Transient chaos in a closed chemical system

Transcription

1 Transient chaos in a closed chemical system Stephen K. Scott*) School of Chemistry, University of Leeds, Leeds LS2 9JT, United Kingdom Bo Peng Department of Chemistry, West Virginia University, Morgantown, West Virginia Alison S. Tomlin School of Chemistry, University of Leeds, Leeds LS2 9JT United Kingdom Kenneth Showaltera) Department of Chemistry, West Virginia University, Morgantown, West Virginia (Received 5 September 1990, accepted 4 October 1990) Complex oscillations and even aperiodicity can exist as transient phenomena in closed chemical systems. These effects are illustrated through the analysis of a simple, isothermal chemical model based on mass action kinetics for autocatalytic feedback, involving the conversion of a reactant to a final product via three intermediate species. The use of the socalled pool chemical approximation and of pseudo-steady-state analyses for such systems is indicated and discussed, particularly with relevance to real chemical situations where small perturbations due to extraneous noise are inevitably present. I. INTRODUCTION Chemical reactions in thermodynamically closed systems (those with no exchange of mass with their surroundings) cannot show indefinitely sustained exotic behavior such as oscillations or multistability. Such features can, however, arise as transient phenomena and may be sustained for significant periods of time.lm3 The modeling of such effects,& whether for specific reactions or for more generalized, prototype schemes, frequently makes use of the socalled pool chemical approximation. Here, the concentrations of the major reactants are treated as constants (i.e., reactant consumption is ignored). This allows the powerful methods of bifurcation analysis to be employed, which can reveal in great detail the conditions for the onset of qualitative changes in behavior (bifurcations). At the next stage, the consumption of reactants can sometimes be simply reintroduced, and the reactive intermediates are assumed to respond to a slowly varying pool chemical concentration, following the slow decay in a pseudosteady state. This method has recently been successfully employed to study the onset and development of oscillations in a very simple model based on chemical feedback and mass action kinetics. ~ Here we study an extension of that model, -* a threevariable, isothermal scheme capable of supporting complex periodic and aperiodic responses. The model involves the conversion of a relatively stable precursor reactant P to a final product D via three intermediate species A, B, and C: P-+A, rate = k,p; P + C-+A + C, rate = k,pc; A-B, rate = k,a; (01 CC) A + 2B+3B, rate = k, ab 2; (1) B-C, rate = k, 6; (2) C-+D, rate = k,c. (3) a> To whom correspondence should be addressed. W) There are two feedback, or autocatalytic processes in this scheme. The species B is involved in a direct, cubic autocatalysis in step ( 1 ), while C participates in a degenerate autocatalysis through step (C!). II. RATE EQUATIONS AND DIMENSIONLESS VARIABLES The governing rate equations can be written simply by applying mass action principles to the above scheme, giving dp z-= - kop - kcpc, (1.1) $- = k,p + k,pc - k,u - k, ab 2, (1.2) --$=k,a+k,ab -k2b, (1.3) $==k,b-k,c. (1.4) It is convenient to write these equations in dimensionless form where 4 d7= -p/b + y), (2.1) s=p(tc+y) -afl --a, (2.2) **=,p2+a+ dr, (2.3) +p-y7 (2.41 a = (k, k,/k : > a, fi = (k, /k, ) 1 2b, y=(k,k:/k,k;) %, /i=kk,p/k3 are the dimensionless concentrations species, r=k,,t of the four chemical 1134 J. Chem. Phys. 94 (2), 15 January /91/ $ American Institute of Physics

