# Analysis of a remarkable singularity in a nonlinear DDE

Nonlinear Dynamics, Jul 2017

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. We investigate this situation in three ways: (1) harmonic balance, (2) Melnikov’s integral, and (3) adding damping to regularize the singularity.

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Matthew Davidow, B. Shayak, Richard H. Rand. Analysis of a remarkable singularity in a nonlinear DDE, Nonlinear Dynamics, 2017, 1-7, DOI: 10.1007/s11071-017-3663-2