Extending slow manifold near generic transcritical canard point
Canards [\textit{E. Benoit} et al., Collect. Math. 32, 37--119 (1981; Zbl 0529.34046)] are defined as trajectories of singularly perturbed systems of differential equations that follow a repelling portion of the corresponding slow manifold [\textit{N. Fenichel}, J. Differ. Equations 31, 53--98 (1979; Zbl 0476.34034)] for a considerable amount of time. Canard dynamics may emerge when a steady state in the system coincides with a bifurcation point of the critical manifold thereof, as is the case when the associated layer problem undergoes a saddle-node bifurcation at a generic fold point [\textit{M. Krupa} and \textit{P. Szmolyan}, SIAM J. Math. Anal. 33, No. 2, 286--314 (2001; Zbl 1002.34046)]. An alternative mechanism is related to a loss of normal hyperbolicity at points of self-intersection of the critical manifold. Scenarios where these points are of transcritical or pitchfork type were studied by Krupa and Szmolyan [\textit{M. Krupa} and \textit{P. Szmolyan}, Nonlinearity 14, No. 6, 1473--1491 (2001; Zbl 0998.34039)] via the so-called blow-up technique, or geometric desingularisation [\textit{F. Dumortier} and \textit{R. Roussarie}, Mem. Am. Math. Soc. 577, 100 p. (1996; Zbl 0851.34057)]; however, no detailed analysis of the potential for canard dynamics was attempted there. Here, the authors investigate the emergence of canards in a family of planar fast-slow systems in a neighbourhood of a transcritical point. They show that, under certain non-degeneracy assumptions on the associated local normal form for that family, a maximal canard will exist for a unique critical value of a parameter which is introduced by the requisite normal form transformation, and they derive an asymptotic expansion for that critical value. Moreover, they identify a second value for the same parameter which separates the regime where the incoming attracting branch of the corresponding slow manifold undergoes passage along the fast flow of the system from the regime where a connection exists between the incoming and outgoing attracting branches of that manifold. Their analysis is similar in spirit to that in [Zbl 0998.34039], and relies on a detailed blow-up of the transcritical point, in combination with a variety of dynamical systems techniques which include centre manifold theory and a Melnikov computation. Finally, a simple illustrative example is presented.
scientific article; zbMATH DE number 6827834