Bursting phenomena in a simplified Oregonator flow system model

Abstract

We investigate a model for the Belousov–Zhabotinskii reaction in a continuous flow, stirred tank reactor. The model consists of a system of three ordinary differential equations derived from a more complicated five‐dimensional model introduced by Janz, Vanecek, and Field [J. Chem. Phys. 73, 3132 (1980)]. It is basically a simplified, irreversible, batch reactor Oregonator with flow terms and with instantaneous dependence of the stoichiometric parameter f on the amount of brominated organic substance. Over a range of physical parameters (e.g., lower flow rates) the system exhibits ’’bursts’’ of oscillations. As in some experiments [e.g., Marek and Svobodova, Biophys. Chem. 3, 263 (1975)], the observer sees several spikes followed by an interval of quiescence (IQ) which is subsequently followed by a resumption of the spikes, etc. As Janz et al. found, bursting occurs as f sweeps slowly back and forth through values for which the batch reactor subsystem exhibits bistability and hysteresis. The CSTR solution alternates between the batch model’s stable periodic solution, during the oscillatory phase, and the batch model’s stable steady state of low oxidation, during the IQ. Analytically and numerically, we describe these subsystem solutions as functions of f; bistability arises as a subcritical Hopf bifurcation. Also, we estimate quantitatively features of the CSTR bursts, such as IQ duration, and their dependence on parameter values. For other parameter ranges we find qualitatively different bursting phenomena. In one example, the CSTR system is excitable; there is a stable steady state and the response to an adequate perturbation is an excitation burst of several pulses and then a return to the steady state. At considerably higher flow rates, there are repetitive, single‐spike bursts with IQ’s of high oxidation; such patterns resemble those calculated by Showalter, Noyes, and Bar‐Eli [J. Chem. Phys. 69, 2514 (1978)] and some responses observed experimentally by Schmitz, Graziani, and Hudson [J. Chem. Phys. 67, 3040 (1977)]