2 Scott eta/.: Transient chaos in a closed chemical system 1135 is the dimensionless time, and K= (kok,(k,kc9(k,/k,9 2, 6=k,/k,, a= k,/k,, p = k,k,/k, (k, k, 9 are groups of rate constant ratios. These definitions differ in one respect from those chosen in our earlier paper. The parameter K has a different form here, such that only the parameter,ll involves the concentration of the precursor reactant P. (In terms of the parametrization in Peng et a1.,9 the present K is the group K/,U in the previous work.) This modification allows the effects of reactant consumption to be incorporated with only one time-dependent parameter. In the present paper, we consider the specific parameter values ~=65, cr=5.oxlo-, and 6 = Furthermore, all computations will take the following initial conditions: po = 0.2; a, =po = ycj = d P Ill. RESULTS A. Pool chemical behavior and pseudosteady-state analysis We obtain the pool chemical form of the present model by settingp = 0 in Eq. (2.1). Then the precursor concentration becomes a constant, with,u =,LL~ for all time 7: The remaining three equations for the intermediate species allow a steady-state solution & l s =&U/(1 -PI, (3.1) ass -K,U(l -,U9/[1-2pf (1 +&,U ]. (3.2) The dependence of &, on the dimensionless concentration of the precursor is shown in Fig. 1 (a). The steady state exists only for,u < 1, with a branched-chain runaway as,u + 1. If we now admit a slow decay of the precursor concentration (sop is positive but small compared with unity), we might expect the system to evolve in a pseudosteady-state manner, with,u decreasing in time and the intermediate concentrations adjusting rapidly to the instantaneous value of,ll(7). If this is the case, we can substitute for y from Eq. (3.1) into Eq. (2.1) to give d, z== -p,uk/(l -,U). (4) This integrates, with p = p. at r = 0, to give (/-Lo -p9 -ln(pu,/p) = --Kr. (5) Equation (5) gives the pseudosteady consumption of the precursor implicitly, with the corresponding intermediate concentration histories then obtained by substituting this into Eqs. (3.1) and (3.2), or we may take the result p = p /(K + fi) and substitute into Eq. (5) to give,llo - [P/(K + b? 1 - In.1 (K + ~)po/~ 1 = - pk7. (6) Figures l(b) and l(c) show the predicted development of ~(7) andp(r) for the parameter values chosen previously. The actual evolution of the autocatalyst concentration for three different values ofp is shown in Fig. 2. In each case, there is some initial transient evolution from the initial conditions. This can involve a dramatic spike, especially for small decay rates. This initial transient cannot be matched by the above pseudosteady-state analysis. The subsequent behavior differs in the three cases. With p = 10 -, the solu- FIG. 1. (a) Steady-state concentration & as a function of precursor p for the pool chemical model. Hopfbifurcation points& shown by circles. (b) Pseudosteady-state concentration ppbs as a function of time according to Ekq. (5). (c) Pseudosteady-state concentration &, as a function of time according to Eq. (6). Initial conditionsp, = 0.2, a, = a,, = y,, = 0; parameter values K = 65, CT= 5.0~ 10m3, S = tion computed numerically from the full equations follows the pseudosteady locus closely [Fig. 2 (a) 1. With the much faster decay forp = 1 I: Fig. 2 (c) 1, the agreement is not at all good and the pseudosteady approximation is clearly invalid. For the intermediate decay rate p = 10d3, the behavior is

3 1136 Scott eta/.: Transient chaos in a closed chemical system the early oscillatory responses shown in the inset of this figure. Although later oscillations have a typical period-l waveform (with simply a slowly varying amplitude and period), between r = 3.5 and 4.5, the system shows evidence of a period-2 oscillation. To understand the origin of the departure from the simple predictions above, we return to the pool chemical model and examine the local stability of the steady-state solution in Eqs. (3.1) and (3.2) Q O-4XT B. Local stability analysis Steady-state solutions are not always stable. Perturbations may grow in time, leading to a departure from the steady state, possibly to oscillatory states. In the pool chemical model, the loss of local stability occurs at points of Hopf bifurcation. The location of the conditions for such events cari be determined analytically, as given in the Appendix. The two Hopf bifurcation points for the present parameter values are marked in Fig. l(a) as,ut2:,up = 0.175,,uz = For values of the precursor concentration in between these, the steady state is locally unstable. These are Q 0.10,140 0.k o.i50 o.i55 c 0.05 (b). a m * mo*...* 0. * Q 0.8 FIG. 2. Temporal evolution of p computed from Eqs. (2) for (a) p = LOX 10m6; (b) 1.0X IO- ; and (c) 1.0. Other parameter values and initial conditions are the same as Fig. 1. Dashed line shows the pseudosteady-state value & predicted by F!q. (6) in each case. quite different again [Fig. 2 (b ) I. During the initial transient motion, the system moves to the pseudosteady locus, but at 7~3, there is a departure from this path and the onset of oscillations. The latter mode lasts until 7~ 36, after which the system moves back onto the pseudosteady curve. There are also some intriguing details revealed in the blowup of j0 lu FIG. 3. (a) Bifurcation diagram showing the Poincark section as a function of parameterp for pool chemical form. (b) Transient Poincart section as a function of instantaneous value ofp(t) for the closed system with reactant = 1.0X 10W3).

