Bifurcations of nonlinear oscillations and frequency entrainment near resonance

A unified approach to the Poincare–Andronov global center bifurcation and the subharmonic Melnikov bifurcation theory is developed using S. P. Diliberto’s integration of the variational equations of a two-dimensional system of autonomous ordinary differential equations and a Lyapunov–Schmidt reduction to the implicit function theorem. In addition, the subharmonic Melnikov function is generalized to the case of subharmonic bifurcation from an unperturbed system whose free oscillation is a limit cycle. Thus, results on frequency entrainment are obtained when an external periodic excitation is in resonance with the frequency of the limit cycle. The theory is applied to the subharmonic bifurcations of two coupled van der Pol oscillators running in resonance.