Square wave periodic solutions of a differential delay equation

Abstract

We prove the existence of periodic solutions of the differential delay equation $\varepsilon\dot{x}(t) + x(t)= f(x(t-1)), \varepsilon>0$ under the assumptions that the continuous nonlinearity $f(x)$ satisfies the negative feedback condition, $x \cdot f(x) < 0, x \neq 0$, has sufficiently large derivative at zero $|f'(0)|$, and possesses an invariant interval $I \ni 0, f(I) \subseteq I$, as a dimensional map. As $\varepsilon \to 0+$ we show the convergence of the periodic solutions to a discontinuous square wave function generated by the globally attracting 2-cycle of the map $f$.