the stirred tank forced - university of minnesota i. g. kevrekidis et al. the study of such problems...

ck,,,kd .t$,,&ce,~ Science, Vol. 41, No. 6. pp. 1549-1560, 1986. Printed in Great Britain.

0009~2509/66 S3.00 + 0.00 Pergamcm Journala Lid.

THE STIRRED TANK FORCED

I. G. KEVREKIDIS,? R. ARIS and L. D. SCHMIDT Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis,

MN 55455, U.S.A.

(Receiued 10 December 1985)

Abstract-We examine the response of a continuous stirred tank reactor (CSTR), in which a simple irreversible reaction A -D B occurs, to periodic variations of the coolant temperature. The amplitude and the frequency of the forcing are used as control parameters. The remaining operating conditions are chosen so that the unforced CSTR exhibits a single stable oscillation; this means that for small forcing amplitudes the system behaves like a typical forced oscillator: its response consists of alternating entrainment and quasi- periodicity. As the forcing amplitude grows, more characteristic traits of the CSTR and the particular forcing variable become apparent. and a complicated picture develops, involving the co-existence of multiple periodic, quasi-periodic and even chaotic oscillations, period doublings and global bifurcations. We track several of these traits numerically, placing special emphasis on the mechanisms by which such features branch out from the well-defined low-amplitude region. Several algorithms based on shooting methods for boundary value problems are used in this task, and some appropriate ways of tackling the simple initial value problem of simulation are also employed. In particular, the need for efficient algorithms to tackle global bifurcations is stressed. Such algorithms seem to be indispensable tools in the systematic study of the forced CSTR and, more generally, of periodically forced oscillators.

The qualitative traits discovered for the forced CSTR are compared with other known results for model systems, and several questions present themselves as possible subjects of further research, both for the CSTR and for a wide class of periodically forced and/or coupled reactor models.

INTRODUCMON

The study of the CSTR permeates the history and the development of the analysis of chemical reactor dynamics. Interest in the dynamics of the CSTR per se aside, this ubiquitousness lies in that this simple and important system is the favourite example, a sort of common reference point used in illustrating new methods (both algebraic and numerical) whose scope and applicability often extend beyond reacting systems to dynamical systems in general. Though various aspects of CSTR behaviour had been treated before by Denbigh and others, it is with Amundson and Bilous’s 1955 paper that the dynamical systems’ viewpoint first comes to the fore. This was developed in a series of increasingly complex situations in the work of Amundson and his collaborators which may be followed in his selected papers (Aris and Varma, 1980) and will not be reviewed here. From Poore’s paper (1973), through the now classical analysis of Uppal et al. (1974, 1976) to the numerical investigations of Doedel and Heinemann (1983), and from the singu- larity theory work of Balakotaiah and Luss (e.g. 1982) to the work of Jorgensen et al. (1984) on the CSTR with two consecutive reactions this pervasiveness becomes manifest.

Our research is focused on the study of coupled chemical oscillators, and the qualitative and quantitat- ive analysis of phenomena that arise in the dynamic response of such systems. As a useful intermediate step in this effort, we study periodically forced systems; in

7 Present address: CNLS and T-Division, Los Alamos National Laboratory, Los Alamos, NM 87544, U.S.A.

these, one of the interacting frequencies- the forcing frequency-is an operating parameter, and this provides a convenient handle on the conceptual and computational tracking of the nonlinear dynamics of these systems, precisely because it is not affected by the nonlinearities.

In the past we have described several-mostly computational-methods that we have developed and/or applied in our study of periodically forced systems (Kevrekidis et al., 1984, 1985, 1986). We have studied some common qualitative features in the dynamics of such systems, and have tried to justify some of these features on theoretical grounds (Aronson et al., 1986). In all this previous work we restricted ourselves to the study of two-dimensional systems [the Brusselator (Glansdorff and Prigogine, 19711, a surface reaction scheme (Takoudis et al., 1981), the CSTR] both for reasons of computational and representational convenience, but also because, in terms of dimensionality, these systems are sufficiently complicated to exhibit the phenomena we are interested in: quasi-periodicity, chaos, global manifold interactions. The one extra dimension of the forcing variable is enough to create these complications.

We have also consistently chosen the operating parameter values for the unforced systems so that they exhibit one single stable oscillation in their phase planes at zero forcing amplitude; we have thus studied the interaction of a natural oscillation with an external imposed one.

