Oscillations and tipping points in a model thermokinetic reaction

Abstract

We consider a model thermokinetic reaction that is both autocatalytic and temperature sensitive. The mechanism consists of two reactions. The first is a reversible reaction with a temperature-dependent rate, and the second involves a cubically autocatalytic process. We demonstrate the presence of up to five steady states in the system, dependent on parameters, and we illustrate the fact that the system can exhibit tipping-point behaviour, in which small changes to initial conditions in an experiment could lead to radically different long-term outcomes. We prove that oscillatory behaviour is not possible over large regions in the parameter space; outside these regions, however, oscillations have been found, and we present a simple and robust method for computing them. Small-amplitude oscillations can arise through Hopf bifurcations at an equilibrium point. In addition, large-amplitude oscillations can bifurcate directly from global structures that are associated with equilibria behaving as saddles. This is illustrated with some numerical solutions of these highly nonlinear equations.