Blue sky catastrophe in relaxation systems with one fast and two slow variables

We establish conditions under which three-dimensional relaxational systems of the form

$$dot x = f(x,y,mu ),varepsilon dot y = g(x,y),x = (x_1 ,x_2 ) in mathbb{R}^2 ,y in mathbb{R},$$

where 0 ≤ ε ? 1, |µ| ? 1, and f, g ∈ C ^∞, exhibit the so-called blue sky catastrophe [the appearance of a stable relaxational cycle whose period and length tend to infinity as µ tends to some critical value µ_*(ε), µ_*(0) = 0].