Uniqueness of relaxation oscillations: A classical approach

In a recent paper (Kosiuk and Szmolyan (2011)), the authors discuss a relaxation oscillator which apparently has not been put in standard Lienard form. They use geometric perturbation theory to analyze this model. Their main result is the existence and uniqueness of a periodic solution for small values of two parameters, $\delta$ and $\varepsilon ,$ and the behavior of this solution as $\delta$ and $\varepsilon$ tend to zero. We show how standard ode methods can be used to give shorter and more direct proofs of these results. Along the way we give a new proof of a more general result.