Analysis of nonlinear systems near Hopf bifurcation with periodic disturbances

In this paper, we study the effects of periodic perturbations on a smooth nonlinear system possessing a subcritical Hopf bifurcation. The goal is to obtain the analytic relations between the region of attraction of the nominal equilibria near the bifurcation and the amplitude and frequency of the perturbation. First, via a smooth coordinate transformation, we transform the nonlinear system into a normal form, in which the dynamics of the center manifold and those of the stable manifold are decoupled in the lower order terms. Then, we study the stability of the dynamics on the normal form by using two methods. First, we obtain the periodic solution by using the harmonic balance, and we analyze the linear stability of the periodic solutions. Then we construct Lyapunov functions to evaluate the domain of attraction of the periodic orbits. The Lyapunov method gives more conservative estimate of the critical bifurcation parameters than linear stability analysis.