The Forced van der Pol Equation I: The Slow Flow and Its Bifurcations

The forced van der Pol oscillator has been the focus of scientific scrutiny for almost a century, yet its global bifurcation structure is still poorly understood. In this paper, we present a hybrid system consisting of the dynamics of the trajectories on the slow manifold coupled with "jumps" at the folds in the critical manifold to approximate the fast subsystem. The global bifurcations of the fixed points and periodic points of this hybrid system lead to an understanding of the bifurcations in the periodic orbits (without canards) of the forced van der Pol system.