Four-Dimensional Canards and Their Center Manifold

Abstract

We consider four-dimensional slow–fast systems, which can be represented either by one-dimensional slow vector field or three-dimensional fast vector field and denoted as $\mathbb {R}^{1+3}$, or $\mathbb {R}^{2+2}$, or $\mathbb {R}^{3+1}$. In each of these cases, the corresponding system can be well analyzed using blowing up the system and a time-scale reduction technique. Moreover, for each of these cases, by constructing a local model, the existence of a singular-limit solution (that are usually called Canards) is established. Some sufficient conditions for the existence of the canards are provided in this notice. What kind of four-dimensional canards are there?

The author thanks Professors V. Sobolev and A. Korobeinikov for the invitation to the conference and Professor J. Llibre for his generous support of the author’s visit. This work is partly supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911.