects of changing the forcing amplitude while maintaining the forcing frequency constant at w wf . In later sections we examine the eects of varying the frequency at constant value of the forcing amplitude. 3.1. Eect of amplitude forcing on the chaotic attractor A complete one-parameter stroboscopic bifurcation diagram of the forced system is shown in Fig. 4 where the Y-axis represents the stroboscopic dimensionless concentration of the reactant A. On the scale of Fig. 4 the system looks like an alternation of periodic regimes interrupted by chaotic-like strips, via period adding mechanism. The system evolves from a chaotic-like strip (region I) to a period-3 attractor that persists throughout region II. At the end of this region the system bifurcates to chaotic strips, (region III), themselves interrupted by periodic windows, and emerges at the end of this region as a period-2 attractor. The period-2 attractor persists throughout the region IV. At high forcing amplitudes and beyond the values of Am 4:86 the system is fully entrained as the forcing term (Eq. (7)) dominates and the system emerges as a period one attractor in region V. The behavior of the system is in fact more complex and an enlargement of portions of Fig. 4 is needed to identify the ®ner structures of the system behavior. 3.1.1. Region I (Am 0.00±0.12): interior catastrophes The limiting case of this region, enlarged in Fig. 5(a), is the autonomous chaotic system Am 0:0. A fully developed chaotic attractor characterizes the behavior of the system for forcing amplitudes up to
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Fig. 4. One-parameter stroboscopic diagram for the amplitude forcing.
Fig. 5. (a) Enlargement of region I of Fig. 4 for the range Am 2 0:0; 0:12; (b) corresponding Lyapunov exponents spectrum.
Am 0:0204 (close to point P1 ). Beyond this forcing amplitude the fully developed chaotic attractor is suddenly interrupted and gives rise to a three-banded chaotic attractor (point P2 ). The banded chaotic attractor bifurcates to a period-5 attractor (seen as period-3 attractor on the ®gure, since two periods are very close to each other) that bifurcates to a strip of chaos. The strip of chaos bifurcates again to a period-4 attractor and so on. The system alternates then between chaotic and periodic regimes by reverse period adding bifurcation and emerges as a period-3 attractor at Am 0:105 (close to point P3 ). It looks then that periodic windows, i.e., frequency locking occurs at small forcing amplitudes. In order to characterize accurately the nature of the emerging attractors, Lyapunov exponents are computed for this region. These exponents are one of the most practical indicators of chaotic behavior. A chaotic attractor is characterized by at least one positive Lyapunov exponent. The technique and the algorithm of Wolf et al. [43] were used to compute them. Fig. 5(b) shows the maximum component (excluding zero) of the Lyapunov exponents as function of the amplitude. The spectrum becomes negative for small values of amplitude Am 0:0295 indicating the termination of the chaotic regime and the emergence of a periodic attractor. The mechanism of bifurcation from the fully developed chaos to the three banded chaotic attractor is investigated by examining the two attractors P1 Am 0:020 and P2 Am 0:0205, shown in Fig. 5(a). The two points lie, respectively, in the fully developed and the banded chaotic attractors. Fig. 6(a) shows the
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phase plane for both the two attractors. It can be seen that the fully developed chaos lies completely inside the three-banded chaotic attractor. This is also con®rmed in Figs. 6(b) and (c) showing the two-dimensional Poincare maps for the two attractors. The attractor P2 of Fig. 6(c) goes through an ÔexplosionÕ while keeping the same structure as the original attractor P1 (Fig. 6(b)). Topologically then, a jump in size of the chaotic attractor has occurred. But since the new emerging attractor lies inside the original one, this bifurcation is a continuous change in the bifurcation mechanism, i.e., an interior catastrophe. The bifurcation, on the other hand, from chaotic attractors to periodic windows can be examined by considering, for instance, the point P3 Am 0:103 of Fig. 5(a). Dynamic simulations at this point are shown in Fig. 7(a). The third-iterate for the dimensionless concentration Y n 3 vs Y n (Fig. 7(a)) shows that the curve approaches the diagonal and almost becomes tangent at three distinct points. The chaotic behavior is destroyed then through type I intermittency mechanism as studied by Pomeau and Manneville [44]. As soon as the curve becomes tangent to the diagonal the chaotic attractor disappears and each of the three tangent point generates one stable point of the node form and another of the saddle form, i.e., saddle± node bifurcation. Fig. 7(b) shows time trace for the intermittency point P3 Am 0:103, while Fig. 7(c) shows oscillations just prior to the intermittency (i.e., chaotic) at Am 0:101. It can be seen that the average time interval of nearly periodic behavior of Fig. 7(b), has declined in Fig. 7(c), giving rise to more frequent bursts that are characteristics of a chaotic behavior.
Fig. 6. Transition from fully developed chaos (P1 , Fig. 5(a)) to three-banded chaos (P2 , Fig. 5(a)): (a) phase plane: P1 attractor ± solid; P2 attractor ± dashed; (b) two-dimensional Poincare map for point P1 ; (c) two-dimensional Poincare map for point P2 .
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Fig. 7. Transition from chaotic attractor to period-3 attractor (P3 of Fig. 5(a)): (a) third-iterate map (the trajectory is tangent to the diagonal in three points); (b) time trace at the point of intermittency Am 0:103; (c) time trace just prior to intermittency Am 0:101.
3.1.2. Regions III and IV: reverse period adding and quasi-periodicity Throughout region II the system is dominated by a period-3 attractor. Beyond the point corresponding to Am 1:45 (end of region II of Fig. 4), the system alternates between periodic and chaotic regimes via reverse period adding mechanism (region III), until it is fully entrained at high amplitudes (region IV and V of Fig. 4, enlarged in Fig. 8(a)). The ®nal bifurcation to the harmonic trajectory (fully entrained) is investigated by choosing the point P4 (Fig. 8(a)) corresponding to Am 4:81. Dynamic simulations (phase plane and two-dimensional Poincare map) shown in Figs. 8(b) and (c) suggest a quasi-periodic behavior. The maximum Lyapunov exponent is zero, further con®rming the quasi-periodic nature of the attractor. The system bifurcates then from the quasi-periodic attractor to the harmonic period-one attractor through Hopf bifurcation. This is in agreement with the conjecture made by McKarnin [45] that all tori that do not become phase-locked undergo a Hopf bifurcation at a period-one Hopf curve. The results of the amplitude forcing of the reactive system has shown that chaos can be controlled for even small values of the forcing amplitude. The practical relevance of these results is that this phenomenon is practically unavoidable given the small values at which chaotic regimes terminate and periodic regimes emerge, and also given the large width (in terms of amplitude) of the emerging periodic windows (Fig. 4). The eect of forcing on the performances of the reactive system can be seen in Fig. 9 showing the variations of the average conversion of the reactant (A) with the forcing amplitude. Starting from a conversion of 0.8666 corresponding to the unforced system Am 0, it can be seen that the conversion follows
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in general, a decreasing trend. Occasional bursts can be observed but in no case does the conversion increase higher than that of the unforced system. The amplitude forcing has, thus, a deteriorating eect on the performances of this reactive system. In Section 3.2, we study the eects of changes in forcing frequency while the amplitude is ®xed. 3.2. Eect of frequency forcing on the chaotic attractor The previous results have shown that the amplitude forcing of the chaotic behavior (for constant forcing frequency wf gives rise to six regions of chaotic behavior. These regions are, respectively, · region A1 : Am 2 [0, 0.025] (region I of Fig. 4, enlarged in Fig. 5(a)), · region A2 : Am 2 [0.052, 0.065] (region I of Fig. 4, enlarged in Fig. 5(a)), · region A3 : Am 2 [0.078. 0.100] (region I of Fig. 4, enlarged in Fig. 5(a)), · region A4 : Am 2 [1.640, 1.664] (region III of Fig. 4), · region A5 : Am 2 [1.696, 1.736] (region III of Fig. 4), · region A6 : Am 2 [1.975. 2.044] (region III of Fig. 4). In this section, we examine forcing each of these regions (for constant values of Am ) by changing the forcing frequency w, relative to the original forcing frequency wf . For each of this six regions and for each constant value of Am within these regions, stroboscopic maps are constructed as a function of the ratio
Fig. 8. Transition from quasi-periodic to period-1 attractor: (a) enlargement of the stroboscopic map of Fig. 4; (b) phase plane for point P4 ; (c) two-dimensional Poincare map for point P4 .
