Analysis of a remarkable singularity in a nonlinear DDE

Abstract: In this work we investigate the dynamics of the nonlinear DDE (delay-differential equation) x''(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, 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.