Square wave periodic solutions of a differential delay equation

Abstract

We prove the existence of periodic solutions of the differential delay equation εx˙(t)+x(t)=f(x(t−1)),ε>0 under the assumptions that the continuous nonlinearity f(x) satisfies the negative feedback condition, x⋅f(x)<0,x≠0, has sufficiently large derivative at zero |f′(0)|, and possesses an invariant interval I∋0,f(I)⊆I, as a dimensional map. As ε→0+ we show the convergence of the periodic solutions to a discontinuous square wave function generated by the globally attracting 2-cycle of the map f.