Uncertain dynamics in nonlinear chemical reactions

PCCP

Uncertain dynamics in nonlinear chemical reactions Jichang Wang,a Hongyan Sun,b Stephen K. Scottc and Kenneth Showalterb a

Department of Chemistry and Biochemistry, University of Windsor, Windsor, Ontario, Canada N9B 3P4 b Department of Chemistry, West Virginia University, Morgantown, WVU 26506-6045, USA c School of Chemistry, University of Leeds, Leeds, UK LS2 9JT Received 8th September 2003, Accepted 24th October 2003 First published as an Advance Article on the web 5th November 2003

The three-variable autocatalator exhibits chaotic dynamics in an open system configuration. When the model is modified to include a dissociation reaction of the autocatalytic species, B ! X + Y, followed by a recombination reaction, X + Y ! C, an unusual sensitivity to initial conditions is displayed. The modified system exhibits an effectively infinite number of asymptotic attractors, some chaotic, some periodic, and some stationary, which are accessible by suitable initial conditions or perturbations. We also investigate the modified model in a closed system configuration and find that initial conditions may significantly affect the temporal evolution of the system.

1. Introduction The study of nonlinear chemical kinetics has flourished in the past three decades.1–4 Complex dynamical behaviors, such as periodic, quasiperiodic, and chaotic oscillations, have been observed in both continuous flow stirred tank reactors (CSTRs)5,6 and as transients in batch reactors.7 Such dynamical behavior has challenged long-held notions concerning reproducibility in chemical systems having the same experimental conditions, such as reactant concentrations and temperature. In chaotic chemical systems, for example, the temporal evolution displays an extreme sensitivity to initial conditions as well as to perturbations during the course of the reaction.3 Hence, we now know that it is impossible to exactly reproduce the dynamical evolution of a reaction exhibiting chaotic dynamics. However, we also know that when such a system eventually settles into its asymptotic chaotic behavior, the statistical properties of the attractor, such as the Lyapunov exponents and the correlation dimension, are expected to be the same for a given set of conditions.8 In the case of coexisting multiple attractors, such as bistability in the iodate oxidation of arsenite,9 the initial conditions determine the selection of the asymptotic state of the system. The phase plane is divided into two different basins of attraction, and the relative areas of these give the probabilities of attaining each attractor for random initial conditions. There is generally a finite probability for the realization of each attracting state (steady or oscillatory) in bistable systems, where the basins are separated by simple basin boundaries.10 More recent studies have shown that the boundary separating these regions can be fractal, making the outcome of initial conditions much less certain, since the fractal basin boundary has an infinitely deep mixing of the basins. There is an even more complicated scenario, that of riddled basins, in which the entire basin of a given attractor is riddled with ‘‘ holes ’’ that lead to the competing attractor.11–13 A different type of multistablity has been recently characterized in a coupled Lorenz system, in which an infinite number of attractors can be realized, depending on the initial conditions or external perturbations.14 Since small perturbations cause the system to evolve to completely new attractors with different statistical properties, such systems are said to exhibit 5444

‘‘ uncertain destination dynamics ’’. Such dynamics may underlie the behavior reported in the chlorite–thiosulfate reaction, where the system displays a marked irreproducibility in the clock-reaction induction period, which varies from reaction to reaction in an apparently random manner.15 Nagypal and Epstein16 also observed similar behavior in the chlorite–iodide reaction, particularly when the ratio of [ClO2]/[I2] was within a specific range. The inability to reproduce the dynamical behavior for the same set of experimental conditions, despite care to ensure reproducibility, suggests that the inherent dynamics of these systems is playing some role. The stochastic behavior of such systems has been characterized in terms of micro- and macroscopic fluctuations coupling with the deterministic chemical kinetics. In this paper, we describe how deterministic chemical kinetics can exhibit an extreme sensitivity to fluctuations, such that qualitatively different dynamical behaviors would be explored in experimental chemical systems where such fluctuations are unavoidable.