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Fig. 9. Variations of the average conversion of reactant (A) with the forcing amplitude.
w=wf . The dierent bifurcation mechanisms are then depicted. Since the computational costs required to generate the stroboscopic maps for the whole range can be prohibitive, the investigation was limited to forcing frequencies up to ®ve times the original forcing frequency wf , i.e., 0 6 w=wf 6 5. The results of the numerical investigation are summarized in Fig. 10(a) showing the dierent bifurcation mechanisms in the diagram (amplitude Am vs the forcing frequency w=wf . It can be seen from this ®gure that frequency forcing (with constant Am ) of any chaotic regime of regions A1 , A2 and A3 cannot produce any periodic behavior and hence chaotic attractors are expected in these regions. The situation is quite dierent for regions A4 , A5 and A6 . These regions can be considered as high amplitude regions. It can be seen from Fig. 10(a) that frequency forcing of these regions leads to an alternation of chaotic and periodic regimes up to a certain value of forcing frequency. Beyond that the system is characterized by chaotic behavior only.
Fig. 10. (a) Bifurcation mechanisms in the excitation diagram (amplitude±frequency) for six regions; (b) detailed excitation diagram, for frequency forcing of region A4 Am 2 1:640; 1:664 (numbers in parentheses indicate the periodicity of periodic regimes); (c) indicate a chaotic regime.
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Fig. 11. One-parameter stroboscopic diagram at the chaotic attractor Am 1:648.
Fig. 10(b) shows the detailed results for forcing region A4 Am 2 1:640; 1:664. This ®gure shows that starting from the center of forcing of the chaotic attractor w=wf 1 the system alternates between chaotic and periodic strips through period adding until high values of forcing frequency w=wf > 2:93 where chaos is the only attractor to the system. Figs. 11(a), (b) and (c) show the detailed bifurcation mechanisms, for example, for the value of Am 1:648 lying in the same region A4 . The limiting case for this system is the autonomous chaotic system w=wf 0. As the forcing frequency w increases, a period-1 attractor emerges that persists until w=wf 0:605 (point P5 , Fig. 11(a)). It then bifurcates to a tiny strip of chaos (P5 ) itself destroyed to give rise to a period-2 attractor. The period-2 attractor bifurcates to chaos at P6 (Fig. 11(a)). This point corresponds to the center of forcing w=wf 1. The tiny chaotic strip bifurcates to a period-3 attractor that bifurcates to chaos at P7 (Fig. 11(a)) and so on. It can be seen that as the forcing frequency increases (Figs. 11(b) and (c)), the periodic windows decrease (P9 , P10 , P11 ) and chaos dominates. At forcing frequency w=wf > 2:93 the system is dominated by chaos (Figs. 11(b) and (c)). To examine the ®ner structure of the system, parts of the stroboscopic map of Fig. 11(a) were enlarged in Fig. 12(a). They correspond to w=wf 2 1:23; 1:46. It can be seen that the period-3 attractor bifurcates to a chaotic attractor (point P7 ), itself interrupted by periodic windows until it emerges as period-4 attractor. The mechanism of destruction of chaos is once again through type I intermittency as illustrated in Fig. 12(b) showing the dynamics at point Q of Fig. 12(a). It can be seen that the third iterate map approaches the diagonal at three distinct points.
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Fig. 12. (a) Stroboscopic map (Fig. 11(a)) enlarged for 1:23 6 w=wf 6 1:46; (b) third-iterate map at point (Q). The trajectory is tangent to the diagonal in three points. P7 is a period-7 attractor.
Contrary to amplitude forcing, where periodic windows appear at very small forcing amplitudes, the frequency forcing of a chaotic attractor does not always stabilize the system. It was noticed for the studied system that frequency locking occurs only when high values of amplitudes are forced. The eect of the frequency forcing on the performances of the reactive system is seen in Fig. 13, showing the variations of the average conversion with the forcing frequency at Am 1:648 for which the conversion is 0.8401. The limiting case for this diagram is the original unforced chaotic attractor w 0 for which the conversion is 0.8662. It can be seen from this ®gure that the conversion pro®le is not monotonic and has an overall increasing trend. Moreover, for forcing frequencies w=wf in the range of [2.7, 4.5], the average conversion reaches values that are higher than that of the chosen forced attractor w=wf 1, point P). The maximum value of 0.8670 reached by the conversion at w=wf 4:05 (point Q) is even slightly higher than that of the original unforced system (conversion 0.8662). Figs. 14(a) and (b) show the time traces for the maximum conversion point (point Q) and the forced attractor (point P). The two ®gures shown in the same scale indicate clearly that the ¯uctuations of the chaotic oscillations around the mean are smaller in Fig. 14(b) than in Fig. 14(a).
Fig. 13. Variations of the average conversion of reactant (A) with forcing frequency. Point (P) is the chosen forced attractor and point (Q) is the point of maximum conversion.
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Fig. 14. Time traces showing the conversion for: (a) the chosen forced attractor (point P); (b) the point of maximum conversion (point Q).
