A note on the forced Van der Pol equation

Abstract

Abstract It is well known that the forced Van der Pol equation e u +(u 2 −1) u +u=α (where u = d u d t and e→0 ) is a singularly perturbed differential equation having exceptional solutions (called canards) for some values of α(e) . It was already proved that there are exactly two distinct complex canards solutions (v+,α+) , (v−,α−) of the transformed equation ev d v d u =(1−u 2 )v+α−u , which are bounded in wide infinite sectors centered at u=0 and containing u=1 . In this Note, explicit asymptotic approximations for α+−α− as e→0 and, as n→∞ , of the an (of the asymptotic series α = ∑ a n e n corresponding to α ) will be proved. The main tool of the proof is the analytic continuation of the solutions v± in a subset of C containing [−1+X l e 1/3 ,+∞[ ⊂ R .