Delay, resonance and the Lambert W function

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.

submitted for a publication; ver2. added proof to make a conjecture into a theorem, updating references and minor corrections; ver3. added Appendix and corrections of typos