Singularity induced bifurcation and the van der Pol oscillator

In parameter dependent differential-algebraic models (DAEs) of the form x˙=f and 0=g, it has been shown recently that the generic codimension one local bifurcations are the well-known saddle node and Hopf bifurcations and a new bifurcation called the singularity induced bifurcation. The latter occurs generically when an equilibrium of the DAE system crosses the singular surface of noncausal points. In this paper, it is shown that when singularly perturbed models of the form x˙=f and ∈y˙=g are considered, the singularity induced bifurcation in the slow DAE system corresponds to oscillatory behavior in the singularly perturbed models. As an example, it is proved that the oscillations in the classical van der Pol oscillator arise when a stable equilibrium undergoes the singularity induced bifurcation in the slow DAE system, which in turn corresponds to the occurrence of supercritical Hopf bifurcations in the singularly perturbed models