On the presence of limit-cycles in a model exothermic chemical-reaction: Sal'nikov's oscillator with 2 temperature-dependent reaction-rates

L. K. Forbes, M. R. Myerscough, Brian Gray

Research output: Contribution to journalArticlepeer-review


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 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.

Original languageEnglish
Pages (from-to)591-604
Number of pages14
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number1895
Publication statusPublished - 9 Dec 1991



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