Canards, folded nodes, and mixed-mode oscillations in piecewise-linear slow-fast systems (Q2832108)

Canard trajectories constitute one of the most interesting dynamical features that may be observed in multiple-scale systems of differential equations; such trajectories originate close to attracting slow manifolds [\textit{N. Fenichel}, J. Differ. Equations 31, 53--98 (1979; Zbl 0476.34034)] in the system and subsequently remain in the vicinity of some repelling slow manifold for significant amounts of time [\textit{E. Benoît} et al., Collect. Math. 32, 37--119 (1981; Zbl 0529.34046)]. Canard dynamics has been studied extensively in the past three decades, both from a theoretical and an applied point of view, with a definite focus on smooth slow-fast systems of ordinary differential equations and their applications in mathematical neuroscience [\textit{J. E. Rubin} and \textit{D. Terman}, in: Handbook of dynamical systems. Volume 2. Amsterdam: Elsevier. 93--146 (2002; Zbl 1015.34048)]. In parallel, it has been shown that the canard phenomenon can be replicated in the context of planar piecewise linear systems [the first author et al., Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 469, No. 2154, Article ID 20120603, 18 p. (2013; Zbl 1355.34093)]; however, a generalisation to the three-dimensional case has been attempted only sporadically [the fourth and the last author, Discrete Contin. Dyn. Syst. 33, No. 10, 4595--4611 (2013; Zbl 1280.34060)]. That case seems particularly relevant, as the canard phenomenon in smooth slow-fast systems with two slow variables and one fast variable provides a fundamental generating mechanism for a complex type of dynamics known as mixed-mode oscillation (MMO), in which oscillatory time courses of small amplitudes alternate with large-amplitude ones [the first author et al., SIAM Rev. 54, No. 2, 211--288 (2012; Zbl 1250.34001)].NEWLINENEWLINEIn the present article, the authors strive to bridge the gap between the smooth and piecewise-linear settings by proposing a canonical family of three-dimensional piecewise linear systems that allows for the generalisation of several concepts from the theory of smooth canards; examples include folded singularities, strong and weak canard trajectories, and the maximal canard. Here, the authors focus on the scenario where a folded node [\textit{M. Wechselberger}, SIAM J. Appl. Dyn. Syst. 4, No. 1, 101--139 (2005; Zbl 1090.34047)] is present, showing that the maximum number of canard trajectories in the system is controlled by a winding number, as is the case in the smooth setting; however, and somewhat surprisingly, they find that small-amplitude oscillation may also occur in the presence of a folded saddle. They emphasise the similarities between the corresponding singular dynamics in the two settings; moreover, they substantiate a classical observation made in [\textit{N. Arima}, \textit{H. Okazaki} and \textit{H. Nakano}, ``A generation mechanism of canards in a piecewise linear system'', IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E80-A, No. 3, 447--453 (1997)], namely, that true canard trajectories can only be obtained if a three-piece linear approximation is introduced near the relevant fold point of the critical manifold. In particular, they establish an analogue of the ``blow-up'' transformation [\textit{F. Dumortier} and \textit{R. Roussarie}, Mem. Am. Math. Soc. 577, 100 p. (1996; Zbl 0851.34057)] that unfolds the resulting canard dynamics in the smooth setting by keeping the fold region ``open'' in the singular limit. Finally, the authors construct an example that incorporates a (linear) global return, which allows them to observe robust MMOs in the context of their family of canonical systems.