4. Eect of the position of the center of forcing In this section, we present an analysis of the eect of the position of the center of forcing on the dynamics of the non-autonomous system. For this purpose we study the forcing of the same chaotic attractor h 3049 by changing the location of the center of forcing around the nominal value of r 2. As a ®rst step, the relative location of the center of forcing is displayed in Fig. 15(a) showing the continuity diagram of the autonomous system for h 3049 and the rest of the other unchanged parameters. The system admits two Hopf points located, respectively, at r 0:7785 and r 2:0624. The periodic branch emanating from the largest Hopf point loses its stability through period doubling. The period doubling sequence, better seen on the one-dimensional Poincare map (Fig. 15(b)), culminates into a banded and then a fully developed chaos for r 2 1:99; 2:003. The periodic branch regains its stability at r 1:989. From these diagrams it can be seen that a chaotic regime of the autonomous system is being forced as long as the center of forcing is located in the range r 2 1:99; 2:003. In order to assess the eect of the position of the center of forcing, a one-parameter stroboscopic bifurcation diagram is constructed when the center of forcing r is moved closer to the smallest Hopf point but still in the chaotic region, i.e., r 1:99 Am sin wt. The forcing frequency is kept unchanged. It can be seen from the stroboscopic map of Fig. 16(a) that the global structure of the
Fig. 15. (a) Continuity diagram for the autonomous system with h 3049 b 4=15; k 225; q 20; k 4=450 (solid ± stable, dash ± unstable, ®lled circles ± stable periodic branch, empty circles ± unstable periodic branch); (b) Poincare bifurcation diagram for the region r 2 1:98; 2:04.
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Fig. 16. (a) One-parameter stroboscopic bifurcation diagram when the center of forcing is r 1:99; (b) enlargement of the diagram for Am 2 0; 0:12.
Fig. 17. (a) One-parameter stroboscopic bifurcation diagram when the center of forcing is r 2:5; (b) enlargement of the diagram for Am 2 0; 0:25; (c) enlargement of the diagram for Am 2 4:5; 5:5.
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forced system is unchanged. The same ®ve regions depicted earlier in Fig. 4 can also be found in this ®gure. The ®ner structure enlarged in Fig. 16(b) for forcing amplitudes Am 2 0; 0:12 is also unchanged, except that the frequency locking occurs at smaller values of forcing amplitudes Am 0:0184 compared to Am 0:0204 when the center of forcing is r 2. Similar observations can be made about the ®ner structure of the other regions. Moreover, stroboscopic maps constructed for various values of the center of forcing within the chaotic region did not indicate any dramatic change in the behavior of the dynamics of the forced system. The only noticeable results are the change in the values of the forcing amplitudes at which the dierent bifurcation mechanisms explained in the previous section occur. The insensitivity of the ®ner structure of the forced system is an indication of a quite robust behavior of the system with respect to the position of the center of forcing within the chaotic regime. When on the other hand the center of forcing is moved away from the chaotic regime and closer to the larger of the two Hopf points, the behavior of the emerging forced system changes substantially. Fig. 17(a) shows the stroboscopic bifurcation diagram when the center of forcing is chosen to be r 2:5 Am sin wt. While the overall structure of the emerging system is kept relatively unchanged, the ®ner structure is however altered. Looking at the enlargement of the diagram for small forcing amplitudes Am 2 0; 0:25 (Fig. 17(b)) it can be seen that starting from a low periodic regime the forced system undergoes a period doubling sequence at Am 0:081 leading to a banded chaos and then to a fully developed chaos at Am 0:122. The fully developed chaos persists until Am 0:0161, where a sudden apparition of a ®ve period attractor is observed. The emergence of the periodic attractor is due to an interior catastrophe similar to the one discussed in the previous section. Each branch of the ®ve periodic attractors goes through a period doubling sequence that culminates in chaos and then a three periodic attractor emerges at Am 0:215. From there the period attractor will undergo a bifurcation similar in its overall structure to the regions II±V found earlier in Fig. 4. Notably it can be seen from Fig. 17(c) that at high forcing amplitude the system bifurcates to period-1 attractor through torus bifurcation. 5. Conclusions This paper has carried out a numerical investigation of the eects of periodically forcing a two-phase CSTR where a Gray±Scott autocatalytic reaction with decay is taking place. The investigation has shown that regular regimes can emerge from the original chaotic autonomous system when the feed ¯ow rate is perturbed either in amplitude or in frequency. Dierent mechanisms for the transition between chaotic regions and periodic windows have been identi®ed and analyzed, including period adding and tangent bifurcation. An excitation (amplitude±frequency) diagram was constructed for the system. The diagram has shown that, for the studied system, it is not always possible to control chaos by frequency forcing. Stabilization of chaos is only possible by frequency forcing of regions of high amplitudes with a relatively low forcing frequency. An analysis of the eect of the location of the center of forcing has revealed a noticeable robustness of the dynamics of the forced system for a variation of the location of the center of forcing in the region yielding chaotic behavior. When the center of forcing is moved away from this region and closer to the Hopf points of the autonomous system, a noticeable change in the dynamics is observed especially at low forcing amplitudes. Numerical simulations have shown that the reaction conversion deteriorates with the increase in the forcing amplitude but it can be improved by the selection of appropriate forcing frequency. References [1] Alhumaizi K, Aris R. Chaos in a simple two-phase reactor. Chaos, Solitons & Fractals 1994;4:1985. [2] Elnashaie SSEH, Abasaeed AE, Ibrahim G. On the practical relevance of bifurcation and chaos in chemical and biochemical reaction engineering. Fractals 1997;5:549. [3] Teymour F, Ray WH. The dynamic behavior of continuous polymerization reactors. VI Complex dynamics in full scale reactors. Chem Eng Sci 1992;47:4133. [4] Choi KY, Ray WH. The dynamic behavior of ¯uidized bed reactors for solid catalysed gas phase ole®n polymerization. Chem Eng Sci 1990;41:2261.
