Large relaxation oscillation in slow–fast excitable Brusselator oscillator

In general, critical manifold loses normal hyperbolicity at folded, transcritical and pitchfork singularities. There is another situation where normal hyperbolicity of critical manifold fails, namely, the alignment of the tangent and normal bundles at the unbounded part of critical manifold. In this case, how to reveal the attracting or repelling natures of unbounded critical manifold is essential to detect the birth of relaxation oscillations. In this article, after the compactification of the unbounded critical curve and then blowing-up the resulting degenerate line, we find that return mechanism exists at the $O(1/\varepsilon )$-region of the critical curve in a slow–fast excitable Brusselator oscillator. By so doing the birth of relaxation oscillation near the unbounded critical curve in this model is demonstrated. In addition, we reveal the continuation process from Hopf small-amplitude cycle to large relaxation oscillation of size $O(1/\varepsilon )$ in the blown-up space. This may be the counterpart of canard explosion in unbounded situation. All the theoretical predictions are verified by numerical simulations.