Bogdanov-Takens resonance in time-delayed systems

TL;DR

This paper investigates how time delays in dynamical systems can lead to a resonance phenomenon, where small external forces cause large responses near a Bogdanov-Takens bifurcation, revealing unique oscillatory behaviors.

Contribution

It introduces the concept of Bogdanov-Takens resonance in time-delayed systems and analyzes how delays influence resonance phenomena under periodic forcing.

Findings

Resonance occurs at specific delay values near bifurcation points.

Small forcing amplitudes can induce large oscillations.

The phenomenon differs from classical resonance in key aspects.

Abstract

We analyze the oscillatory dynamics of a time-delayed dynamical system subjected to a periodic external forcing. We show that, for certain values of the delay, the response can be greatly enhanced by a very small forcing amplitude. This phenomenon is related to the presence of a Bogdanov- Takens bifurcation and displays some analogies to other resonance phenomena, but also substantial differences.