Duck cycles of difference-differential equations (Q1358599)

Consider the differential-delay equation \[ dN/dt=\lambda \left[ 1+\frac{a}{\varepsilon} N(t-1) - \frac{1}{\varepsilon} N(t-h)\right]N\tag \(*\) \] for \(h<1/3\) and sufficiently small \(\varepsilon\). The authors introduce an operator \(\pi\) whose fixed points define a relaxation oscillation of \((*)\) with period near \((1+h)/2\). They state a theorem on properties of \(\pi\) which imply the existence of at least one periodic solution to \((*)\) and note that the proof follows from \textit{A. Ju. Kolesov} and \textit{Ju. S. Kolesov} [Russ. Acad. Sci., Izv., Math. 41, No. 1, 75-94 (1993); translation from Izv. Russ. Akad. Nauk, Ser. Mat. 56, No. 4, 790-812 (1992; Zbl 0792.34067)]. The authors state a theorem which implies that all these periodic solutions are unstable.