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[5] Elshishini SS, Elnashaie SSEH. Digital simulation of industrial ¯uid catalytic cracking unit I. Bifurcation and its implications. Chem Eng Sci 1990;45:553. [6] Elnashaie SSEH, Elshishini SS. Digital simulation of industrial ¯uid catalytic cracking units IV. Dynamic behavior 1993;48:567. [7] Elnashaie SSEH, Abasaeed AE, Elshishini SS. Digital simulation of industrial ¯uid catalytic cracking units V. Static and dynamic bifurcation. Chem Eng Sci 1995;50:1635. [8] Elnashaie SSEH, Elshishini SS. Dynamic modeling, bifurcation and chaotic behavior of gas±solid catalytic reactors. London: Gordon and Breach; 1996. [9] Qin F, Wolf EE. Vibrational control of chaotic self sustained oscillations during CO oxidation on a Rh±SiO2 catalyst. Chem Eng Sci 1995;50:117. [10] Vleeshouwer PHM, Garton RD, Fortuin MH. Analysis of limit cycles in an industrial OXO reactor. Chem Eng Sci 1992;47:254. [11] Scheintuch M, Adaje J. Comparison of multiplicity patterns of a single catalytic pellet and a ®xed catalytic bed for ethylene oxidation. Chem Eng Sci 1990;45:1897. [12] Elnashaie SS, Fouad MMK, Elshishini SS. The use of mathematical modeling to investigate the eect of chemisorption on the dynamic behavior of catalytic reactors. Partial oxidation of o-xylene in ¯uidized beds. Math Comput Modell 1990;13:11. [13] Elnashaie SSEH, Wagialla KM, Helal AM. The use of mathematical and computer models to explore the applicability of ¯uidized bed technology for highly exothermic reactions II. Oxidative dehydrogenation of ethylbenzene to styrene. Math Comput Modell 1991;15:43. [14] Ibrahim G, Teymour FA, Elnashaie SSEH. Periodic and chaotic behavior of substrate-inhibited enzymatic reactions with hydrogen ions production. Appl Biochem Biotech 1995;55:175. [15] Elnashaie SSEH, Ibranhim G, Teymour FA. Chaotic behavior of an acetylcholinestrase enzyme system. Chaos, Solitons & Fractals 1995;6:993. [16] Homan U, Schadlich HK. The in¯uence of reaction orders and of changes on the total number of moles on the conversion in a periodically operated CSTR. Chem Eng Sci 1986;41:2733. [17] Silveston PL, Hudgins RR, Adesina AA, Ross GS, Feimer JL. Activity and selectivity control through period composition forcing over Fischer-Tropsh catalysts. Chem Eng Sci 1986;41:923. [18] Gray P, Scott SK. Autocatalytic reactions in the isothermal, continuous stirred tank reactor. Isola and other forms of multiplicity. Chem Eng Sci 1983;38:29. [19] Gray P, Scott SK. Autocatalytic reactions in the isothermal, continuous stirred tank reactor. Oscillations and instabilities in the system A 2B ! 3B; B ! C. Chem Eng Sci 1984;39:1087. [20] Gray P, Scott SK. Archetypal response patterns of open chemical systems with two components. Philos Trans Sci 1990;A332:69. [21] Scott SK, Tomlin AS. Period doubling and other complex bifurcations in nonisothermal chemical systems. Philos Trans R Soc London 1990;A332:51. [22] Peng B, Scott SK, Showalter K. Period doubling and chaos in a three-variable autocatalotor. J Phys Chem 1990;94:5243. [23] Gray P, Scott SK. Chemical oscillation and instabilities: nonlinear chemical kinetics. Oxford: Oxford University Press; 1990. [24] Troy WC. Mathematical analysis of the orgenator model of the Belousov±Zhabotinskii reaction. In: Field RJ, Burger M, editors. Oscillations and traveling waves in chemical systems. New York: Wiley; 1985. [25] Scott SK. Chemical chaos. Oxford: Clarendon Press; 1991. [26] Lorenz EN. Deterministic nonperiodic ¯ow. J Atoms Sci 1963;20:130. [27] Shinbrot T, Grebogi C, Ott E, Yorke JA. Using small perturbations to control chaos. Nature 1993;363:411. [28] Shinbrot T. Synchronization of coupled maps and stable windows. Phys Rev E 1994;50:3230. [29] Alekseev VV, Loskutov AY. Control of a system with a strange attractor through periodic parameter action. Sov Phys Dokl 1987;32:1346. [30] Lima R, Pettini M. Suppression of chaos by resonant parametric perturbations. Phys Rev Lett A 1990;41:726. [31] Li YX, Halloy J, Martiel JL, Wurster B, Goldbeter A. Suppression of chaos by periodic oscillation in a model for cyclic AMP signaling in dictyostelium cells. Experientia 1992;48:603. [32] Ajbar A, Elnashaie SSEH. Controlling chaos by periodic perturbations in nonisothermal ¯uidized-bed reactor. AIChE J 1996;42:3008. [33] Mankin JC, Hudson JL. Oscillatory and chaotic behaviour of a forced exothermic chemical reaction. Chem Eng Sci 1984;39:1807. [34] Kevrekidis IG, Schmidt LD, Aris R. Some common features of periodically forced reacting systems. Chem Eng Sci 1986;41:1263. [35] Cordonier GA, Schmidt LD, Aris R. Forced oscillations of chemical reactors with multiple steady-states. Chem Eng Sci 1990;45:1659. [36] Tambe SS, Kulkarni BD. Intermittency route to chaos in a periodically forced model reaction system. Chem Eng Sci 1993;48:2817. [37] Elnashaie SSEH, Absahar ME. Chaotic behavior of periodically forced ¯uidized bed catalytic reactors with consecutive exothermic reactions. Chem Eng Sci 1994;49:2483. [38] Abasaeed AE, Elnashaie SSEH. The eect of external disturbances on the performances and chaotic behavior of industrial FCC units. Chaos, Solitons & Fractals 1997;8:1941. [39] Alhumaizai K, Elnashaie SSEH. Eect of control loop con®guration on the bifurcation behavior and gasoline yield of industrial ¯uid catalytic cracking (FCC) units. Math Comput Modell 1997;25:37. [40] Abasaeed AE, Elnashaie SSEH. On the chaotic behavior of externally forced industrial ¯uid catalytic cracking units. Chaos, Solitons & Fractals 1998;9:455. [41] Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems and bifurcations of vector ®elds. New York: Springer; 1983. [42] Kubicek M, Marek M. Computation methods in bifurcation theory and dissipative structures. New York: Springer; 1983.
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[43] Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov exponents from a time series. Physica 1985;16D:285. [44] Pomeau Y, Manneville P. Intermittent transition to turbulence in dissipative dynamical systems. Commun Math Phys 1980;74:189. [45] McKarnin MA, Schmidt LD, Aris R. Response of nonlinear oscillations to forced oscillations: three chemical reaction systems. Chem Eng Sci 1988;43:2833.
