On the Presence of Limit-Cycles in a Model Exothermic Chemical Reaction: Sal'nikov's Oscillator with Two Temperature-Dependent Reaction Rates

• Published in: Proceedings of the Royal Society. A, Mathematical and physical sciences Mathematical and physical sciences, 1991-12, Vol.435 (1895), p.591-604
• Main Authors: Forbes, Lawrence K., Myerscough, Mary R., Gray, Brian Frederick
• Format: Article
• Language: English
• Subjects: Ambient temperature; Chemical reactions; Limit cycles; Mathematical constants; Oscillation; Oscillators; Phase plane; Reaction kinetics; Steady states; Temperature dependence
• ISSN: 1364-5021; 0962-8444; 1471-2946; 2053-9177
• DOI: 10.1098/rspa.1991.0163

This paper investigates a model chemical reaction in which a single substance undergoes a two-stage process of decay, first producing an intermediate species and finally giving a product chemical. Each of the two stages involves only simple first-order reaction kinetics, but the governing rate parameter for each of the two reactions is temperature dependent. The reaction vessel is assumed to be well stirred, and the rate of each reaction is governed by Arrhenius kinetics, although with a different activation energy for each process. The mathematical behaviour of the system is therefore described by a coupled system of two highly nonlinear ordinary differential equations for the concentration of the intermediate species and the temperature, arising from the rate equation and the energy conservation equation. This simple model is capable of predicting oscillatory behaviour in the concentration of the intermediate chemical and in the temperature. We present the Hopf condition for the emergence of these limit cycles from a homogeneous steady state, and then continue these solutions numerically into regions of the parameter space in which oscillations of very large amplitude can occur. The presence of multiple limit cycles is detected and discussed. An extension of Bendixson’s criterion is used to show that oscillatory behaviour is only possible in a certain confined region of the parameter space.