Global bifurcations and phase portrait of an analytically solvable nonlinear oscillator: Relaxation oscillations and saddle-node collisions

A second-order nonlinear differential equation whose general solution can be expressed in terms of elementary functions is studied. Analytical expressions describing both the phase portrait and global bifurcations for all values of the parameters are given. With the advantage of being exactly solvable, this equation models essentially the same physical phenomena as the well-known van der Pol's equation. In particular, the relaxation limit of the oscillatory regime where systems periodi- cally undergo fast switching between two different kinds of slow motions can be explained on the basis of our model as a consequence of the presence of a saddle-node bifurcation.