Periodically Forced Hopf Bifurcation

We study a periodically forced system of ODEs near a point of Hopf bifurcation, where the forcing frequency ωF is near the Hopf frequency ωH and the forcing amplitude ε is small. 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 ωF . In other words, we examine the influence of the forcing frequency ωF on the number of periodic solutions to the forced system with frequency ωF . This problem is complicated because of the existence of three small parameters: the amplitude of the forcing ε, the deviation of the bifurcation parameter from the point of Hopf bifurcation λ, and the relative deviation of the forcing frequency from the Hopf frequency ω = 1− ωH ωF . The problem of periodically forced Hopf bifurcation has been studied by many authors [1, 2, 4, 6, 10, 12, 13, 14]. In Bogoliubov and Mitropolsky [2], there are numerous examples of periodically forced systems whose approximate solutions are obtained via multiple methods such as perturbation and spectral methods. However,