A detailed, largely descriptive but nevertheless pioneering effort by Tomita and co-workers on the forced Brusselator (Tomita, 1982) was significant in

1549

1550 I. G. KEVREKIDIS et al.

the study of such problems and in attracting our interest to the subject. We felt that forcing the CSTR was the appropriate trial target in our first attempt of a detailed two-parameter study, despite the stiffness of the governing equations that arises from the Arrhenius temperature dependence and renders the computational efforts more costly. Thanks to the generosity of the University of Minnesota Computer Center and of N.S.F., and to the capabilities of the Minnesota Cray-1 we were able to handle a part of this problem. There remained the choice of the operating conditions for the autonomous system, around which we would periodically force the CSTR. In this we were guided again by the Chemical Engineering literature; a pioneering paper on the forced CSTR was written by Sincic and Bailey in 1977. More recently, a valuable study of the effect of varying the forcing amplitude has been presented by Mankin and Hudson (1984) in which a period doubling cascade to chaos as well as the co-existence of periodic and chaotic oscillations was shown. We felt that imbedding this type of one- parameter study in the two-parameter variation of both the forcing amplitude and frequency might be a way of understanding the system more completely, and we have constructed a diagram that might be called an excitation spectrum: a diagram in which changes in the qualitative response of the system are depicted as functions of the amplitude and the frequency of the forcing. For easier reference we have scaled the forcing frequency u with the natural frequency of the unforced system we = 2x/T,,. We also sought a suitable scaling for the amplitude of the forcing, and this required the autonomous bifurcation diagram.

27re autonomous bifurcation diagram As in both the Mankin-Hudson and the Sincic-

Bailey papers, we chose the coolant temperature as our forcing variable. The equations for the forced CSTR can then be written for infinite activation energy in the form

x1 = -x1 +Da exp(x,) (1 -xi)

A* = -x,+BDaexp(x,)(l -x1) (1)

+~(T,+acoswt-x,)

where x1 is a dimensionless concentration of reactant A, x2 is a dimensionless temperature and Da, B, fi and T, are parameters. At the midpoint of the forcing ( T, = 0), the autonomous system (a = 0) has a stable limit cycle of period T, = 1.09499598 surrounding an unstable steady state (source). It is then reasonable to expect that for small forcing amplitudes the system (independently of the chosen forcing variable) will behave like a typical forced oscillator, and further- more, that its behaviour will be “close” to that of the autonomous system (Chenciner, 1984). This is indeed the case, since the autonomous system can be smoothly obtained as a limiting case of the forced system as the forcing amplitude OL + 0. This can actually be used in obtaining the behaviour of the forced system to the desired degree of approximation by perturbation

analysis of eq. (lj around the autonomous system (small a). Such analysis provides good initial guesses for finite forcing computations, but lacks further interest since the qualitative response of forced oscillators in this region is fairly well understood. As the forcing amplitude becomes larger, however, this simple identity of a “forced oscillator” is no longer appropriate: the instantaneous values of the forcing variable- the coolant temperature-wander further away in parameter space, and it becomes important to know what the autonomous system would exhibit if the forcing was frozen at each of these instantaneous values. In other words, the uutonomom bifurcation diugrmn with respect to the parameter that will be used as the forcing variable is of crucial importance to this work.

There exist several well-documented methods and algorithms to obtain such one-parameter diagrams. It is easy to follow the stability of the steady states of eq. (1) as T, varies and observe two turning points (T,, = -0.749635, source to saddle, and T,, = -0.010088, saddle to sink) as well as a Hopf bifurcation point at TGH = 0.063036 where the limit cycle branch originates (see Fig. 1). With the midpoint of the forcing at T, = 0 it is evident that when the forcing amplitude reaches (z, = TGH we are not forcing a full-time oscillator any more. During part of the forcing cycle we force an oscillator, and during the other part forcing a steady state. When a exceeds Tst. we force a part-time three steady-state phase plane, too. These developments should become qualitatively apparent in our excitation diagrams. We thought it thus appropriate to

Stable steady states

t

--- Soddles I0 . . . Sources

l 0 0 Stable limit cycles

000 Unstable limit cycles

T 5-

-.7496X -.0512’4546 -.010088

Fig. 1. The autonomous bifurcation diagram for the CSTR under Mat&in-Hudson case I conditions: Da = 0.085, B

= 22, j3 = 3, T, = 0.

The stirred

scale the forcing amplitude a with a, = TCh so that at a,=cz/uO= 1 the identity “forced oscillator” is lost.

