The transient route to resonance

TL;DR

This paper investigates the transient behavior of the damped harmonic oscillator under periodic driving, revealing complex transient dynamics through visualization of winding numbers, which are often overlooked in steady-state analyses.

Contribution

It introduces a novel visualization method of winding numbers to characterize transient dynamics in the driven harmonic oscillator, highlighting complexities beyond steady-state solutions.

Findings

Transient dynamics are more complex than steady-state analysis suggests

Winding number visualizations reveal intricate transient behaviors

Transient effects may be significant in certain applications

Abstract

The periodically driven harmonic oscillator with damping is one of the most elementary and trusted models in physics and normally applied in its steady state, disregarding specific initial conditions and associated transients. For example, steady state solutions are already sufficient to describe resonance. In some applications transient solutions might play a role though, and a review of these might be useful. In order to characterize the transients of the harmonic oscillator, its winding number is graphed in dependence of initial conditions. These visualizations show a complexity of the transients of the harmonic oscillator that are not readily evident from the inspection of the model alone.