Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

International audienceWe generalize the results on the existence of an over-stable solution of singularly perturbed differential equations to the equations of the form εẍ − F (x, t,ẋ, k(t), ε) = 0. In this equation, the time dependence prevents from returning to the well known case of an equation of the form εdy/dx = F (x, y, a, ε) where a is a parameter. This can have important physiological applications. Indeed, the coupling between the cardiac and the respiratory activity can be modeled with two coupled van der Pol equations. But this coupling vanishes during the sleep or the anesthesia. Thus, in a perspective of an application to optimal awake, we are led to consider a non autonomous differential equation