4 Scott etal.: Transient chaos in a closed chemical system 1137 both supercritical Hopf points and at each a stable limit cycle emerges and grows to surround the unstable steady state. The motion of the system on this limit cycle gives period-l oscillations in the concentrations of all three intermediates. Just as steady states may become unstable, so can limit cycles undergo subsequent bifurcations. In systems with three or more variables, such as the present, these bifurcations may lead to oscillatory states that correspond to complex oscillations, or even to chaos. The behavior in the present case within the range of steady-state instability (,LQ<,u<~T) can be summarized in terms of the PoincarC section as a function ofp [Fig. 3 (a) 1. This is constructed by plotting the value of the autocatalyst concentration fi every time y passes through a minimum. For values of,u such that the evolution is a simple period-l oscillation, such a process yields a single point-such responses occur for /.~~-+<0.143 and for 0.157<~&. Over the range C/A < 0.157, the period-l limit cycle is unstable and higher periodicity is found. This proceeds via a period-doubling cascade, leading to chaotic behavior. Various periodic windows are found, before a reverse, period-halving sequence reestablishes the period-l state. For further details, see Peng et al9 We can also construct a (transient) Poincare section as a function of the internal bifurcation parameter,u(7) from the full time series corresponding to Fig. 2 (b ). This is shown in Fig. 3 (b). Comparing this with the section for the pool chemical approximation, we see that there is a close match at the lower values of,u (corresponding to later times as,u, the precursor concentration, is decreasing through the reaction). The delayed onset of oscillations, which first become visible in the time series at some p significantly less than,ut, is typical of systems with slowly varying parameters and has been discussed for a reduced version of the present model by Kordylewski et al. and for another chemical scheme by Baer et all3 Kordylewski et al. found that this delay is extremely sensitive to small perturbations in the system and undertook a preliminary analysis of the effect of noise on the onset and development of oscillations in such cases. In that study, the pseudo-two-variable model could exhibit only period- 1 oscillations and it is of interest to consider the influence of noise for the present model for which higher periodicities exist. C. Influence of noise on system with reactant consumption In any real chemical system, there are inevitable extraneous influences that will cause generally small, but nonzero perturbations on the evolution. One source might be small temperature fluctuations that, in turn, affect the instantaneous values of some-of the rate constants for the reac- 0.8 Q2 f? -I )$ ; : ;-.L~ i FIG. 4. Evolution in presence of noise (p = 1.O x 10m5). (a) Autocatalyst concentration /3 as a function of time r. From left to right, the time series shows transitions from period-1 to period-2, period-2 to period-4 to transient chaos, transient chaos to period-8, period-4 to period-2, and period-2 to period-l. Transition from the nonoscillatory pseudosteady state to period-l and the reverse transition occur at and (not shown). (b) Poincare section as a function of p(r) and (c) as a function of 7:

5 1138 Scott eta/.: Transient chaos in a closed chemical system tion of interest. We have investigated various representations of such noise. Here we discuss the form which imagines that the parameter S is the most responsive to such effects. We have found, however, that the basic responses are not sensitive to the particular form chosen. For our present purposes, Eq. (2.4) has been modified to S(dy/&) = (@- y) (1 + E sin UT), (5) where E represents the magnitude of the noise and w its frequency. The most interesting cases are those with small noise effects [so that the forcing through Eq. (5) is not likely to bring additional changes of behavior in its own right]. We present here results for E = lo- and take w = 60 and a decay ratep = A range of these parameter values has been studied, including a tenfold decrease in amplitude and tenfold increases and decreases in frequency, and the results described below are typical of these values. We have also found that random forcing as well as random fluctuations imposed on the concentration y result in similar behavior. Figure 4( a) shows the evolution of the autocatalyst concentration fi with time. Period-l oscillations begin almost immediately as,u(7) passes through the value equal to,ur. Many more changes in oscillatory waveform can be seen within the time series, corresponding to the dynamic bifurcations of the pool chemical model emerging in the present full computation with reactant consumption. To bring this out more clearly, the PoincarC section as a function of,u( T) is shdwn in Fig. 4(b)-and the same data, as a function of time, are given in Fig. 4 (c). These dynamic bifurcation diagrams bear very close comparison with Fig. 3 (a) and reveal clearly that the system is evolving through many of the most complex periodic states accessible in the pool chemical formulation. We may thus, in a qualified sense, talk of this closed system evolving through a period of transient aperio: dicity. IV. DiSCUSSION The results presented above are of importance in that they demonstrate that closed chemical systems should be capable of displaying even the most complicated dynamic behavior, even in the presence of the inevitable effects of reactant consumption. Such responses are, of course, strictly transient; however, they can be long lived. Even in the absence of any experimental noise, the present computations reveal a period-doubling and period-halving change in waveform. When small perturbing effects are introduced, the response can include a dense sampling of the complete period doubling and remerging Feigenbaum sequence. If the decay rate is sufficiently slow, one may expect to even find evidence of the periodic windows of, say, period-5 within the evolution. In this case, then, the system with reactant consumption is essentially following the bifurcation structure of the pool chemical model. The latter thus provides a most useful reference and can be readily analyzed through the various path-following algorithms available. We may also suspect that the model with added noise is closer to reality than the perfectly stirred, noise-free formulation. The particular method of perturbing our model employed in Sec. III C is a simple periodic forcing. It is well known that such a driving term can in itself lead to a complex bifurcation structure. 14*15 We have thus investigated only low amplitude effects and have studied a wide range of frequencies-and additional forms of adding noise. In each case, the resulting bifurcation sequence, as evidenced by the PoincarC section-bifur ; I I I I (al 7 CL I I I (b) Q u s 0.4 (cl 7 FIG. 5. Sensitivity to initial conditions. (a) -Two time series (solid and dashed curves) within the period-2 region with initial values of,b differing byo.l% (p = 1.0X 10-s). (b) Two timeseries initiated within thechaotic region and (c) initiated within the period-2 region that then evolve through the chaotic region. I

