Y. Zhang and M. Golubitsky
Periodically forced Hopf bifurcation
SIAM J. Appl. Dynam. Sys.
10 (2011) 1272-1306.
We study a periodically forced system of ODEs near a Hopf bifurcation
point where the forcing has small amplitude epsilon and frequency
omega_F near the Hopf frequency omega_H. We assume that
in this system only the forcing frequency is varied and we determine
all small amplitude periodic solutions of the forced system that have
frequency omega_F. In other words, we aim to examine the influence
of the forcing frequency omega_F on the number of
periodic solutions to the forced system with frequency omega_F. This
problem is complicated because of the existence of three small
parameters: the amplitude of the forcing epsilon, the deviation
of the bifurcation parameter from the point of Hopf bifurcation lambda,
and the relative deviation of the forcing frequency from the Hopf frequency
omega = 1- omega_H/omega_F. Our results are presented in terms of
bifurcation diagrams of amplitude of periodic solution versus omega
for fixed epsilon and lambda.