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Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations. (English) Zbl 1322.34067

The authors study the birth of canard trajectories at the interface between small-amplitude limit cycles and large-scale canard dynamics – or, in their words, of “large” small-amplitude cycles – in a family of planar fast-slow systems of ordinary differential equations. While canard dynamics in the plane has traditionally been considered in the context of a codimension-1 Hopf bifurcation [M. Krupa and P. Szmolyan, J. Differ. Equations 174, No. 2, 312–368 (2001; Zbl 0994.34032)], they assume a singularity of codimension 3 that is unfolded by three parameters in addition to the singular perturbation parameter. Correspondingly, two blow-up transformations [the third author, NATO Adv. Sci. inst. Ser. C Math. Phys. Sci. 408, 19–73 (1993; Zbl 0791.58069)] are performed, a “primary” one to unfold the singularity and a “secondary” one to resolve the resulting fast-slow structure; the study thereof is greatly facilitated by the introduction of appropriate normal forms [P. Bonckaert et al., J. Dyn. Differ. Equations 23, No. 1, 115–139 (2011; Zbl 1215.34042)]. Finally, the authors determine the cyclicity of the limit periodic set in the vicinity of which the birth of canard cycles occurs. The requisite analysis, which is based on the authors’ previous publication [J. Differ. Equations 255, No. 11, 4012–4051 (2013; Zbl 1316.34060)], is performed with great care, and is presented in detail, making the presentation largely self-contained. Significantly, their results hence contribute towards a complete characterisation of codimension-3 singularities in planar fast-slow systems; moreover, they may generalise to other settings. Given its technical character, it seems unsurprising that the article should preferably be read in its entirety for its facets to be fully appreciated.

MSC:

34E17 Canard solutions to ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C45 Invariant manifolds for ordinary differential equations
34C26 Relaxation oscillations for ordinary differential equations
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References:

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