## Abstract

The simplest model of thermokinetic oscillations in a closed, chemical system requires only two first-order reaction steps $(0) P \rightarrow A \text{rate} = k_0p,$ $(1) A \rightarrow B \text{rate} = k_1(T) a.$ Step (0) is assumed to be thermoneutral and its rate constant to not depend on the temperature (i.e. to have zero activation energy). Step (1) is an exothermic process, and the rate constant k$_1$ has an Arrhenius temperature dependence k$_1 = A_1e^{-E_1/RT}.$ The governing reaction rate and energy-balance equations written in dimensionless form in terms of the reduced concentration $\alpha$ of the intermediate A and the temperature rise $\theta$ are: $d\alpha/d\tau = \mu e^{-\gamma\tau}-\kappa\alpha f(\theta)$ and $d\theta/d\tau = \alpha f(\theta)-\theta$, where $\mu$, $\gamma$ and $\kappa$ are parameters and the function f($\theta$) has the form $f(\theta) = \exp[\theta/(1+\epsilon\theta)].$ All of the dynamical behaviour of these equations can be determined qualitatively and, to leading order, quantitatively from the pool chemical approximation, $\gamma \rightarrow 0$. Oscillatory solutions emerge as $\mu$ and $\kappa$ are varied, from points of Hopf bifurcation. These are located analytically and additional expressions are derived for calculating the growth in amplitude and period as the system moves away from these points. Both stable and unstable limit cycles (supercritical and subcritical bifurcations) can be found, provided $0 < \epsilon < \frac{2}{9}$. For $\epsilon$ in the range $\frac{2}{9} < \epsilon < \frac{1}{4}$ only stable limit cycles exist. Oscillatory solutions cannot be observed if $\epsilon \geqslant \frac{1}{4}$.

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