Asymptotic behaviour of solutions near a turning point: The example of the Brusselator equation

The Brusselator equation is an example of a singularly perturbed differential equation with an additional parameter. It has two turning points: at x=0 and x=-1. We study some properties of so-called canard solutions, that remain bounded in a full neighbourhood of 0 and in the largest possible domain; the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions. For this purpose we need a study of behaviour of the solutions near a turning point; here we prove that, for a large class of equations, if 0 is a turning point of order p, any solution y not exponentially large has, in some sector centred at 0, an asymptotic behaviour (when ε→0) of the form ∑Yn(x/ε′)ε′n, where ε′p+1=ε, for x=ε′X with X large enough, but independent of ε. In the Brusselator case, we moreover compute a Stokes constant for a particular nonlinear differential equation.