Bifurcations in Forced Response Curves

Abstract

We consider small amplitude periodic forcing (with forcing frequency $\omega_F$) of a system of differential equations near a point of Hopf bifurcation (with Hopf frequency $\omega_H$). We follow Zhang and Golubitsky [SIAM J. Appl. Dyn. Syst., 10 (2011), pp. 1272--1306] and consider only those small amplitude periodic solutions to the forced system whose frequency is $\omega_F$; that is, the 1:1 phase-locked or entrained solutions. These authors assume sinusoidal forcing of a normal form Hopf system when $\omega_F$ is close to $\omega_H$, and they classify the existence and multiplicity of the entrained solutions. The forced response curve is a bifurcation diagram showing the amplitude of the entrained solutions as the forcing frequency is varied. Zhang and Golubitsky showed that there are six kinds of forced response curves with distinguished bifurcation parameter $\omega = \omega_H - \omega_F$. In this paper we show that there are 41 possible bifurcation diagrams when stability in addition to multiplicity of the periodic solutions in the forced response curves is included.