Three types of nonlinear resonances

TL;DR

This paper classifies and analyzes various nonlinear resonances in a weakly damped Duffing oscillator, revealing new resonance types and explaining complex resonance structures through bifurcation theory.

Contribution

It uncovers a novel resonance type involving isolas of periodic solutions and links these to observed complex resonance patterns in weak damping regimes.

Findings

Identification of three resonance types, including a new isolated resonance.

Resonance tongues mapped in parameter space showing complex structures.

Connection between isolated resonances and observed patchy resonance tongues.

Abstract

We analyse different types of nonlinear resonances in a weakly damped Duffing oscillator using bifurcation theory techniques. In addition to (i) odd subharmonic resonances found on the primary branch of symmetric periodic solutions with the forcing frequency and (ii) even subharmonic resonances due to symmetry-broken periodic solutions that bifurcate off the primary branch and also oscillate at the forcing frequency, we uncover (iii) novel resonance type due to isolas of periodic solutions that are not connected to the primary branch. These occur between odd and even resonances, oscillate at a fraction of the forcing frequency, and give rise to a complicated resonance `curve' with disconnected elements and high degree of multistability. We use bifurcation continuation to compute resonance tongues in the plane of the forcing frequency vs. the forcing amplitude for different but fixed values of the damping rate. In this way, we demonstrate that identified here isolated resonances explain the intriguing structure of "patchy tongues" observed for week damping and link it to a seemingly unrelated phenomenon of "bifurcation superstructure" described for moderate damping.