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

Abstract

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].