Synchronisation vs. resonance: Isolated resonances in damped nonlinear oscillators

Abstract

We describe differences between synchronisation and resonance, and analyse different types of nonlinear resonances in a weakly damped Duffing oscillator using bifurcation theory techniques. In addition to previously reported (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. Our analysis shows that identified here isolated resonances explain the intriguing “intermingled tongues” that were observed for weak damping and misinterpreted as (synchronisation) Arnold tongues in Paar and Pavin (1998). What is more, isolated resonances link “intermingled tongues” to a seemingly unrelated phenomenon of “bifurcation superstructure” described for moderate damping in Parlitz and Lauterborn (1985).