Richard J. Field

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.

Chaos in chemistry and biochemistry

Mathematics of coupled and forced oscillators, W.R. Derrick experimental description of chaos in the Belousov-Zhabotinsky reaction, J-C. Roux modelling and interpretation of chaos in the Belousov-Zhabotinsky reaction, L. Gyorgyi chaos in forced and coupled chemical oscillators, M. Marek chaos in electrochemical systems, J. Hudson chaos in surface reactions, M. Eiswirth and C. Ertl chaos in biochemical systems - the peroxidase reaction as a case study, R. Larter from periodicity to chaos in biochemical systems, A. Goldbeter new theoretical approach for describing physico-chemical reaction dynamics with chaotic behaviour, V. Gontar.

Complex Effects Arising in Smoke Plume Simulations due to Inclusion of Direct Emissions of Oxygenated Organic Species from Biomass Combustion

Oxygenated volatile organic species (oxygenates), including HCOOH, H 2 CO, CH 3 OH, HOCH 2 CHO (hydroxyacetaldehyde), CH 3 COOH, and C 6 H 5 OH, have recently been identified by Fourier transform infrared measurements as a significant component of the direct emissions from biomass combustion. These oxygenates have not generally been included in the hydrocarbon-based initial emission profiles used in previous photochemical simulations of biomass combustion smoke plumes. We explore the effects of oxygenates on this photochemistry by using an established initial emission hydrocarbon profile and comparing simulation results obtained both with and without addition of the above six oxygenates. Simulations are started at noon and carried out for 30 hours in an expanding Lagrangian plume. After an initial transient period during which [NO x ] falls rapidly, conditions within the oxygenated smoke plume are found to be strongly NO x -sensitive, and the simulated final species profile is thus strongly dependent upon the Δ[NO]/Δ[CO] initial emission profile. Oxygenate addition results in very significant and complex effects on net O 3 production, as well as on the relative amounts of long-lived HO, and NO, reservoir species (H 2 O 2 , organic hydroperoxides, HNO 3 , and peroxyacetyl nitrate (PAN)) that are mixed into the surrounding atmosphere. Oxygenates may either increase or decrease net O 3 production (depending upon the initial Δ[NO]/Δ[CO]). However, they always increase H 2 O 2 and organic hydroperoxide production as a result of increased rates of radical + radical reactions. These effects spring largely from accelerated removal of NO, from the smoke plume due to increased radical concentrations resulting both from photolysis of oxygenates (mainly CH 2 O) and from their relatively high reactivity. Predicted concentrations of H 2 O 2 , Δ[O 3 ]/Δ[CO], Δ[NH 3 ]/Δ[CO], and Δ[HCOOH]/Δ[CO] are compared with some available measured values.

Composite double oscillation in a modified version of the oregonator model of the Belousov–Zhabotinsky reaction

A number of nonmonotonic behaviors appear when the Belousov–Zhabotinsky reaction is run in a flow system (CSTR) which are not observed when the reaction is run in a closed system. Among these behaviors is composite double oscillation in which nearly identical bursts of oscillation are separated by regular periods of quiescence. Here we use a modified version of the oregonator model of the Belousov–Zhabotinsky reaction to simulate composite double oscillation. Our modification involves the addition of a new variable which is related to the amount of brominated organic material present in the system. This new variable changes slowly on the time scale of the oscillations and controls the value of f, the stoichiometric factor of step 5 in the oregonator. Thus the behavior of the modified oregonator in CSTR mode when flowrates are moderate can be rationalized in terms of the properties of the unmodified oregonator in a closed system. We show that composite double oscillation is a hysteresis phenomenon occurrin...

Relative abundance and structure of chaotic behavior: the nonpolynomial Belousov-Zhabotinsky reaction kinetics.

We report a detailed numerical investigation of the relative abundance of periodic and chaotic oscillations in phase diagrams for the Belousov-Zhabotinsky (BZ) reaction as described by a nonpolynomial, autonomous, three-variable model suggested by Gyorgyi and Field [Nature (London) 355, 808 (1992)]. The model contains 14 parameters that may be tuned to produce rich dynamical scenarios. By computing the Lyapunov spectra, we find the structuring of periodic and chaotic phases of the BZ reaction to display unusual global patterns, very distinct from those recently found for gas and semiconductor lasers, for electric circuits, and for a few other familiar nonlinear oscillators. The unusual patterns found for the BZ reaction are surprisingly robust and independent of the parameter explored.