The fate of the limit cycle branch can best be followed by the series of diagrams in Fig. 2. These have been considerably distorted to show more clearly the way in which the trajectories move around. In fact, they require great care in computation for the speed of the point (x1 (t), x2 (t)) along the trajectory can vary by several orders of magnitude and the trajectories have a sharp corner near xi = 1 and then turn downwards passing close to the highest temperature steady state. Figures 2(b) and 2(d) are drawn to scale and show why the distortion in the topologically equivalent Figs 2(B) and 2(D) is necessary to show how the trajectories lie. This figure can be followed more easily by the sequence of phase portraits A, . . . , F corresponding to an increasing sequence of values of T,. In region A there is only the low-temperature steady state, but at the turning point which marks the transition to region B a saddle-node pair is born at a higher temperature, the saddle moving towards the low-temperature steady state n and the node, which is unstable, turning into an unstable focus. The stable manifold of the saddle has two parts: s, which comes from f, and s’, which comes from the boundary. One branch of the unstable manifold, u’, goes to n while the other, u, also goes to n by going upwards and to the right descending close to the vertical xi = 1 and swinging in to n underneath s. It is because many of these curves lie extremely close to one another that it is almost impossible to show it to scale as in Fig. 2(b). In the transition 7 between B and C, a pair of limit cycles is born and splits into a stable and an unstable pair, the unstable enclosing the stable. It is the unstable limit cycle, U, which dies homoclini- tally at TGh [Fig. 2(h)] by becoming of infinite period, whilst the stable one, S, shrinks in size and dies at the Hopf point TGH. The infinite period cycle, T in Fig. 2(h), is dilIicult to compute and demands special

tank forced 1551

techniques which will be discussed elsewhere. It is the intersection of the unstable and stable manifolds, s and u, of the saddle. In the region D, the still unstable high- temperature steady state f is surrounded by the stable limit cycle S, and the branch u of the unstable manifold of the saddle winds itself on to S. The stable manifold, s, now comes in from the boundary and now lies below the unstable u. As T, increases, S shrinks onto the unstable focus [Fig. 2(E)], which gains its stability at the Hopf point. This picture is consistent with the criteria for the transitions at the Hopf and homoclinic points. The Hopf is well known; for the homoclinic, see Guckenheimer and Holmes (1983, Theorem 6.1.1, p. 292).

The importance of the autonomous bifurcation diagram is now apparent. At a, = 0.16003, we start perturbing a phase plane with one stable limit cycle and three steady states (one of them stable). At a, = 0.81296, we lose the limit cycle at the low end of the forcing cycle and we start perturbing a homoclinic orbit. At a, = 1, we lose the limit cycle at the high end of the forcing cycle when it shrinks into a single stable steady state at the Hopf bifurcation. We expect these changes in the autonomous system phase plane to cause visible modifications in the dynamics of the forced system at or around comparable values of the forcing amplitude. It is already interesting to observe that Mankin and Hudson find a transition to quasi- periodicity around a, = 0.563174 and a different chaotic state around a, = 3.65. It is also already evident that this will be a complicated system to elucidate, precisely because of the co-existence of all these transitions in the autonomous bifurcation diagram.

THE MEtWODS AND THE ALGORITHMS In a periodically forced system there are no steady

states. If the system has two degrees of freedom the possible forms of behaviour are periodic, quasi-

Fig. 2. Bifurcation diagram for the autonomous system with typical phase planes.

The stirred tank forced 1553

THE EXCITATION SPEffRUM

We now study in some detail the features of the excitation spectrum depicted in Fig. 3, a segment of the excitation diagram, or, in other words, a fragment of the “big picture”. What we observe for small forcing amplitudes is typical of a forced oscillator and the main phenomenon of interest is entrainment. When the ratio of the forcing to the natural frequency is close to being rational, the response of the forced system becomes periodic: its period is an integer multiple of the forcing period. This phenomenon is called entrainment. It occurs in horn-like regions -which are also termed entrainment regions, resonance horns or Arnol’d ton- gues. These regions have their tip at u = 0 on rational values of w/w0 = p/q, where p and q are prime integers. In the resonance horn with its tip at w/o0 = p/q we find an attracting period p, that is, a periodic trajectory with a period p times that of the forcing. This oscillation appears stroboscopically as p discrete points among which the phase point rotates at success- ive iterations of the stroboscopic map. The actual trajectories have q loops in their phase plane projection and q peaks in their time plots for simple systems (see Fig. 4).

The asymptotic shape of the resonance horns can be calculated near the tip; they become increasingly sharper as the prime integers p and q grow. We expect such an entrainment region for each rational value of w/oO. Since these regions initially grow with CY > 0 the question of their fate and possible interactions for

large forcing amplitudes arises naturally. We will address this problem later on. It becomes already evident that the “complete” description of the system is impossible due to the infinity of the rational numbers. This will limit us to the study of the major resonance horns, that is, resonances with small p and q. Of these resonances, those with p,q < 4 (the so-called forbid- den resonances) are special (Chenciner, 1984). Those with q 2 5 are qualitatively similar.

