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The simplest model of thermokinetic oscillations in a closed, chemical system requires only two first-order reaction steps (0) P → A rate = k 0 p , (1) A → B 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 1 e –E1/RT . The governing reaction rate and energy-balance equations written in dimensionless form in terms of the reduced concentration α of the intermediate A and the temperature rise θ are : d α /dז = μ e - γ ז ─ kαf(θ) and d θ /dז = αf(θ) ─ θ , where μ, γ and k are parameters and the function f(θ) has the form f(θ) (θ) = exp [ θ/(1 + ∊θ )]. All of the dynamical behaviour of these equations can be determined qualitatively and, to leading order, quantitatively from the pool chemical approximation, γ = 0. Oscillatory solutions emerge as μ and k 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 < ∊ < 2/9. For ε in the range 2/9 < ∊ < 1/4 only stable limit cycles exist. Oscillatory solutions cannot be observed if ∊ ≽ 1/4.

The behaviour of the prototype cubic autocatalator A + 2B → 3B rate = k 1 ab 2 , B → C rate = k 2 b , when reaction is coupled with diffusion from a surrounding reservoir of constant composition is investigated. For indefinitely stable catalysts (i. e. for k 2 = 0) the model exhibits ignition, extinction (wash-out) and hysteresis. The range of conditions over which multiple stationary states are found decreases as the concentration of the autocatalyst b ex in the reservoir increases. Finally ignition and extinction points merge in a cusp catastrophe with the consequent loss of multiplicity. Away from the critical points, the ultimate approach to the stationary state is, in general, governed by an exponential decay. The characteristic relaxation time for this approach lengthens as ignition or extinction points are approached, thus displaying ‘slowing down’. Eventually, non-exponential time-dependences are also found. With finite catalyst lifetimes ( k 2 > 0), the dependence of the stationary-state composition on the diffusion rate or the size of the reaction zone shows more complex patterns. Five qualitatively different responses can be found: (i) unique, (ii) isola, (iii) breaking wave, (iv) mushroom and (v) breaking wave + isola. The stationary-state profile for the distribution of the autocatalytic species B now allows multiple internal extrema (the onset of dissipative structures). The cubic autocatalator also provides the simplest, yet chemically consistent, example of temporal and spatial oscillations in a reaction-diffusion system.

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The influence of a direct, uncatalysed reaction A → B rate = k u a on the stationary-state behaviour of the cubic ‘autocatalator ’ A + 2 B → 3 B rate = k 1 a b 2 , B → C r a t e = k 2 b in an isothermal, well-stirred, open reactor is studied by singularity methods. It is shown that the dependence on the mean residence time of the extent of conversion of A in the stationary state has five different patterns possible. Three of these are similar to those found in the absence of the uncatalysed step: unique, isola and mushroom. The two extra patterns, a breaking wave (a single, S-shaped hysteresis loop) and breaking wave plus isola, require non-zero k u . The uncatalysed reaction also effects the behaviour at long residence time where the system tends to complete conversion. This kinetic scheme represents the simplest, isothermal chemical model with a full unfolding of a winged-cusp singularity and requires only the minimum number of parameters. Sustained oscillatory behaviour, arising from Hopf bifurcations, is also exhibited and is again influenced by the uncatalysed step. The main results can be summarized as follows. k u = 0 multiple stationary states ( isola , mushroom ) and oscillations ; 0 < k u < 1 200 k 1 a 0 2 multiple stationary states ( isola , mushroom , breaking wave , breaking wave and isola ) and oscillations ; 1 200 k 1 a 0 2 < k u < 1 195 k 1 a 0 2 multiple stationary states ( isola , mushroom , breaking wave , breaking wave and isola ) , no oscillations ; 1 195 k 1 a 0 2 < k u < 1 27 k 1 a 0 2 multiple stationary states ( breaking wave only ) , no oscillations ; 1 27 k 1 a 0 2 < k u no multiple stationary states , no oscillations , .