www.elsevier.nl/locate/chaos
Stabilization of chaotic behavior in a two-phase autocatalytic reactor Abdelhamid Ajbar * Department of Chemical Engineering, College of Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia Accepted 1 March 2000
Abstract The eect of feed forcing on the chaotic behavior found by Alhumaizi and Aris (Chaos, Solitons & Fractals 1994;4:1985) in the k1 k2 autocatalytic reaction of Gray and Scott A 2B ! 3B; B ! C, taking place in a two-phase reactor, is numerically investigated. It is shown that by periodic forcing of the feed ¯ow rate, the chaotic behavior of the reactor can be controlled for even small forcing amplitudes or frequencies. The behavior of the forced system alternates between chaotic and period behavior via period adding, quasiperiodic and intermittent bifurcations. The results of the stabilization of the chaotic behavior are summarized in an amplitude±frequency excitation diagram showing the dierent bifurcation mechanisms. A numerical investigation has also revealed that the reaction conversion can be improved by an appropriate selection of the forcing frequency. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction Models of chemically reactive systems are known to exhibit a variety of non-linear phenomena ranging from steady-state multiplicity to complex quasi-periodic as well as strange chaotic and non-chaotic behavior. Bifurcation and chaotic behavior exhibited by a number of these models are not only of theoretical importance, but also these phenomena are rather important in a number of industrial chemical and biochemical processes [2]. Some of the important examples include reactors for the production of polymers and co-polymers either in liquid phase or in gas-phase catalytic processes [3,4], the industrial ¯uid catalytic cracking (FCC) units [5±8] for the cracking of gasoil to high octane number gasoline, the catalytic oxidation of CO in car exhaust systems [9], the catalyzed OXO reactions involving the sequential conversion of ole®ns to aldehydes and aldehydes to alcohol [10], the highly exothermic catalytic partial oxidation reactions (e.g., ethylene to ethylene oxide, o-xylene to phthalic anhydride) [11±13] as well as a number of biochemical systems such as substrate inhibited enzymatic reactions [14,15]. There are many practical reasons for interest in studying the chaotic behavior in chemically reactive systems. As pointed out by a number of authors [16,17], a process operating around oscillatory trajectories may be more advantageous in terms of mean productivity or selectivity as compared to operating the process at steady-states. On the other hand, large ¯uctuations or instabilities in chemical reactors can pose safety hazards. Knowledge of how to predict and stabilize this behavior is necessary for proper design and operation. One necessary requirement for any chemical reaction system to exhibit `exotic' behavior such as multiplicity or sustained oscillations is that it should have a Ôfeedback mechanismÕ. Some intermediate species
*
Fax: +966-1-467-8770. E-mail address: [email protected] (A. Ajbar).
0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 0 5 4 - 0
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or products of the process must be able to in¯uence the rate of earlier steps. This eect may be thermal as in the classical case of ®rst-order exothermic reaction where the temperature rise, consequent on heat release, in¯uences the reaction rate constant. The feedback may also be chemical as for example in non-monotonic kinetics or in autocatalytic reactions, where at least one product of reaction increases the reaction rate and thus its own rate of production. One such system is the cubic autocatalysis with catalyst decay k1
A 2B ! 3B; k2
B ! C;
with rate r1 k1 AB2 ;
with rate r2 k2 B
1 2
taking place in an isothermal continuous stirred tank reactor (CSTR). This system has been studied extensively by Gray and Scott and co-workers [18±22]. Using a two-dimensional continuous model, the authors showed that this reactive system can display a whole range of non-linear phenomena ranging from multiplicity to self-sustained periodic solutions. The considerable interest in studying autocatalytic reactions stem from the fact that these reaction models are at the core of some common reactions such as hydrogen and carbon monoxide oxidation [23], and oreganator reaction [24]. Moreover, the richness of the two-variable Gray±Scott model is another evidence that low-dimensional deterministic chemical systems can describe much of the complex behavior found in more complicated ones [25]. The original two-variable Gray±Scott model was further developed by a number of investigators. It was found for instance that complex periodic and aperiodic behavior can result when coupling the cubic autocatalytic reaction with self heating feedback element [21] or when the reaction is coupled with an equivalent isothermal feedback step [22]. In their contribution, Alumaizi and Aris [1] studied the autocatalytic reactions (1) and (2) in a two-phase reactor, consisting of two isothermal well-stirred tanks separated by a semipermeable membrane (Fig. 1). The autocatalytic reaction (1) occurs only in one tank, Ôthe reactorÕ to where the autocatalyst B is fed directly. The reactant A diuses through the membrane from the other compartment ÔreservoirÕ. This two-phase reactor allows the analysis of the reactive system using a three-variable model, with the concentration of the reactant in the reservoir providing the third variable in this extended Gray and Scott model. The authors showed that the modi®ed reactive system can exhibit complex chaotic and nonchaotic behavior. This also con®rms that chaotic chemical systems can be obtained when two or more gradientless system such as CSTR are connected to each other via heat or mass transfer. In this paper, we analyze the dynamics of the chaotic behavior found by the authors [1], when the reactive system is subjected to periodic perturbations in feed conditions, such as the feed ¯ow rate. Chaotic attractors show extreme sensitivity to initial conditions as nearby process trajectories diverges
Fig. 1. Schematic diagram of the two-phase autocatalytic reactor. Fig. 2. Poincare bifurcation diagram of the autonomous system (point (C) is the chosen attractor for forcing).
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exponentially. This characteristic, known as the Ôbutter¯y eectÕ [26], is a formidable obstacle to an adequate control of chaotic behavior. The application of periodic parameter perturbations to stabilize unstable orbits of a chaotic attractor is one of the ideas that have been developed, in recent years, for means to control chaotic behavior [27,28]. While the eects associated with these perturbations are generally dicult to predict, nevertheless this technique is easy to implement. Furthermore in some cases [29±32] dramatic changes in the fundamental characteristics of the chaotic attractor occur at very small values of the forcing amplitude or frequency, which are almost impossible to avoid in practice. It should be noted that the eect of feed disturbances on the behavior of chemically reactive systems having periodic and quasi-periodic attractors has been investigated by a number of authors [33±40]. It was clearly demonstrated that for certain values of amplitude and frequency of the forcing variable, the periodic and quasi-periodic attractors of the autonomous system can turn into chaotic attractors. In this paper the opposite problem is tackled, starting from a chaotic attractor of the autonomous system, it is shown that a variety of periodic and aperiodic behavior can be obtained by appropriate forcing of the feed conditions.
2. Process model and presentation techniques The autonomous model of the two-phase reactive system is described by the following three ordinary dierential equations. The detailed derivation of the model was carried out by Alhumaizi and Aris [1] dx z ÿ x x ÿ ÿ xy 2 ; ds k h
3
dy b ÿ y xy 2 ÿ ky; ds h
4
q
dz zÿx r ÿ 1 ÿ z: ds k h
5
The model state variables are, respectively, the dimensionless concentrations of reactant A, intermediate product B and reactant A in the reservoir x
A ; A0
y
B ; A0
z
A : A0
6
The rest of the model parameters are the dimensionless concentration of the feed of B, b B0 =A0 , the ratio of the tank volumes q v=V , the ratio of the feed ¯ow rates r q=Q, the dimensionless decay rate constant k, the dimensionless mass transfer resistance k and the reactor residence time h. In this paper, we study the eects of periodically forcing the ratio r of feed ¯ow rates. The forcing takes the following simple form: r r0 Am sin ws;
7
where r0 is the baseline value of r and Am and w are the dimensionless forcing amplitude and frequency, respectively. Chaotic behavior in the autonomous model was found for a wide range of model parameters. For example, for the following values of parameters: b
4 ; 15
k 225;
r0 2;
q 20;
k
4 : 450
8
Fig. 2 shows the Poincare bifurcation diagram for h 2 3038; 3050. It can be seen that the autonomous system goes through a series of period doubling that culminates into banded and then fully developed chaos for h 2 3044:95; 3049:55. In this paper, we study the forcing of the chaotic attractor at h 3049:00 (point C) that lies in the chaotic strip of the autonomous system depicted in Fig. 2. Figs. 3(a) and (b) show the phase plane and the
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Fig. 3. Dynamic characteristics of the chosen forced attractor: (a) phase plane; (b) two-dimensional Poincare map.