2. Model We investigate the dynamics of a model based on the threevariable autocatalator,17 which we expand by introducing two intermediate species, X and Y, that are produced and consumed in successive reactions: P!A

ð1Þ

PþC!AþC

ð2Þ

A!B

ð3Þ

A þ 2B ! 3B

ð4Þ

B!XþY

ð5Þ

XþY!C

ð6Þ

C!D

ð7Þ

The model describes a chemical system that converts a reactant P to a final product D through seven steps and five intermediate species, A, B, C, X, and Y. Reactions (5) and (6) provide a ‘‘ buffer step ’’ in which reagents X and Y are produced and consumed through the same processes. The system

Phys. Chem. Chem. Phys., 2003, 5, 5444–5447 This journal is # The Owner Societies 2003

DOI: 10.1039/b310923b

without these steps has been thoroughly studied and is known to exhibit a wide variety of dynamical behaviors, including period-doubling and mixed-mode chaos.17,18 We study here how the kinetics of X and Y in reactions (5) and (6) gives rise to new and unusual dynamics in a chemical system. An open system is first considered in which a continuous supply of reactant P maintains the system in a nonequilibrium state. We then consider the dynamics of a closed system where reactant P is consumed. The dimensionless rate equations governing the evolution of the five intermediate species are given by da ¼ k þ mc ab2 a dt db ab2 þ a b ¼ dt s dc xy c ¼ dt d dx ¼ b kxy dt dy ¼ b kxy dt

ð8Þ ð9Þ ð10Þ ð11Þ ð12Þ

1=2 2 1=2 k3 k4 1=2 k4 k4 k7 A, b ¼ B, c ¼ C, k3 k52 k3 k52 1=2 1=4 1=2 1=4 k5 k3 k5 k3 X, and y ¼ Y are the x¼ k6 k4 k6 k4 dimensionless concentrations of species A, B, C, X, and Y, k1 k4 1=2 k2 P0 k3 k3 P0 , m ¼ ,s¼ ,d¼ , respectively, and k ¼ k5 k3 k7 k5 k7 1=4 1=2 k3 and t ¼ k3 are dimensionless parameters k ¼ ðk5 k6 Þ k4 derived from the original reaction scheme.17 Because the rates of production and consumption of X and Y are equal, any difference between x and y will remain throughout the reaction. This persistent concentration difference suggests an implicit ‘‘ constant of the motion" in the dynamical system. As shown below, this constant is a function of the initial conditions and any subsequent perturbations experienced by the system. We describe how the existence of such a constant gives rise to qualitatively different behavior for slightly different initial conditions even though all other operating conditions are exactly the same. In addition, the evolution of the system may be completely altered when it is subjected to a perturbation during the course of the reaction. where a ¼

db ab2 þ a b ¼ dt s dc yðy þ cÞ c ¼ dt d dy ¼ b kyðy þ cÞ dt

ð14Þ ð15Þ ð16Þ

where the new parameter c represents the difference between x and y. The value of c during the course of the reaction depends on the initial conditions and any subsequent perturbations that change the value of x y. The significance of the parameter c is illustrated in Fig. 1(a), which depicts a bifurcation diagram of the system response with respect to the difference between x and y. The parameters s, d, k, m and k, as well as the initial values of the other concentration variables are held constant. The points shown in the figure represent the maximum of x in each oscillatory period. We see that transitions from simple to complex to chaotic oscillations are displayed as c is varied. This result demonstrates that a slight difference in the concentrations of the reagents X and Y may not only lead to quantitatively but qualitatively altered dynamical behavior. The transition from regular to chaotic behavior with the variation of c is also illustrated by the value of the largest Lyapunov exponent, Fig. 1(b), in which each value was determined after the system evolved to its asymptotic state.19 Examples of the time evolution of the system with different initial values of x and y are shown in Fig. 2, where the parameter values are the same as in Fig. 1. The initial compositions

