Richard M. Noyes

Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction

The chemical mechanism of Field, Koros, and Noyes for the oscillatory Belousov reaction has been generalized by a model composed of five steps involving three independent chemical intermediates. The behavior of the resulting differential equations has been examined numerically, and it has been shown that the system traces a stable closed trajectory in three dimensional phase space. The same trajectory is attained from other phase points and even from the point corresponding to steady state solution of the differential equations. The model appears to exhibit limit cycle behavior. By stiffly coupling the concentrations of two of the intermediates, the limit cycle model can be simplified to a system described by two independent variables; this coupled system is amenable to analysis by theoretical techniques already developed for such systems.

A modified Oregonator model exhibiting complicated limit cycle behavior in a flow system

A reversible Oregonator model has been used to simulate recent experimental measurements by Schmitz, Graziani, and Hudson of complicated oscillations by a Belousov–Zhabotinsky system in a stirred tank reactor. The experimental observations indicate chaotic behavior of the small amplitude oscillations occurring between major excursions, but our computer simulation with a small error parameter apparently generates a true limit cycle with six relative maxima before the pattern repeats. The differences between experiment and simulation suggest the chaotic behavior observed experimentally may result from fluctuations too small to measure in any other way. The computations also indicate that reversibility of the reaction of bromate with bromide is important in a continuously stirred tank reactor under conditions such that the (unstable) steady state has a very low concentration of bromide ion.

An amplified Oregonator model simulating alternative excitabilities, transitions in types of oscillations, and temporary bistability in a closed system

The original Oregonator model of five irreversible steps and one stoichiometric parameter has been amplified to seven irreversible steps. The amplified model simulates excitability of a reduced steady state induced by silver ions and of an oxidized steady state induced by bromide ions. It also simulates how oscillations during a single run might change from reduction pulses to oxidation pulses. Finally, it simulates a temporary bistability persisting for over two hours in a closed system. If independent measurements determine the rate of enolization of the organic substrate and the rate of reaction of that substrate with the oxidized form of the catalyst, this model may be able to simulate semiquantitatively a large fraction of reported behaviors of catalyzed homogeneous Belousov–Zhabotinsky systems. It will not model systems in which Br2 accumulates or in which there are high‐frequency small‐amplitude oscillations around an oxidized steady state. Procedures are suggested for simulating such behaviors.

A discrepancy between experimental and computational evidence for chemical chaos

Hudson and Mankin studied the oscillatory Belousov–Zhabotinsky reaction in a flow reactor and found ranges of a few percent of residence time within which behavior was a chaotic mixture of the regular behaviors on either side of that range. Our efforts to model that system computationally indicate that any range of chaotic behavior is less than one part per million of residence time. The discrepancy may mean that the experiments were influenced by the peristaltic pumping or by uncontrolled fluctuations of some other variable. It may also mean that the computational model was too simplified to reproduce the chaotic effects inherent in the real system. The only resolution we can suggest is that random fluctuations in a parameter like residence time may induce chaotic behavior over a range of average residence times that is much wider than the amplitudes of the fluctuations themselves.

Relevance of a two‐variable Oregonator to stable and unstable steady states and limit cycles, to thresholds of excitability, and to Hopf vs SNIPER bifurcations

A stiffly‐coupled Oregonator model based on the two independent variables Y and Z has been examined in detail with the use of the stoichiometric factor as a single disposable parameter. The trajectories and periods of the stable limit cycles can be generated with unanticipated accuracy from simple linear differential equations. Transitions between stable limit cycles and stable steady states takes place by means of subcritical Hopf bifurcations. Unstable limit cycles and thresholds of excitability can be recovered by integrating the equations of motion backward in time; such procedures cannot be applied so easily for models based on more than two independent variables. We have examined claims of experimental evidence for saddle‐node infinite period (SNIPER) bifurcations and have concluded that all currently available evidence is equivocal. Until unambiguous criteria can be established for identifying SNIPER bifurcations in real systems, and until chemical mechanisms have been proposed which generate such ...

