Analysis of chemical kinetic systems over the entire parameter space: II. Isothermal oscillators

Two simple but similar isothermal kinetic schemes are studied by using recently developed qualitative techniques with the aim of finding all of the qualitatively distinct behaviour that the system can exhibit. The first scheme, which can be regarded as one of the simplest isothermal models of interest, is based on the `quadratic' autocatalytic step $A + B \rightarrow 2B \text{rate} = k_q ab,$ where a and b are the concentrations of A and B respectively, combined with a nonlinear termination step of the form $B \rightarrow C \text{rate} = k_t b/(1+rb),$ where r is some parameter. The second scheme is a simplified (by treating one parameter as a constant) version of a model that has been proposed for the oxidation of carbon monoxide (CO). This scheme has a termination step of the same form as above, and like the `quadratic' model, the highest nonlinearity apart from this term is quadratic. For the quadratic model we find 10 qualitatively distinct phase portraits and, with our choice of bifurcation parameter, 22 distinct bifurcation diagrams. The most complex phase portraits contain 3 steady states and 2 limit cycles or 2 steady states and 3 limit cycles. For the CO model we find 8 distinct phase portraits but leading to only 5 separate bifurcation diagrams. In both models much of the behaviour occurs over only extremely small regions in parameter space.