All these resonances are interspersed in a “sea” of quasi-periodic behaviour. If the operating parameters Q and w do not lie in some resonance horn (something practically impossible to ascertain) the system response will contain two incommensurate frequencies with irrational ratio: the forcing frequency and some descendant of the natural frequency slightly altered due to the nonzero forcing. The fact that it is practically impossible to distinguish between quasi-periodicity and a complicated high-order resonance @,q large), along with the thinness of the corresponding horns makes high-order resonances increasingly difficult to compute and observe. As we have noted elsewhere (Kevrekidis et al., 1984), independently of whether the observable response is periodic or quasi-periodic there always exists an invariant torus, that is, a surface in the system’s phase space R2 x S’ such that if a trajectory starts on it it will remain on this surface for all positive time. This doughnut-like surface can be visualized in the quasi-periodic case because ultimately all trajectories get attracted by it and wind around it without

Qr 15

14

13

12

11

10

9

8

7

6

5

4

3

2

1

I 1 I I I I I I 2/3_ .f1/5 1.0 4/3 1.5 2.0 2.5

Fig. 3. A segment of the excitation diagram. Two-parameter bifurcation curves and special points are marked and discussed in the text.


7.0 (al

-Ns

6.0

x2 N

LAZI

Y

4.0 0.6 0.9 1.0

Xl DIMENSIONLESS TIME Xl

Fig. 4. Subharmonic saddle period 3 at ar = 0.5. w/w0 = 1.498. (a) Stroboscopic view of a transient starting close to the unstable period 1 (Fl) and approaching gradually the attracting period 3 (N3). (b) Phase plane projection of the subharmonic period 3. Notice that there are three periodic points and two loops (we are in the 3/2 resonance). There are also two peaks in the time plot of the period 3 (c). The X twos wt projection is

shown in (d).

ever returning to the same spot. The projection of this surface on the x1-x2 plane gives a space-filling curve (see Fig. 5).

In the case of entrainment, the entrained periodic trajectories will also lie on this surface. Entrained periodic trajectories are usually referred to as subharmonic orbits or simply subharmonics. The stability of subharmonic solutions is governed by their Floquet

1.0

x, 0.9

I I I 20 40 60 80

DIMENSIONLESS TIME

x2

or characteristic multipliers FMI, the eigenvalues of the linearization of the stroboscopic map [the matrix [aF/ax], eq. (6)] around them. For this system there are two free Floquet multipliers (the third multiplier, which is constrained to be unity, has been automati- cally discarded by considering the map and not the entire trajectories). The trajectories are stable when both of these multipliers have absolute value (or norm)

8.0

x2 6.0

4.0 0.8 0.9 1.0

Xl

6.5

6.0 -

5.5 -

x2 5.0-

1 i

Fig. 5. Quasi-periodicity at ur = 0.2 and w/w0 = 0.75. (a) Transient system response; the stroboscopic points are marked. (b) Phase plane projection of the transient gives a space-filling curve. (c) Stroboscopic view of the torus and (d) computed invariant circle with 80 radially spaced mesh points and linear interpolation. (e) Shows the location of the nonzero entries in the ;sr5z Jacobtan of the computatron (see Kevrekrdis et al.,


less than one. When they cross the unit circle in the complex plane a bifurcation occurs. The bifurcation associated with entrainment is a turning point, where a stable oscillation collides with an unstable one and they disappear. The moment this collision occurs, one of the free multipliers has a value of 1. It is interesting to observe the sequence of events as we make a one- parameter cut through a resonance horn (Fig. 6). At the onset of entrainment a pair of limit cycles (a stable node and a nonstable saddle) appears as periodic points on the invariant circle. As w/o0 changes, these points move away from each other until they meet their next neighbours at the other entrainment boundary. The torus retains its integrity throughout this interval, and when entrainment ends, quasi-period&y appears again smoothly. During entrainment the invariant circle (the stroboscopic cut of the invariant torus) is patched together with the unstable manifolds of the saddle-type periodic points which lie on the unstable oscillation.

It is now evident that in order to obtain an excitation diagram we need to be able to continue entrainment limits in two-parameter space. For this purpose, we impose one condition on the stability of the entrained

1 6

x2 .