two-dimensional Poincare map for this point. The maximum Lyapunov exponent is 0.122 further con®rming the chaotic nature of this attractor. The investigation of the behavior of the forced system is suitably carried out using a stroboscopic Poincare map [41]. Instead of following the whole trajectory, stroboscopic points are taken for every forcing period at discrete time intervals. Then transient motions appear on the map as scattered dots while the emergence of a periodic attractor of order n would be seen as jumps between n ®xed points. The stroboscopic map eectively reduces the dimension of attractors in the phase space by one-dimension. It is evident that the simplicity of the presentation is lost when the strobing period, i.e., forcing period is not exactly determined. High accuracy is then required for the determination of the forcing frequency wf 0:001563709 chosen to be the frequency of period-1 attractor of the autonomous system at h 3040 (Fig. 2). A shooting method [42] was used to compute this forcing frequency accurately. 3. Results and discussion In the ®rst part of the investigation we study the eects of changing the forcing amplitude while maintaining the forcing frequency constant at w wf . In later sections we examine the eects of varying the frequency at constant value of the forcing amplitude. 3.1. Eect of amplitude forcing on the chaotic attractor A complete one-parameter stroboscopic bifurcation diagram of the forced system is shown in Fig. 4 where the Y-axis represents the stroboscopic dimensionless concentration of the reactant A. On the scale of Fig. 4 the system looks like an alternation of periodic regimes interrupted by chaotic-like strips, via period adding mechanism. The system evolves from a chaotic-like strip (region I) to a period-3 attractor that persists throughout region II. At the end of this region the system bifurcates to chaotic strips, (region III), themselves interrupted by periodic windows, and emerges at the end of this region as a period-2 attractor. The period-2 attractor persists throughout the region IV. At high forcing amplitudes and beyond the values of Am 4:86 the system is fully entrained as the forcing term (Eq. (7)) dominates and the system emerges as a period one attractor in region V. The behavior of the system is in fact more complex and an enlargement of portions of Fig. 4 is needed to identify the ®ner structures of the system behavior. 3.1.1. Region I (Am 0.00±0.12): interior catastrophes The limiting case of this region, enlarged in Fig. 5(a), is the autonomous chaotic system Am 0:0. A fully developed chaotic attractor characterizes the behavior of the system for forcing amplitudes up to
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Fig. 4. One-parameter stroboscopic diagram for the amplitude forcing.
Fig. 5. (a) Enlargement of region I of Fig. 4 for the range Am 2 0:0; 0:12; (b) corresponding Lyapunov exponents spectrum.
Am 0:0204 (close to point P1 ). Beyond this forcing amplitude the fully developed chaotic attractor is suddenly interrupted and gives rise to a three-banded chaotic attractor (point P2 ). The banded chaotic attractor bifurcates to a period-5 attractor (seen as period-3 attractor on the ®gure, since two periods are very close to each other) that bifurcates to a strip of chaos. The strip of chaos bifurcates again to a period-4 attractor and so on. The system alternates then between chaotic and periodic regimes by reverse period adding bifurcation and emerges as a period-3 attractor at Am 0:105 (close to point P3 ). It looks then that periodic windows, i.e., frequency locking occurs at small forcing amplitudes. In order to characterize accurately the nature of the emerging attractors, Lyapunov exponents are computed for this region. These exponents are one of the most practical indicators of chaotic behavior. A chaotic attractor is characterized by at least one positive Lyapunov exponent. The technique and the algorithm of Wolf et al. [43] were used to compute them. Fig. 5(b) shows the maximum component (excluding zero) of the Lyapunov exponents as function of the amplitude. The spectrum becomes negative for small values of amplitude Am 0:0295 indicating the termination of the chaotic regime and the emergence of a periodic attractor. The mechanism of bifurcation from the fully developed chaos to the three banded chaotic attractor is investigated by examining the two attractors P1 Am 0:020 and P2 Am 0:0205, shown in Fig. 5(a). The two points lie, respectively, in the fully developed and the banded chaotic attractors. Fig. 6(a) shows the
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phase plane for both the two attractors. It can be seen that the fully developed chaos lies completely inside the three-banded chaotic attractor. This is also con®rmed in Figs. 6(b) and (c) showing the two-dimensional Poincare maps for the two attractors. The attractor P2 of Fig. 6(c) goes through an ÔexplosionÕ while keeping the same structure as the original attractor P1 (Fig. 6(b)). Topologically then, a jump in size of the chaotic attractor has occurred. But since the new emerging attractor lies inside the original one, this bifurcation is a continuous change in the bifurcation mechanism, i.e., an interior catastrophe. The bifurcation, on the other hand, from chaotic attractors to periodic windows can be examined by considering, for instance, the point P3 Am 0:103 of Fig. 5(a). Dynamic simulations at this point are shown in Fig. 7(a). The third-iterate for the dimensionless concentration Y n 3 vs Y n (Fig. 7(a)) shows that the curve approaches the diagonal and almost becomes tangent at three distinct points. The chaotic behavior is destroyed then through type I intermittency mechanism as studied by Pomeau and Manneville [44]. As soon as the curve becomes tangent to the diagonal the chaotic attractor disappears and each of the three tangent point generates one stable point of the node form and another of the saddle form, i.e., saddle± node bifurcation. Fig. 7(b) shows time trace for the intermittency point P3 Am 0:103, while Fig. 7(c) shows oscillations just prior to the intermittency (i.e., chaotic) at Am 0:101. It can be seen that the average time interval of nearly periodic behavior of Fig. 7(b), has declined in Fig. 7(c), giving rise to more frequent bursts that are characteristics of a chaotic behavior.
Fig. 6. Transition from fully developed chaos (P1 , Fig. 5(a)) to three-banded chaos (P2 , Fig. 5(a)): (a) phase plane: P1 attractor ± solid; P2 attractor ± dashed; (b) two-dimensional Poincare map for point P1 ; (c) two-dimensional Poincare map for point P2 .
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Fig. 7. Transition from chaotic attractor to period-3 attractor (P3 of Fig. 5(a)): (a) third-iterate map (the trajectory is tangent to the diagonal in three points); (b) time trace at the point of intermittency Am 0:103; (c) time trace just prior to intermittency Am 0:101.
