Analysis of a remarkable singularity in a nonlinear DDE

TL;DR

This paper explores the complex dynamics of a nonlinear delay-differential equation, revealing the emergence of infinite limit cycles as delay increases, and employs multiple analytical methods to understand this singular behavior.

Contribution

It introduces a detailed analysis of a nonlinear DDE's singularity at zero delay using harmonic balance, Melnikov's integral, and damping regularization techniques.

Findings

Infinite limit cycles appear for any positive delay

Amplitude of cycles diverges as delay approaches zero

Regularization methods help understand the singularity

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.