Delayed Dynamics with Transient Resonating Oscillations

TL;DR

This paper introduces a new delay differential equation with a Gaussian factor that exhibits transient resonating oscillations, analytically solvable with Lambert W, and demonstrates how oscillation amplitude varies with delay.

Contribution

It presents a novel delay differential equation with exponential Gaussian feedback that is analytically solvable and exhibits resonant transient oscillations without external inputs.

Findings

Transient oscillations appear and disappear with increasing delay.

Amplitude of oscillations peaks at a certain delay value.

Equation is analytically tractable using Lambert W function.

Abstract

Recently, we have studied a delay differential equation which has a coefficient that is a linear function of time. The equation has shown the oscillatory transient dynamics appear and disappear as the delay is increased between zero to asymptotically large delay. We here propose and study another equation that shows similar transient oscillations. It has an extra exponential gaussian factor on the delayed feedback term. It is shown that this equation is analytically tractable with the use of the Lambert $W$ function. This equation is also studied numerically to confirm some of the properties inferred from the analytical solution. We also have found that the amplitude of transient oscillation changes and goes through a maximum as we increase the value of the delay. In this sense, the proposed equation is one of the simplest dynamical equations that brings out a resonant behavior without any external oscillating inputs.