B D

+

Y 3/Z

QvqJ

3.00 r (b)

0.96

Xl 0.92

0.68

0.65 i 1.49

=a /- -------

b

-. -2 5 --_ NS

N1

--------H s,

1.494 q.496 1.502 W/W0

Xl

Fig. 6. (a) One-parameter cut through a resonance horns (q. = 0.5 across the 3/2 resonance). At A we have a quasi-periochc attractor. At B, three saddle nodes appear on the invariant circle. Crossing through the horn as w/w,, varies, the saddles S, and the nodes N, move apart (C) and recombine at D with different pairing. Arrows indicate the rotation orientation of a phase point. A bifurcation diagram in w(b) shows the change of pairing that occurs between the left and right entrainment

limits.

trajectories [eq. (7), FM1 = 11. In a typical numerical experiment we would look for the left-hand side of the p/q resonance horn for small forcing amplitude by looking for a period p with a multiplier at unity and an initial guess for w/o0 slightly smaller than p/q. Good initial conditions can also be obtained from perturbation analysis of eq. (l), as we mentioned above. We would then make a one-parameter cut for a constant forcing amplitude through the resonance horn that would bring us to the right-hand side entrainment limit. We would then continue these turning points in two-parameter space with a full Newton algorithm. Curves like this are shown in Fig. 3 with the indication SN, (for right-hand side) and SN: (for left-band side), where i is the period of the entrained trajectories in multiples of T, and SN stands for saddle-node bifurcation. These lines start at zero forcing amplitude at w/we = p/q (points H, in Fig. 3). We know that for sufficiently low amplitude these horns are separate and do not overlap. It becomes immediately evident that for high amplitudes these horns haoe to overlap. This means two things: first, that periodic solutions with different periods co-exist, and second, that when this happens, the torus- on which frequency locking occurs for small ~---has already somehow broken. It is impossible to have a torus with periodic trajectories of different periods locked on it. The apparent intersection of various SN, lines in Fig. 3 are not bifurcation points; the appearance in saddle-node bifurcations of a pair of period 3’s at point a, for example, is independent of the existence of the period 4 entrained solutions.

The problem is that the breaking of the torus on which both the period 3 and the period 4 oscillations were locked for small forcing amplitudes is missed by our computations: it is a global bifurcation since it involves the interaction of global manifolds; it may give rise to chaotic behaviour which, unlike a period doubling cascade, we would miss by simply monitoring the stability of the periodic solutions. We also observe that a number of the curves SN; in Fig. 3 veer over towards the left-hand side of the diagram, to small frequencies (large periods) around forcing amplitude a, = 1. We have a plausible explanation for this trend.

As we follow SN curves, however, it is inevitable that in a two-parameter study we will encounter the so- called m&a-bifurcations (or codimension two bifurcation points) where a qualitative change occurs. These are again local bifurcations since they can be observed by monitoring the linearization and the higher deriva- tives of the stroboscopic map at the periodic point. On line SN1, for example, we encounter the point BI 1, where the periodic trajectory has a double Floquet multiplier at unity. Such a degenerate point is called a Bogdanou point (Bogdanov, 1981) and so we named it B1 r, where the subscript stands for the order of the resonance horn on which the point is observed. We also observe such Bogdanov points on the right-hand sides of the 2/l resonance (Bzl), the 2/3 resonance (BzJ) and on the 3/4 resonance, which we have not marked in Fig. 3. Such degenerate points are usually


qualitatively analysed in the literature, so that we know what to expect in their neighbourhood for the particular example we are interested in. At B1 f two lines branch from SN1. On SSI , we again have a turning point bifurcation, but now the second free multiplier is greater than unity. This means that as we cross SS, an unstable (repelling, source) period 1 oscillation hits another unstable (saddle) period 1 oscillation and they annihilate each other. It is useful to consider this phenomenon as it appears on one-parameter diagrams (Fig. 7). At point b in the l/l resonance horn we have an unstable focus period 1 (descending from forcing the autonomous unstable steady state) surrounded by an invariant circle (descending from forcing the autonomous stable limit cycle). On this invariant circle we find locked a stable node oscillation and an unstable saddle period 1 oscillation. As we cross SS1 (d in Fig. 3) the unstable source period 1 hits the saddle period 1 that was initially on the torus. This leaves us only with the stable period 1, which, as we now change w/oO, loses its stability as it crosses the Hopf bifurcation line HF1 at e. When this happens, its two (now complex) multipliers leave the unit circle at an angle 8 which varies from 0” at B1 1 to 180” at Mll. Continuing this now unstable period 1 down in a, we recover, at zero amplitude, the original, autonomous unstable steady state. This suggests a similarity between lines SS1 and HFI: on both of these lines, something descending from the autonomous unstable steady state (the unstable period 1) hits something descending from the autonomous stable limit cycle (the torus on HF1, the saddle on the torus on SS1). In other words, both of these lines owe their existence to the Hopf bifurcation point Tc._, in the autonomous bifurcation diagram (Fig. 1). Our forcing started crossing this value of T, as soon as Q_ became larger than 1. Hence these two lines

originate, are an indication of the fact that the

or, eq&alently, the point BI1 from which they

Xl

a

b

Fig. 7. Schematic diagram of bifurcations related with the l/l resonance horn. Initially (a), or point b in Fig. 3, we have a source pe?iod 1, a torus and a stable as well as a saddle period 1 locked on it. Going up in amplitude, the source hits the saddle (b) and we are left with the now focus stable period 1. If we subsequently change o this period 1 will lose stability to a torus (d, e)_ At low amplitude, changing o/o0 will cause the saddle to hit the node ( f ) resulting in a situation similar to (e).