An alternative to the stoichiometric factor in the Oregonator model

The original Oregonator model to the Belousov–Zhabotinsky oscillating reaction used three composition variables, five rate constants, and one stoichiometric factor involving five irreversible reaction steps. As a result of objections to the stoichiometric factor by Noszticzius, Farkas, and Schelly, we have developed an alternative model of the same complexity based on six irreversible steps. The revised model eliminates some stoichiometric difficulties, and by minor modification of one of the six steps it handles situations in which the final oxidation state of the reduced bromine is +1, 0, or −1. An alternative skeleton model called the Explodator and proposed by Noszticzius, Farkas, and Schelly should not be considered a viable alternative to the Oregonator unless a computational prerequisite and an experimental kinetic prerequisite can both be satisfied.

Oscillations in chemical systems. XII. Applicability to closed systems of models with two and three variables

Most computations on models of chemical oscillators have maintained constant concentrations of major reactants while concentrations of intermediates were allowed to vary. Such simplified computations are applicable to closed chemical systems only if reactants are depleted by small fractions during each cycle. Existing models that involve only two intermediate species are generally unsatisfactory for modeling closed systems. Thus, the Lotka mechanism (which does not generate a true limit cycle) can not model even infinitesimally small oscillations in a closed system unless the rate constant for predator–prey interaction is very large. The Brusselator model can not model closed system oscillations unless the various rate constants are confined to very restricted ranges. Any other model with only two intermediates must contain a step at least third order in those intermediates. By contrast, the Oregonator model with three variables and only first‐ and second‐order processes, can model a closed system in whic...

Unstable supersaturated solutions of gases in liquids and nucleation theory

Limiting supersaturations for dissolved gases manifested by gas evolution oscillators and by direct experiments cannot be accounted for by the application of classical nucleation theory (CNT). The theory predicts bubbles containing 104–105 molecules at nucleation, with Helmholtz energies of ca. 104kT per bubble, much too high for homogeneous nucleation to occur spontaneously in a finite time. We investigate alternative unstable structures (‘blobs’) which do not have well-defined interfaces, which may exist transiently at the point of nucleation as the precursors of true bubbles and which circumvent the need for a large Helmholtz energy for their formation. Effects due to global or local depletion of the solution concentration at nucleation are also considered.

A model for imperfect mixing in a CSTR

When a chemical reaction is carried out in a continuously stirred tank reactor, the behavior may be significantly affected by the efficiency with which the entering chemicals are mixed with the main contents of the reactor. We have developed a model for this effect which assumes that a feed of premixed chemicals remains for a while in totally segregated packets before they are rapidly and perfectly mixed with the rest of the system. The time of this initial segregation is affected by the efficiency of stirring in the reactor. The model has been tested by computations on a mechanism developed by Roelofs et al. for a reaction which would oscillate even in a closed system. It has also been tested by computations on the rapid autocatalytic oxidation of cerous ion by bromate in the presence of a small amount of bromide. The results are qualitatively consistent with effects observed experimentally and in computations with other models including a somewhat similar one by Kumpinsky and Epstein. More quantitative ...

A model illustrating amplification of perturbations in an excitable medium

The oscillatory Belousov–Zhabotinskii reaction can be modelled approximately by five irreversible steps: A + Y → X (M1), X + Y → P (M2), B + X → 2X + Z (M3), 2X → Q (M4), Z →ƒY. (M5). These equations are based on the chemical equalities X = HBrO2, Y = Br–, Z = 2Ce(IV), and A = B = BrO–3. If the rate constants kM1 to kM4 are assigned by experimental estimates from oxybromine chemistry, the kinetic behaviour of the model depends critically upon the remaining parameters kM5 and ƒ. When ƒ does not differ too greatly from unity, and when kM5 is not too large, the steady state is unstable to perturbation and the system oscillates by describing a limit cycle trajectory.When ƒ and kM5 lie outside the range of instability, the steady state is stable to very small perturbations. However, the steady state may still be excitable so that perturbation of the control intermediate Y by a few percent will instigate a single excursion during which concentrations of X, Y, and Z change by factors of about 105 before the system returns to the original steady state. This ability of a small perturbation of the steady state to trigger a major response by the system is just the type of behaviour necessary to explain the initiation of a trigger-wave by a heterogeneous “pacemaker” as has been observed by Winfree. The same type of excitability of a steady state has important implications for the understanding of biochemical control mechanisms.