Limit-cycle behaviour in a model chemical reaction: the Sal’nikov thermokinetic oscillator

Abstract

The Sal’nikov thermokinetic oscillator is a model chemical reaction consisting of a two-stage decay of some chemical species. The first stage is a straightforward first-order decay process at constant temperature, but the second stage is exothermic, and is assumed to be governed by Arrhenius kinetics. When the containing vessel is well-stirred, the kinetic rate equation for the reaction along with an equation expressing conservation of energy leads to a system of two ordinary differential equations describing the behaviour of the process. The equations are coupled and highly nonlinear, and their solution gives the temperature inside the vessel and the concentration of the intermediate chemical species. Under certain circumstances, sustained periodic oscillations in these two quantities are possible. In this paper, I give a rigorous proof that these remarkable oscillations are not possible for certain combinations of the defining physical parameters. A numerical solution technique for obtaining periodic oscillations in the system is then presented; the method gives results of great accuracy, it automatically determines the stability of the solution, and is capable of computing unstable periodic orbits. Results of extensive numerical investigation are presented, and the occurrence of unstable limit cycles and multiple solutions is discussed in detail.