autonomous bifurcation diagram had a Hopf bifurcation close to the midpoint of the forcing. We must also remember that now that CL, is greater than 1 we have already started to perturb the autonomous homoclinic orbit (that is, the coolant temperature takes instantaneous values at both sides of T,_J. This, we believe, is the reason for the leftward veering of the resonance boundaries: they tend towards larger periods (smaller frequencies), an indication of homocli- nicity. In other systems that also had a Hopf bifurcation in their autonomous diagram (the Brusselator, the surface reaction scheme) points like Bll did exist for the forced case. Since these systems did not also have a homoclinic orbit close by, their resonance horns rose smoothly and did not turn to the left. In the forced Brusselator we actually found some of them to close off smoothly at high forcing amplitudes.

This qualitative behaviour is indicative of the importance of the autonomous bifurcation diagram to the qualitative picture that the forced system response presents_ Another indication of the interference of the homoclinic orbit is, for example, the chaotic response that we observe at point 1 (Fig. 8). This does not come from a sequence of period doublings but from global bifurcations, and is therefore difficult to track.

Let us return to more tractable features of Fig. 3. The line of Hopf bifurcations HF1 is important to the fate of the resonance horns we described above. Along this line there exist points on which the multipliers of

a

Fig. 8. Change of the stroboscopic phase plane structure in the 312 horn. The smooth invariant circle at Q, = 0.5, o/o0 = 1.5 (a) breaks to a chaotic attractor (b), 01~ = 5.3, o/% = 1.5737 that co-exists with a stable period 3. The stable period 3 oscillation (0) has a small basin of attraction, confined by the stable manifold branches of the saddled

period 3 (A).


the bifurcating period 1 oscillation leave the unit circle at precisely some integer root of unity (1 ‘I3 or cos 120” fl sin 120” at R:l,ll/d or f 1 at R$,, etc.). The resonance horns that are born at a, = 0 will interact with these points. The SNa boundary of the 3/2 resonance horn rises and circles around R: 1. If we take a one-parameter cut up in amplitude that passes precisely through R : I , we will find that the saddle subharmonic period 3 oscillations collapse on the period 1 that loses stability at R:, (see Fig. 9). The R:, point is also interesting. An inverted period 4 resonance horn appears to come off the Hopf bifurcation line HF1 at this point (see Fig. 10). At least one of the sides of this resonance horn coincides with the entrainment boundaries of the 4/3 horn that starts at a = 0. This observation should hold true more generally, especially for resonances of order p k 5. When such resonance horns close at large forcing amplitudes, this closing occurs in relation to a Hopf bifurcation of a period 1 oscillation inside them (R:,) or on their boundary (R:,).

The Hopf bifurcation line HF1 ends at a point M 11 where the period 1 oscillation has two Floquet multipliers at -i. What happens in general at such a point, as well as at the R:, point in two-parameter space has -to our knowledge--not yet been studied theoreti- cally. We find a two-sided line of period doubling bifurcations (one multiplier FM, = - 1) starting at that point. One side of it (PD,) extends inside the l/l resonance horn to the left. When we cross it (g) a stable period 1 loses stability to a period 2 (an oscillation that has twice the period of the forcing). Further up in a, this period 2 loses stability to a period 4 (h) (see Fig. 11). On the right of Ml,, we have a line of unstable period doublings PD2, on which an unstable period 1 period doubles (one multiplier at - 1 and one outside the unit circle, FM, x - 1). What happens is that the unstable period 1 (coming from the autonomous unstable steady state) hits the saddle subharmonic period 2 (initially locked on the torus). In this sense,

3.63 I

PD2 is conceptually a continuation of SS1 and HF1 to the right: like them, it is also a descendant of the autonomous Hopf bifurcation point. Further on to the right, we get a different type of bifurcation point on PDz, C21, whose nature is also interesting. Originally, a saddle-node subharmonic line SN2 begins at the tip H21 of the 2/l resonance horn. It reaches a Bogdanov point Bzi (a period 2 oscillation with two multipliers at

eo-

5.9 -

=r 5.8 -

5.7 -

5s -

Fig. 9. High-amplitude bifurcations in the 3/2 resonance horn. (a) A bifurcation diagram across the horn at CC, = 5.8 shows that the pairing of the saddles and nodes now remains the same at both entrainment boundaries. A one-parameter diagram in a., going through R:, (b) shows the saddle

subharmomcs collapsing on the period 1 at that point.

