The duck and the devil: canards on the staircase

Slow-fast systems on the two-torus $T^2$ provide new effects not observed for systems on the plane. Namely, there exist families without auxiliary parameters that have attracting canard cycles for arbitrary small values of the time scaling parameter $\epsilon$. In order to demonstrate the new effect, we have chosen a particularly simple family, namely $\dot x=a-\cos x-\cos y$, $\dot y=\epsilon$, $a\in(1,2)$ being fixed. There is no doubt that a similar effect may be observed in generic slow-fast systems on $T^2$. The proposed paper is the first step in the proof of this conjecture.