Aperiodicity resulting from two-cycle coupling in the Belousov-Zhabotinskii reaction. III: Analysis of a model of the effect of spatial inhomogeneities at the input ports of a continuous-flow, stirred tank reactor

Deterministic chaos is a well‐established phenomenon in continuous‐flow, stirred tank reactor (CSTR) experiments with the oscillatory Belousov–Zhabotinskii (BZ) reaction. However, it has not yet been possible to reproduce the experimentally observed, robust chaos in simulations using realistic models of the homogeneous chemical kinetics of the BZ reaction. That it may be necessary to consider spatial inhomogeneities in modeling the BZ chaos is suggested by the existence of strong stirring effects on the aperiodic behavior and by the difficulty of reproducing chaos under the same conditions in reactors of different physical configuration. Such inhomogeneity might result from a lack of perfect mixing in the CSTR, especially near the inlets, or from diffusion of species at low flow rates from the CSTR reaction mixture into the tips of the inlets. The presence of spatial inhomogeneities allows coupling between essentially independent oscillators, a well‐known source of chaos. Such a model using a realistic re...

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.

Characterization of oscillation and a period‐doubling transition to chaos reflecting dynamic instability in a simplified model of tropospheric chemistry

A simplified box model extracted from tropospheric photochemistry is investigated as the influx of NO (FNO) is increased. A subcritical Hopf bifurcation is encountered as FNO is increased, beyond which the steady state is unstable, and the system evolves to an oscillatory state resulting from alternate dominance of two radical chain processes. The first is an O3-consuming process of net stoichiometry, CO + O3 → {CO2} + {O2}, occurring at high [CO] and [O3] and very low [NOx], but leading to increased [NOx] via FNO as CO and O3 are depleted. As [NOx] thus grows, passes through a maximum, and then declines, the second, an O3-producing process of net stoichiometry, CO + 2{O2} + hν → {CO2} + O3, is dominant. It remains so as O3 and CO accumulate (CO via its influx, FCO) until [NOx] again reaches very low values at high [O3] and [CO] as a result of the removal of NO2 by HO photochemically generated from the increasing [O3]. This alternation occurs because each process affects [NOx] and [HOx] so as to lead to dominance of the other. A period-doubling transition to chaotic oscillation occurs as FNO is increased further. The embedding dimension of the chaos is estimated to be four, and the original six-variable model can be reduced to a four-variable (CO, O3, [NO + NO2] and [HO + HO2]) system that behaves nearly identically to the full six-variable model. While the oscillatory and chaotic periods seem too long (at least weeks) to be observed in real atmospheres, the model displays the nonlinear nature and dynamic instability of tropospheric photochemistry and offers insight into the behavior of and transitions between higher and lower [NOx] states, which may be observable. The importance of the ratios [CO]/[NO2] and [O3]/[NO] to net O3 change is illustrated. The appearance of this instability suggests that predictions based upon the temporal evolution of this or even more complex models of tropospheric chemistry sometimes may be very sensitive to the exact initial conditions of [CO]/[NO2] and [O3]/[NO] prevailing.

Reduction of a Model Describing Ozone Oscillations in the Troposphere: Example of an Algorithmic Approach to Model Reduction in Atmospheric Chemistry

We consider a simplified reaction mechanism from tropospheric chemistryconsisting ofsix chemical species involved in ten dynamic processes. The concentrations of all six speciesundergo temporal oscillation for some parameter values. An asymptoticapproach to reduction of the original six-variable model to a four-variable version is considered together with detailed explanation of the procedure as well as discussion of conditions under whichthe reduction is possible. The reduced system nearly quantitatively represents the oscillating behavior of thefull model and allows elucidation of its basic dynamical features. This approach also can be used to compare various small models of atmospheric chemistry, and to determine their underlying dynamic structure. The method can be systematically appliedto larger atmospheric models.

An NMR Study of the Equilibration of d‐Glucaric Acid with Lactone Forms in Aqueous Acid Solutions

The aqueous solution equilibration of d‐glucaric acid with its lactone forms was studied by NMR with and without acid catalysis. The kinetics of the approach to equilibrium were simulated, and approximate equilibrium and rate constants were obtained.