3.1.2. Regions III and IV: reverse period adding and quasi-periodicity Throughout region II the system is dominated by a period-3 attractor. Beyond the point corresponding to Am 1:45 (end of region II of Fig. 4), the system alternates between periodic and chaotic regimes via reverse period adding mechanism (region III), until it is fully entrained at high amplitudes (region IV and V of Fig. 4, enlarged in Fig. 8(a)). The ®nal bifurcation to the harmonic trajectory (fully entrained) is investigated by choosing the point P4 (Fig. 8(a)) corresponding to Am 4:81. Dynamic simulations (phase plane and two-dimensional Poincare map) shown in Figs. 8(b) and (c) suggest a quasi-periodic behavior. The maximum Lyapunov exponent is zero, further con®rming the quasi-periodic nature of the attractor. The system bifurcates then from the quasi-periodic attractor to the harmonic period-one attractor through Hopf bifurcation. This is in agreement with the conjecture made by McKarnin [45] that all tori that do not become phase-locked undergo a Hopf bifurcation at a period-one Hopf curve. The results of the amplitude forcing of the reactive system has shown that chaos can be controlled for even small values of the forcing amplitude. The practical relevance of these results is that this phenomenon is practically unavoidable given the small values at which chaotic regimes terminate and periodic regimes emerge, and also given the large width (in terms of amplitude) of the emerging periodic windows (Fig. 4). The eect of forcing on the performances of the reactive system can be seen in Fig. 9 showing the variations of the average conversion of the reactant (A) with the forcing amplitude. Starting from a conversion of 0.8666 corresponding to the unforced system Am 0, it can be seen that the conversion follows
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in general, a decreasing trend. Occasional bursts can be observed but in no case does the conversion increase higher than that of the unforced system. The amplitude forcing has, thus, a deteriorating eect on the performances of this reactive system. In Section 3.2, we study the eects of changes in forcing frequency while the amplitude is ®xed. 3.2. Eect of frequency forcing on the chaotic attractor The previous results have shown that the amplitude forcing of the chaotic behavior (for constant forcing frequency wf gives rise to six regions of chaotic behavior. These regions are, respectively, · region A1 : Am 2 [0, 0.025] (region I of Fig. 4, enlarged in Fig. 5(a)), · region A2 : Am 2 [0.052, 0.065] (region I of Fig. 4, enlarged in Fig. 5(a)), · region A3 : Am 2 [0.078. 0.100] (region I of Fig. 4, enlarged in Fig. 5(a)), · region A4 : Am 2 [1.640, 1.664] (region III of Fig. 4), · region A5 : Am 2 [1.696, 1.736] (region III of Fig. 4), · region A6 : Am 2 [1.975. 2.044] (region III of Fig. 4). In this section, we examine forcing each of these regions (for constant values of Am ) by changing the forcing frequency w, relative to the original forcing frequency wf . For each of this six regions and for each constant value of Am within these regions, stroboscopic maps are constructed as a function of the ratio
Fig. 8. Transition from quasi-periodic to period-1 attractor: (a) enlargement of the stroboscopic map of Fig. 4; (b) phase plane for point P4 ; (c) two-dimensional Poincare map for point P4 .
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Fig. 9. Variations of the average conversion of reactant (A) with the forcing amplitude.
w=wf . The dierent bifurcation mechanisms are then depicted. Since the computational costs required to generate the stroboscopic maps for the whole range can be prohibitive, the investigation was limited to forcing frequencies up to ®ve times the original forcing frequency wf , i.e., 0 6 w=wf 6 5. The results of the numerical investigation are summarized in Fig. 10(a) showing the dierent bifurcation mechanisms in the diagram (amplitude Am vs the forcing frequency w=wf . It can be seen from this ®gure that frequency forcing (with constant Am ) of any chaotic regime of regions A1 , A2 and A3 cannot produce any periodic behavior and hence chaotic attractors are expected in these regions. The situation is quite dierent for regions A4 , A5 and A6 . These regions can be considered as high amplitude regions. It can be seen from Fig. 10(a) that frequency forcing of these regions leads to an alternation of chaotic and periodic regimes up to a certain value of forcing frequency. Beyond that the system is characterized by chaotic behavior only.
Fig. 10. (a) Bifurcation mechanisms in the excitation diagram (amplitude±frequency) for six regions; (b) detailed excitation diagram, for frequency forcing of region A4 Am 2 1:640; 1:664 (numbers in parentheses indicate the periodicity of periodic regimes); (c) indicate a chaotic regime.
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Fig. 11. One-parameter stroboscopic diagram at the chaotic attractor Am 1:648.
Fig. 10(b) shows the detailed results for forcing region A4 Am 2 1:640; 1:664. This ®gure shows that starting from the center of forcing of the chaotic attractor w=wf 1 the system alternates between chaotic and periodic strips through period adding until high values of forcing frequency w=wf > 2:93 where chaos is the only attractor to the system. Figs. 11(a), (b) and (c) show the detailed bifurcation mechanisms, for example, for the value of Am 1:648 lying in the same region A4 . The limiting case for this system is the autonomous chaotic system w=wf 0. As the forcing frequency w increases, a period-1 attractor emerges that persists until w=wf 0:605 (point P5 , Fig. 11(a)). It then bifurcates to a tiny strip of chaos (P5 ) itself destroyed to give rise to a period-2 attractor. The period-2 attractor bifurcates to chaos at P6 (Fig. 11(a)). This point corresponds to the center of forcing w=wf 1. The tiny chaotic strip bifurcates to a period-3 attractor that bifurcates to chaos at P7 (Fig. 11(a)) and so on. It can be seen that as the forcing frequency increases (Figs. 11(b) and (c)), the periodic windows decrease (P9 , P10 , P11 ) and chaos dominates. At forcing frequency w=wf > 2:93 the system is dominated by chaos (Figs. 11(b) and (c)). To examine the ®ner structure of the system, parts of the stroboscopic map of Fig. 11(a) were enlarged in Fig. 12(a). They correspond to w=wf 2 1:23; 1:46. It can be seen that the period-3 attractor bifurcates to a chaotic attractor (point P7 ), itself interrupted by periodic windows until it emerges as period-4 attractor. The mechanism of destruction of chaos is once again through type I intermittency as illustrated in Fig. 12(b) showing the dynamics at point Q of Fig. 12(a). It can be seen that the third iterate map approaches the diagonal at three distinct points.
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Fig. 12. (a) Stroboscopic map (Fig. 11(a)) enlarged for 1:23 6 w=wf 6 1:46; (b) third-iterate map at point (Q). The trajectory is tangent to the diagonal in three points. P7 is a period-7 attractor.
