Periodic solutions of a singularly perturbed delay differential equation

A singularly perturbed differential delay equation of the form (1)ε{lunate} over(x, ̇) (t) = - x (t) + f (x (t - 1), λ) exhibits slowly oscillating periodic solutions (SOPS) near the first period-doubling bifurcation point of the underlying map (obtained by setting ε{lunate} = 0). For extremely small values of ε{lunate}, these periodic solutions resemble square waves, which consist of sharp, O (ε{lunate}) transition layers connecting intervals of approximately unit length. In this article, we obtain analytic expressions for these square-wave periodic solutions, by solving the corresponding transition layer equations, and show that they are in excellent agreement with numerical solutions for a range of values of ε{lunate} and λ. We also derive analytic expressions for other periodic solutions which are odd harmonics of the SOPS, and numerically exhibit their instability near the first period doubling bifurcation point of the map. The numerical computations were performed using a high accuracy Chebyshev spectral scheme. We give a brief description together with a study of its accuracy and efficiency.