Delay, resonance and the Lambert W function

TL;DR

This paper explores a new delay differential equation exhibiting resonant oscillations, revealing a novel link between its solutions and the Lambert W function, advancing understanding of delayed feedback dynamics.

Contribution

It introduces a new delay differential equation with resonance phenomena and uncovers a novel connection between its solutions and the Lambert W function.

Findings

Peak power spectrum occurs at specific delay tuning.

Resonant conditions relate to solutions of a transcendental equation.

New insights into nonlinear delayed feedback dynamics.

Abstract

We discuss a new type of delay differential equation that exhibits resonating transient oscillations. The power spectrum peak of the dynamical trajectory reaches its maximum height when the delay is suitably tuned. Furthermore, our analysis of the resonant conditions for this equation has revealed a new connection between the solutions of the transcendental trigonometric equation and the Lambert W function. These results offer fresh insights into the nonlinear dynamics induced by delayed feedback.