Contrary to amplitude forcing, where periodic windows appear at very small forcing amplitudes, the frequency forcing of a chaotic attractor does not always stabilize the system. It was noticed for the studied system that frequency locking occurs only when high values of amplitudes are forced. The eect of the frequency forcing on the performances of the reactive system is seen in Fig. 13, showing the variations of the average conversion with the forcing frequency at Am 1:648 for which the conversion is 0.8401. The limiting case for this diagram is the original unforced chaotic attractor w 0 for which the conversion is 0.8662. It can be seen from this ®gure that the conversion pro®le is not monotonic and has an overall increasing trend. Moreover, for forcing frequencies w=wf in the range of [2.7, 4.5], the average conversion reaches values that are higher than that of the chosen forced attractor w=wf 1, point P). The maximum value of 0.8670 reached by the conversion at w=wf 4:05 (point Q) is even slightly higher than that of the original unforced system (conversion 0.8662). Figs. 14(a) and (b) show the time traces for the maximum conversion point (point Q) and the forced attractor (point P). The two ®gures shown in the same scale indicate clearly that the ¯uctuations of the chaotic oscillations around the mean are smaller in Fig. 14(b) than in Fig. 14(a).
Fig. 13. Variations of the average conversion of reactant (A) with forcing frequency. Point (P) is the chosen forced attractor and point (Q) is the point of maximum conversion.
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Fig. 14. Time traces showing the conversion for: (a) the chosen forced attractor (point P); (b) the point of maximum conversion (point Q).
4. Eect of the position of the center of forcing In this section, we present an analysis of the eect of the position of the center of forcing on the dynamics of the non-autonomous system. For this purpose we study the forcing of the same chaotic attractor h 3049 by changing the location of the center of forcing around the nominal value of r 2. As a ®rst step, the relative location of the center of forcing is displayed in Fig. 15(a) showing the continuity diagram of the autonomous system for h 3049 and the rest of the other unchanged parameters. The system admits two Hopf points located, respectively, at r 0:7785 and r 2:0624. The periodic branch emanating from the largest Hopf point loses its stability through period doubling. The period doubling sequence, better seen on the one-dimensional Poincare map (Fig. 15(b)), culminates into a banded and then a fully developed chaos for r 2 1:99; 2:003. The periodic branch regains its stability at r 1:989. From these diagrams it can be seen that a chaotic regime of the autonomous system is being forced as long as the center of forcing is located in the range r 2 1:99; 2:003. In order to assess the eect of the position of the center of forcing, a one-parameter stroboscopic bifurcation diagram is constructed when the center of forcing r is moved closer to the smallest Hopf point but still in the chaotic region, i.e., r 1:99 Am sin wt. The forcing frequency is kept unchanged. It can be seen from the stroboscopic map of Fig. 16(a) that the global structure of the
Fig. 15. (a) Continuity diagram for the autonomous system with h 3049 b 4=15; k 225; q 20; k 4=450 (solid ± stable, dash ± unstable, ®lled circles ± stable periodic branch, empty circles ± unstable periodic branch); (b) Poincare bifurcation diagram for the region r 2 1:98; 2:04.
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Fig. 16. (a) One-parameter stroboscopic bifurcation diagram when the center of forcing is r 1:99; (b) enlargement of the diagram for Am 2 0; 0:12.
Fig. 17. (a) One-parameter stroboscopic bifurcation diagram when the center of forcing is r 2:5; (b) enlargement of the diagram for Am 2 0; 0:25; (c) enlargement of the diagram for Am 2 4:5; 5:5.
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forced system is unchanged. The same ®ve regions depicted earlier in Fig. 4 can also be found in this ®gure. The ®ner structure enlarged in Fig. 16(b) for forcing amplitudes Am 2 0; 0:12 is also unchanged, except that the frequency locking occurs at smaller values of forcing amplitudes Am 0:0184 compared to Am 0:0204 when the center of forcing is r 2. Similar observations can be made about the ®ner structure of the other regions. Moreover, stroboscopic maps constructed for various values of the center of forcing within the chaotic region did not indicate any dramatic change in the behavior of the dynamics of the forced system. The only noticeable results are the change in the values of the forcing amplitudes at which the dierent bifurcation mechanisms explained in the previous section occur. The insensitivity of the ®ner structure of the forced system is an indication of a quite robust behavior of the system with respect to the position of the center of forcing within the chaotic regime. When on the other hand the center of forcing is moved away from the chaotic regime and closer to the larger of the two Hopf points, the behavior of the emerging forced system changes substantially. Fig. 17(a) shows the stroboscopic bifurcation diagram when the center of forcing is chosen to be r 2:5 Am sin wt. While the overall structure of the emerging system is kept relatively unchanged, the ®ner structure is however altered. Looking at the enlargement of the diagram for small forcing amplitudes Am 2 0; 0:25 (Fig. 17(b)) it can be seen that starting from a low periodic regime the forced system undergoes a period doubling sequence at Am 0:081 leading to a banded chaos and then to a fully developed chaos at Am 0:122. The fully developed chaos persists until Am 0:0161, where a sudden apparition of a ®ve period attractor is observed. The emergence of the periodic attractor is due to an interior catastrophe similar to the one discussed in the previous section. Each branch of the ®ve periodic attractors goes through a period doubling sequence that culminates in chaos and then a three periodic attractor emerges at Am 0:215. From there the period attractor will undergo a bifurcation similar in its overall structure to the regions II±V found earlier in Fig. 4. Notably it can be seen from Fig. 17(c) that at high forcing amplitude the system bifurcates to period-1 attractor through torus bifurcation. 5. Conclusions This paper has carried out a numerical investigation of the eects of periodically forcing a two-phase CSTR where a Gray±Scott autocatalytic reaction with decay is taking place. The investigation has shown that regular regimes can emerge from the original chaotic autonomous system when the feed ¯ow rate is perturbed either in amplitude or in frequency. Dierent mechanisms for the transition between chaotic regions and periodic windows have been identi®ed and analyzed, including period adding and tangent bifurcation. An excitation (amplitude±frequency) diagram was constructed for the system. The diagram has shown that, for the studied system, it is not always possible to control chaos by frequency forcing. Stabilization of chaos is only possible by frequency forcing of regions of high amplitudes with a relatively low forcing frequency. An analysis of the eect of the location of the center of forcing has revealed a noticeable robustness of the dynamics of the forced system for a variation of the location of the center of forcing in the region yielding chaotic behavior. When the center of forcing is moved away from this region and closer to the Hopf points of the autonomous system, a noticeable change in the dynamics is observed especially at low forcing amplitudes. Numerical simulations have shown that the reaction conversion deteriorates with the increase in the forcing amplitude but it can be improved by the selection of appropriate forcing frequency. References [1] Alhumaizi K, Aris R. Chaos in a simple two-phase reactor. Chaos, Solitons & Fractals 1994;4:1985. [2] Elnashaie SSEH, Abasaeed AE, Ibrahim G. On the practical relevance of bifurcation and chaos in chemical and biochemical reaction engineering. Fractals 1997;5:549. [3] Teymour F, Ray WH. The dynamic behavior of continuous polymerization reactors. VI Complex dynamics in full scale reactors. Chem Eng Sci 1992;47:4133. [4] Choi KY, Ray WH. The dynamic behavior of ¯uidized bed reactors for solid catalysed gas phase ole®n polymerization. Chem Eng Sci 1990;41:2261.
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