False bifurcations in chemical systems: canards

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Abstract
A canard is a false bifurcation in which the amplitude of an oscillatory system may change by orders of magnitude while the qualitative dynamical features remain unchanged. Recent theoretical considerations suggest that canards are characteristic of fast-slow dynamical systems and are associated with the stable and unstable manifolds of the phase plane. An alternative characterization of canard behaviour is proposed involving the crossing of an inflection line by a limit cycle growing out from an unstable stationary state. The inflection line comprises the locus of points at which the curvature of any phase plane trajectory is zero. The role of the inflection line in the onset of canard behaviour as well as in the continuity of the transition is examined in a two-variable model for the oscillatory EOE reaction, the Auto-catalator, and the two-variable Oregonator. The approach is also applied to the van der Pol oscillator, the system in which canard behaviour was first examined.

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