3. Results and discussion The dependence of the dynamics on the initial conditions can be understood by examining the rate eqns. (11) and (12), where dx dy ¼ 0. Alternatively, this relationship can be expressed dt dt dðx yÞ ¼ 0. A general solution for this type of differential as dt equation is x y ¼ c, where c represents a constant of the dynamics. Such a constant means that the initial difference between x and y will not change unless a perturbation in either x or y occurs, and that the instantaneous value of one variable can be calculated from the other using the conservation relation x ¼ y + c. Inserting this relation into eqns. (8)–(12) allows the five-variable system to be reduced to four variables: da ¼ k þ mc ab2 a dt

ð13Þ

Fig. 1 Bifurcation diagram (a) and largest Lyapunov exponent (b) plotted as a function of c, the difference between x and y. Parameter values are s ¼ 0.005, d ¼ 0.02, k ¼ 1111.1, m ¼ 166.7, and k ¼ 10.0. The initial values of a, b, and c are 105 and the initial value of x is 10.0.

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Fig. 2 Time evolution of the system described by rate eqns. (8)–(12) for different initial compositions: (a) period-4, x ¼ 0.01, y ¼ 0.01; (b) period-6, x ¼ 0.01, y ¼ 0.21; and (c) chaos, x ¼ 0.01, y ¼ 0.61. The initial values of a, b and c are a ¼ b ¼ c ¼ 105. Other parameter values are the same as in Fig. 1.

are a ¼ b ¼ c ¼ 105, and (a) x ¼ 0.01 and y ¼ 0.01, (b) x ¼ 0.01 and y ¼ 0.21, and (c) x ¼ 0.01 and y ¼ 0.61. In each case the system exhibits qualitatively different behavior, with periodic oscillations in (a) and (b) and chaos in (c). A positive leading Lyapunov exponent (l ¼ 3.1) was calculated for the chaotic behavior in (c). While the final dynamics of the system depends prominently on the initial values of x and y, variations of the initial values of the other species concentrations do not affect the asymptotic behavior. Of course, in the chaotic regime, the initial concentrations of the other species will affect the quantitative behavior such as the detailed features of the time series; however, the qualitative behavior, i.e., the particular strange attractor determined by the system parameters (and the initial values of x and y) is not affected. As described above, the asymptotic dynamics depends on the difference between x and y, which, in turn, is determined by their initial values. The essential role of the difference was tested in two calculations, where the initial values were x ¼ 0.02, y ¼ 0.22 and x ¼ 0.22, y ¼ 0.42, in which identical periodic behavior was observed. The dynamical behavior of the reaction is also susceptible to variations in the difference between x and y during the course 5446


of the reaction, induced by fluctuations or by deliberate perturbations designed to have a desired effect. Fig. 3 shows the behavior of the system when a small perturbation is applied to variable x (Dx ¼ 0.2) at time t ¼ 20. The system exhibits period-4 oscillations before the perturbation, and a transient period of complex oscillations occurs immediately following the perturbation. Eventually the system settles into a new limit cycle with period-6 oscillations. Unlike the behavior observed in perturbation studies of ordinary chemical systems, the system does not relax back to the original period-4 oscillations but remains in the period-6 state (with no additional perturbations). Perturbation methods have proven to be useful in characterizing reaction mechanisms, where valuable information can be collected as the system relaxes from a perturbed state to its stable state in the concentration phase-space.20–22 In contrast, an irreversible change occurs in the system studied here, which relaxes to a completely new state when a perturbation is applied to x or y. Detailed calculations have demonstrated that the new state of the perturbed system is determined exclusively by the amplitude of the perturbation and there is no dependence on phase. The above characterization of ‘‘ uncertain destination dynamics ’’ pertains to the system with the ‘‘ pool chemical approximation ’’, i.e., the concentration of the initial reactant P is assumed to be a constant that does not vary in time. This approximation is commonly assumed in order to maintain a system in a state far from equilibrium. It could be experimentally realized with a selective membrane that allows P to diffuse into the reactor from a reservoir. In a batch reactor, however, the concentration of the reactant P would decrease in time as it is consumed. Since the initial concentration of P is absorbed into the parameters k and m in eqns. (8)–(12), the values of these parameters would decrease in a closed system as P is consumed. We can therefore consider the closed system in terms of the open system with different values of P0 . Fig. 4 shows a bifurcation phase diagram for the P0 vs. c parameter plane. Within the parameter range shown, the system exhibits period-1, period2, and period-4 oscillations, as well as chaos, denoted by P1, P2, P4, and C, respectively. Periodic oscillations can also be found within the region of chaos, and the corresponding parameter windows are designated by symbols in this region. It is clear that a decrease in P0 , corresponding to the consumption of reactant in a batch reaction, would cause the system to pass through a period-doubling bifurcation to chaotic behavior and then a reverse period-doubling bifurcation to yield simple oscillations. A transition from period-4 to period8 oscillations can also be observed but only within a very narrow range of P0 values. Steady state behavior is also observed in this system for values of P0 smaller than 0.87 or larger than 1.17.

