Relevance of a two-variable Oregonator to stable and unstable steady states and limit cycles, to thresholds of excitability, and to Hopf vs SNIPER bifurcations

Abstract

A stiffly-coupled Oregonator model based on the two independent variables Y and Z has been examined in detail with the use of the stoichiometric factor as a single disposable parameter. The trajectories and periods of the stable limit cycles can be generated with unanticipated accuracy from simple linear differential equations. Transitions between stable limit cycles and stable steady states takes place by means of subcritical Hopf bifurcations. Unstable limit cycles and thresholds of excitability can be recovered by integrating the equations of motion backward in time; such procedures cannot be applied so easily for models based on more than two independent variables. We have examined claims of experimental evidence for saddle-node infinite period (SNIPER) bifurcations and have concluded that all currently available evidence is equivocal. Until unambiguous criteria can be established for identifying SNIPER bifurcations in real systems, and until chemical mechanisms have been proposed which generate such bifurcations, observations should be interpreted by existing models based on Hopf bifurcations. Experimental behaviors are listed which can and cannot be simulated in terms of two independent composition variables.