6 Scott eta/.: Transient chaos in a closed chemical system 1139 cation diagram for instance, is qualitatively the same and corresponds almost quantitatively to that exhibited by the undisturbed pool chemical scheme. The major characteristic of chaos, and that which encourages us to use the word even in the transient responses seen here, is that of sensitivity to initial conditions. This effect can be illustrated in -the following way:, Figure 5 (a) shows the detail of two time series for the system with added noise as above, during a sequence of period-2 responses. The solid curve is that computed earlier, but the dotted line corresponds to a recomputation of the data following a small perturbation (0.1% ) in the initial concentration of fi at the beginning of the figure. This perturbation imposes a phase shift, but otherwise the evolution for both starting conditions is qualitatively identical, showing an insensitivity to initial conditions during this phase of the evolution. In Fig. 5 (b), by contrast, a simi1a.r perturbation of the concentrations during a chaotic phase leads to evolutions that grow apart in time-the separation of the two time series can clearly be seen to grow with time and the responses become uncorrelated by r~325.8 (the perturbation was made at r = 325.0). The perturbation causing the simple phase shift in Fig. 5 (a) corresponds to a period-2 evolution after the system has passed through its chaotic behavior. Figure 5 (c) shows a similar perturbation in the period-2 phase at an earlier time (7 = 319.0), before the chaotic sequence. Initially there is simply a phase shift as before, but this difference in the two time series becomes very significant at the onset of chaos. The two traces enter the chaotic period with effectively different initial conditions and the sensitivity of the system at this point causes the two to separate as in Fig. 5 (b). Thus, the timing of any perturbation, no matter how small, is important in determining whether it will cause only a small phase shift or a significant qualitative alteration of the subsequent evolution. [The detailed responses in the chaotic region also depend on the particular form of noise added to the system; however, the sensitivity to initial conditions shown in Figs. 5 (b) and 5(c) is found with each form and also in calculations carried out with no added noise.] It is also worth noting that the behavior described above is not specific to this model, nor to the chemical feedback processes therein. In a slightly different formulation, the second feedback can occur through self-heating. Scott and Tomlin have analyzed the nonisothermal autocatalator and shown virtually identical pool chemical bifurcation diagrams (and others involving mixed-mode oscillation sequences). In such circumstances, the typical temperature dependence of reaction rate constants and the amplification of such effects by nonlinear kinetics mean that otherwise virtually undetectable temperature excursions (perhaps less than 0.1 K) lead to the dramatic changes from simple period- 1 oscillations reported here. ACKNOWLEDGMENT This work was supported by NATO (Scientific Affairs Division, Grant No. 0124/89) and the National Science Foundation (Grant No. CHE ). APPENDIX: HOPF ANALYSIS The three-variable pool chemical model can be deduced from the present model by setting p = 0 in Eq. (2.1). The corresponding rate Eqs. (2.2)-( 2.4) yield analytical values of p* corresponding to Hopf bifurcation points. A linear stability analysis of Eqs. (2.2)-(2.4) yields the secular determinant - (1 +BLJ --R -~,,Pss P 1 +Pi 2aJL - 1 _ A o u CT l A s s = 0, (Al) where ass and & are given by Eqs. (3.1) and (3.2). Note that the steady state does not depend on (T or 6. From Eq. (2.3), ass =s With Eq. (A2), into the form where A3+AA2-tBil+C=0, c= Pss 0:s or ass& = I% the secular determinant can be expanded (A3) At a Hopf bifurcation, two of the three eigenvalues of the three-variable system are purely imaginary and the third is real and negative. The Hopf bifurcations therefore occur when AB=C which results in where d,p+a,s+a, (A>O,B>O), =o, A, =&l +py+l(l --pzs>, u CT2 A, = (1 +p:)2+$(1 +BZs).$(l -BL) &2 2 +-J- SE, u2 ( 1 +BZs ) 1 1-p; A3=1+PZs+- ~. a ( l+bis > (A4) (-45) Therefore, values of S corresponding to the Hopf bifurcation points can be calculated from Eq. (A5 ) for any particular values of p, a, and K, allowing the implicit determination of J. Chem. Phys., Vol. 94, No January 1991

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