Fig. 3 Time series of the perturbed system. A perturbation of the variable x, Dx ¼ 0.02, is applied at t ¼ 20. Parameter values and initial composition of the system are the same as in Fig. 1.

the parameters describing the operating conditions are held constant. Each of the asymptotic states can be realized by introducing specific initial conditions or perturbations. The parameter that determines the final state of the system is the difference between the concentrations of the two intermediate species X and Y. We have also considered the closed, batch reaction and have found that small variations in the concentration difference between X and Y give rise to significant variations in the evolution of the reaction. We emphasize that small variations in initial conditions for either the open or closed system would be unavoidable in an experimental setting. A chemical system governed by kinetics equivalent to eqns. (1)–(7) would therefore be extremely susceptible to experimental errors in initial conditions or perturbations during the course of the reaction. Fig. 4 Bifurcation sequences in the P0 vs. c parameter plane. Parameter values are s ¼ 0.005, d ¼ 0.02, k ¼ 1000.0, m ¼ 170.0 and k ¼ 10.0. The initial values of a, b, c, and y are 105 and the initial value of x ¼ y + c. Periodic oscillations in the parameter window where the system exhibits chaos are indicated by the symbols, (+) period-5, () period-6, (*) period-8, and () period-10.

Transient chaos in a batch reaction has been studied in the original three-variable autocatalator, where the concentration of reactant P was allowed to decrease exponentially.23 The transient appearance of various complex oscillations in the period-doubling route to chaos was observed as a function of time in the batch reaction model. Exactly the same type of transient complex oscillations are observed in eqns. (1)–(7) when treated as a batch reaction, except now the particular transient behavior depends on the difference between the initial concentrations of x and y. The time evolution corresponds to a path determined by the particular value of c, with P0 decreasing in time, passing through the bifurcation sequences shown in Fig. 4. Significant quantitative and some qualitative changes in dynamical behavior would be observed with changes in c. Hence, in an experimental setting, small changes in the initial conditions or small perturbations during the course of the reaction might give rise to significant variations in the time evolution of the system. The ‘‘ uncertain destination dynamics ’’ in the open system as well as the ‘‘ uncertain transient dynamics ’’ in the closed system arise because the species X and Y are produced and consumed at the same rates. If such a relationship is destroyed, for example, by flow terms in a CSTR, the difference between X and Y will decrease in time and eventually disappear. The evolution of the parameter c is then described by the equation dc ¼ fc, where f represents the mean residence time of the dt species in the reactor. In this case, a perturbation on X or Y would cause only transient changes. If the reaction time scale was significantly smaller than the reactor residence time scale, however, long lived transient behavior, corresponding to ‘‘ uncertain destination dynamics ’’ in the open system, would be observed. In summary, we have described an open chemical system that may exhibit an effectively infinite number of attractors, each with different statistical properties, even though all of

Acknowledgements K.S. thanks the National Science Foundation (CHE-9974336) and the Office of Naval Research (N00014-01-1-0596) and J.W. thanks the NSERC for supporting this research.

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