/ / .’

3.62 -

Qr

3.61 - A

/ SN; SN4

I 8 3.60 - , I 1 I L I

1.363 1.364 1.365 1.366 w/w0 1.367 1.368

Fig. 10. High-amplitude tip of the 4/3 resonance horn around the R:, p”

int on the Hopf bifurcation line HFI _ An inverted resonance horn with its tip at R 11 is obtained.


I 1 I

6.0 - i SN, I

1 ',St PO11 I

5.5- \ I I

ar PQ,, \ 1

PD.

7.v

0.80 0.84 0.88 0.92 0.96 xi

Fig. 11. One-parameter diagram in g, at w/o,, = 1.25. The subharmonic stable period 1 (N,) penod doubles to a stable period 2 (Nz) and this doubles to a period 4 (N4). The cascade, however, stops there, and then is inverted. The period 4 recombines to a period 2 which is then lost to a saddle-node

bifurcation at PQ, 1.

unity) from which-exactly like B I i -there emanates a Hopf bifurcation line HF2 and a saddlesource bifurcation line SS2. Crossing the Hopf bifurcation line HF2. the stable subharmonic period 2 loses stability to a torus, and subsequently hits the saddle subharmonic period 2 on SS1. The two stroboscopic points representing the period 2 oscillation on SS1 come increasingly closer and closer, until they collapse on each other at Czl where line SS+ runs into the period doubling line PD2. Needless to say that a whole new host of resonance horns are expected to emanate from the Hopf bifurcation line HF2, none of which is shown. The picture is arbitrarily truncated at w/o0 = 2.5. Let us now study the one-parameter cut of Mankin and Hudson and try to imbed it in this two- parameter picture. They go up in amplitude at w/o0 = 1.1001100110011. They hit the line HF1 at Q, = OL, = 3.33 and proceed through a cascade to chaos. It is interesting to observe that the locus of loss of stability of a period 2 to a period 4 (the line PQ 1 1 ) is a curve that turns back towards small w. Crossing it at h we period double a period 2 to a period 4. Crossing it again at i, we “de-double” the period 4 into a period 2 (see Fig. 11). There is a similar “shell” inside PQll where the period 4 goes into a period 8, and another shell going to a period 16 contained in that one, precisely the way chaos occurs in the forced Brusselator. The only difference is that here these regions are “tilted” to the left, and seem open-sided. We should note at this point that the unfinished left-hand side SN; lines, as well as PQ,,, etc., become increasingly difficult to compute accurately as the periods become larger, even for a stiff integrator (EPISODE with MF = 21). Double precision (and consequently larger cost) becomes necess-

ary to continue them further accurately. Though it would have been possible to push these lines further to the left with double precision computations, the complexity resulting from perturbing the autonomous homoclinic orbit demands a separate study, and, most probably, techniques that we do not yet command. A saddle-node period 2 bifurcation line SNf lies directly above PQ, 1. In the Mankin-Hudson cut, this line is the reason for the seemingly abrupt end of their first chaotic region: the stable period 2 coming from the “de-doubling” of the period-4 shell is almost immediately lost in this saddle-node bifurcation. This line originates on a bifurcation point Gil exactly similar to Czl on the side of the period 2 resonance horn at the 2/l resonance.

Mankin and Hudson also find a high-amplitude period 10 that period-doubles to give eventually a different chaotic region around cc, = 3.65. Since trajectories for these a and u spend time at small x1 and x2, that is, close to the lower branch of the S-shaped autonomous bifurcation diagram, this is a further complication which we will not discuss here; we note only that we feel these high-amplitude oscillations can be traced back to forcing beyond the two turning points T,, and 76, of the autonomous bifurcation diagram.

We return to features that can be traced back smoothly to the low-amplitude resonance horns. SS1 goes to a second Bogdanov point B; 1 (two multipliers of a period 1 at unity) where it again breaks into a Hopf bifurcation line HF’, and a regular saddle-node line SNf . Going up in a ( j), the unstable source period 1 first becomes stable when the torus collapses on it at HFI , and then hits the saddle subharmonic period 1 on line SNf in a turning point bifurcation. A new set of resonance horns is expected to show up at HF’, (as does the right-hand side of the 2/3 horn at the 1”” point R;,). HF> ends at a new double - 1 point (M> i) and it connects to the right-hand side of the period 2 resonance at 2/3 in a way completely analogous to what happens at Cz 1, here at point Cz3. The right- hand side entrainment boundary SN2 of the resonance at 2/3 goes to another Bogdanov point Bz3 (period 2 with two multipliers at unity) and turns into SSz3 up to Cz3 where it hits the period doubling bifurcation line that emanates from the point M;, . The picture has again been arbitrarily truncated at w/w0 = l/2. The right-hand side of the period 1 resonance at o/o0 = l/2 is barely visible.

