On harmonic resonance in forced nonlinear oscillators exhibiting a Hopf bifurcation

The influence of a periodic forcing on a nonlinear second-orderoscillator close to a Hopf bifurcation point is investigated.The forcing frequency is close to the frequency of the Hopfbifurcation, and the forcing amplitude is assumed to be small.Second-order integral averaging is applied to reduce the givensystem to a planar autonomous system. By a bifurcation and stabilityanalysis of this system, the behaviour of the forced oscillatoris determined. It turns out that two qualitatively differenttypes of behaviour can occur. Either the system has a uniqueattractor, or the system has two competing attractors givingrise to a hysteresis phenomenon, which is known from the Duffingequation. Bifurcation diagrams are presented, and explicit formulaefor the quantities determining the behaviour are given