A symptotic analysis of canards in the EOE equations and the role of the inflection line

Abstract

The Edblom–Orbán–Epstein (EOE) reaction, involving iodate, sulphite and ferrocyanide ions may have oscillations in a continuous stirred tank reactor. A simplification of a 10-variable model, including two state variables, was analysed numerically by Peng et al. ( Phil. Trans. R. Soc. Lond . A337, 275 (1991)). They found that within a very small range of the inflow concentration, the amplitude of of the oscillations varied drastically. Such behaviour is well understood in singular perturbations systems, where it is known as a canard explosion . Apparently, the EOE equations do not belong to this class. We show that the two-dimensional EOE equations can be recast as a singular perturbation problem. An asymptotic expansion of the the canard point is obtained, with very good agreement with the numerical results of Peng et al . To deal with the canard explosion in systems which are not of singular perturbation type, Peng et al . introduce a new definition of the canard point, based on change of curvature of the limit cycle. We discuss the new definition, and show that it may essentially agree with the definition based on a singular perturbation approach, but is much less fit for analytical computations. We show that the discontinuous canard transition suggested by Peng et al . violates a continuity theorem in the theory of ordinary differential equations.