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This historical review of the development of the Oregonator model of the Belousov–Zhabotinsky reaction is based on a lecture Dick Field presented during IrvFest2015—Celebrating a founding father of chaos!, a meeting in commemoration of Irving R. Epstein’s [Formula: see text] birthday. For Dick’s [Formula: see text] birthday festschrift, we focus here on the five papers in the series named “ Oscillations in chemical systems,” published in 1972 [Noyes et al., J. Am. Chem. Soc. 94, 1394–1395 (1972); Field et al., J. Am. Chem. Soc. 94, 8649–8664 (1972); Field and Noyes, Nature 237, 390–392 (1972)] and 1974 [Field and Noyes, J. Chem. Phys. 60, 1877–1884 (1974); Field and Noyes, J. Am. Chem. Soc. 96, 2001–2006 (1974)].

The spontaneous formation of ordered spatial concentration patterns in an unstirred chemical medium, supported by dissipation of chemical free energy, has been considered often since a pioneering suggestion by Turing and early work by Prigogine et al. and more recent work by Ross et al. involving nonequilibrium thermodynamics. The prototype experimental example is the oscillatory Belousov–Zhabotinsky reaction, in which target patterns of outward-moving concentric rings are readily observed. One widely-studied question is whether “microscopic” fluctuations can nucleate these target centers, or whether a catalytic, nucleating heterogeneous center is required. Vidal and Pagola observed spontaneous initiation with no nucleating particles visible at 6-micron resolution; however Zhang, Forster, and Ross argued theoretically that this is impossible in regimes far from Hopf bifurcations. We describe here an explicit mechanism in a “supercritical regime,” following and near to the low-f Hopf bifurcation in a generalized Oregonator model, by which microscopic fluctuations can nucleate activity, and reconcile these results with Zhang et al. Concentrations remain very close to the unstable steady-state values after the system slowly passes through the bifurcation point but before occurrence of the inevitable transition to large-amplitude limit cycle oscillations. Suitably timed small (even microscopic) fluctuations about this supercritical state can sharply accelerate the inevitable onset of large-amplitude limit cycle oscillations, potentially nucleating targets.

The recent numerical work of Lindberg et al. convincingly demonstrates that chemical chaos in a continuous flow, stirred tank reactor (CSTR) can be reproduced by a spatially homogeneous, accurate model of the kinetics of the Belousov–Zhabotinskii(BZ) reaction. However, some problems remain. The chaos in this model and two others, one using an accurate model of the chemical kinetics in conjunction with spatial inhomogeneity resulting from the finite CSTR mixing time and the other using a flawed model of the BZ chemical kinetics, results from coupling of two cycles coexisting within the complex dynamic model. The second cycle in the case of the homogeneous models involves a product of the main chemical limit cycle which is present at a high average concentration. In the Lindberg et al. model this product is assumed to be HOBr. It is clear, however, that a large [HOBr] does not accumulate in the real system because of its rapid reaction with Br−. We suggest that while the Lindberg et al. results are clearly ...

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...

The five‐step Oregonator model of the oscillatory Belousov–Zhabotinskii reaction has been expanded by taking into account the reversibility of the various steps. Forward and reverse rate constants have been assigned to four of the five steps by direct analogy to the detailed Field, Koros, and Noyes model of the Belousov–Zhabotinskii reaction. The rate constants and stoichiometry of the fifth step were parametrized and the stability of the steady state was investigated as a function of both of these parameters and the over‐all distance of the system from equilibrium. It was found that there is a critical distance from equilibrium which must be exceeded before the steady state becomes unstable. Numerical integration of the differential equations resulting from the model indicated that the system executes apparent limit cycle oscillations when the steady state is unstable. Introduction of reversibility into the Oregonator leads to a striking change in the range of values of the fifth step stoichiometric parameter over which oscillations will occur. This change is discussed in terms of the chemistry of the Belousov–Zhabotinskii reaction. Under some conditions the reversible Oregonator shows excitability such that a small finite perturbation of an infinitesimally stable steady state may be greatly amplified before the system returns to rest. An apparently analogous phenomenon appears in the Belousov–Zhabotinskii reaction itself.

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 representation of the BZ kinetics leads to an eight‐variable set of ordinary differential equations. Numerical analysis of these equations by continuation methods and by numerical integration shows the existence of broad regions of chaos and various hysteresis effects involving limit cycles, a steady state and/or a strange attractor. Tristability was found in calculations in a narrow flow rate range. The computed behavior appears for parameter values closely related to the values used experimentally to obtain similar phenomena, and the visual similarity of the computed and experimental strange attractors is striking. The experimental routes to chaos, period doubling, intermittency, and secondary Hopf bifurcations are all reproduced in the simulations. The computed and experimental structures of periodic windows observed within chaotic regions also are very similar.