Analysis of a remarkable singularity in a nonlinear DDE

We investigate the dynamics of the nonlinear DDE (delay-differential equation) d^2x/dt^2(t)+x(t-T)+x(t)^3=0 , where T is the delay. For T=0 , this system is conservative and exhibits no limit cycles. For T>0 , no matter how small T is, an infinite number of limit cycles exist, their amplitudes going to infinity in the limit as T approaches zero. We investigate this situation in three ways: (1) harmonic balance, (2) Melnikov’s integral, and (3) adding damping to regularize the singularity.