A Simple Model for Sustained Oscillations in Isothermal Branched-Chain or Autocatalytic Reactions in a Well Stirred Open System II. Limit Cycles and Non-Stationary States

The oscillatory patterns of behaviour exhibited by the simple kinetic scheme A + B -> 2B; rate = k_1 ab, B -> C; rate = k_2 b/(1 + rb) are examined in detail. For systems with slowly decaying catalysts, such that k_2 ~= k_1 a^2_0, a reduced asymptotic form of the governing equations allows a full analytical treatment. Oscillations begin as the residence time is increased through a point of Hopf bifurcation τ^*_res. The bifurcation is always supercritical, with the amplitudes of the concentration variations increasing from infinitesimally small values. The amplitudes grow initially as (τ_res-τ^*_res)^1/2, and tend to finite limiting magnitudes at very long residence times. At this limit, the oscillations in A have a `saw-tooth' waveform, with B varying in a pulse-like manner. Numerical solutions of the full kinetic equations for a non-zero inflow of the catalyst B reveal how the system approaches a second Hopf bifurcation as the residence time is increased. The oscillatory amplitude now reaches a maximum and then decays back to zero. The applicability of this model to real chemical systems is discussed with particular reference to the gas phase oxidation of carbon monoxide.