Periodic solutions of a singular differential delay equation with the Farey-type nonlinearity

We address the problem of existence of periodic solutions for the differential delay equationεx˙(t)+x(t)=f(x(t-1)),0<ε⪡1,with the Farey nonlinearity f(x) of the formf(x)=mx+Aifx⩽0,mx-Bifx>0,where |m|<1,A>0,B>0. We show that when the map x↦f(x) has an attracting 2-cycle then the delay differential equation has a periodic solution, which is close to the square wave corresponding to the limit (as ε→0+) difference equation x(t)=f(x(t-1)).