Bifurcations in Forced Response Curves

The forced response curve is a graph showing the amplitude of the response of a periodically forced system as the forcing frequency is varied. Zhang and Golubitsky (SIADS, 10(4) (2011) 1272–1306) classified the existence and multiplicity of periodic responses to small amplitude periodic forcing of a system near a Hopf bifurcation point, where the forcing frequency, ωF , is close to the Hopf frequency, ωH . They showed that there are six kinds of forced response curves when viewed as bifurcation diagrams with ω = ωH − ωF as the distinguished bifurcation parameter. In this paper we show that there are 41 possible bifurcation diagrams when stability, as well as multiplicity, of the periodic solutions in the forced response curves is included.