It is interesting to consider the qualitative changes as we go up in a in a resonance horn (see Fig. 8). At the 3/2 resonance, for example, for small a the picture is nice and smooth [Fig. 8(a)]: the unstable manifolds of the subharmonic saddle period 3 points go to the subharmonic node period 3 points, thus constituting an invariant circle. Higher up, however [Fig. 8(b)], while one side of these manifolds is again attracted by a stable period 3 oscillation, the other side is attracted by a chaotic oscillation. The region of attraction of the stable period 3 is very small (just a loop inside the two branches of the stable manifold of the saddle period 3


points). Both sides of these stable manifolds are now attracted to the unstable source period 1 (see Fig. 8). The question then arises naturally: how did the first picture [Fig. 8(a)] continuously deform into the second one [Fig. 8(b)] along the line SNJ? How did both sides of the stable manifolds of the saddles converge to the unstable period l? A homoclinic crossing of manifolds has occurred at some point between k and 1 (a. global bifurcation) in the course of which the smooth manifolds become increasingly irregular, self-intersect and cross. It seems that the chaotic attractor that we observe in Fig. S(b) has resulted from this process. This is a transition to chaos that can be continued smoothly to the zero-amplitude torus if we have the means to track invariant manifolds. The period doublings on PQ, 1 and the chaotic states resulting from them cannot be so continued_ In a way, the chaotic behaviour in the PQ, i shell cannot be predicted, while the one at I is more tractable.

We will not discuss manifold computations here. So far, we can only simulate them. Manifold computations as an extension of torus computations are an entirely open research subject.

DISCUSSION We set out to describe the response of the CSTR to

periodic forcing. We obtained a fraction of what we name the excitation diagram of the system. Several points were accomplished in this effort. First, the importance of continuation in multiple parameters was, once again, pointed out. We imbedded, at least partially, the Mankin-Hudson picture in our two- parameter study. Some of the characteristics that could be expected were identified. Foreknowledge of some fundamental theoretical results (the structure around B1 1 and R:, ) was useful as guidance in our numerical work. Second, we were able to “pin down” some general traits of the excitation diagram and attempt to correlate them with the autonomous bifurcation diagram of the unforced system. Some previous work and comparisons (the Brusselator, the surface reaction scheme, the CSTR under different autonomous conditions) led us to associate positively some points (B B;,, II, etc.) and lines (SNf , SS, , HFI , PD2, etc.) with the Hopf bifurcation in the autonomous bifurcation diagram. This could be verified by periodically

generating bifurcations without such tools. Suffice it to say that at least the second kind is present in all forced oscillator systems, and is untractable through tradi- tional local analysis.

An important note is that the shell-like chaotic region, although located above the l/l resonance, is, qualitatively, the same as the corresponding shells in the forced Brusselator, an intriguing observation.

This has been in some ways an ungratifying problem to work on. For one thing, the stiffness makes it expensive. Even the generous grant of 10h of Cray time was exhausted in preparing the excitation diagram, and that only to the degree of detail shown in Fig. 3. Also, the autonomous system is too complicated at the chosen parameter values to conveniently allow separation of the interesting feature; this is conceptually similar to two singularities sufficiently close to each other so as to interact. The next similar effort should be on a “cleaner” autonomous system, i.e. an autonomous bifurcation diagram with less structure. The stiffness of the equations also causes the attractors to be compressed into a narrow region of the phase space, thus making visualization difficult. A smoother example like the forced Brusselator or the Gray-Scott autocatalytic models (Gray and Scott, 1983) would provide nicer pictures.

Although many questions regarding Fig. 3 are left unanswered, we may say that this effort on the forced CSTR gave us “more than we bargained for”. One can argue that practically all these complicarions should be expected in almost any system with the degrees of freedom and the operating parameters of the one we worked on. We certainly believe that with one more control parameter (the midpoint of the forcing 2-J we could smoothly go from Fig. 3 to the corresponding diagram for the Brusselator.

What Bilous and Amundson started by describing in a few pages, grew to a few papers when Uppal, Ray and Poore achieved some degree of comprehensiveness for the autonomous stirred tank. It is both delightful and unnerving to realize how large a monograph on the description of the forced CSTR would be in order to achieve a comparable degree of comprehensiveness. But from such acorns as Amundson’s, do oak trees grow.

perturbing the Hopf bifurcation itself, a horrendous three-parameter task on which undoubtedly some

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the stirred tank forced - university of minnesota i. g. kevrekidis et al. the study of such problems...

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