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Analysis of a remarkable singularity in a nonlinear DDE

We investigate the dynamics of the nonlinear DDE (delay-differential equation) \(\frac{\hbox {d}^2x}{\hbox {d}t^2}(t)+x(t-T)+x(t)^3=0\), 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...