Exceptional Complex Solutions of the Forced Van Der Pol Equation

Consider the equation "v dv du = (1 ? u 2)v + ? u describing the orbits of the forced van der Pol equation " u + (u 2 ? 1) _ u + u = ; " a small parameter, (1) in the phase plane. The authors show the existence of two exceptional solutions corresponding to two conjugate complex exceptional values of the parameter near = 1 which remain close to the slow curve v = ?1=(u + 1) both in the directions +1 and i1, respectively both in +1 and ?i1. They are unique with this property. The analytic continuation of these solutions in the "-plane around " = 0 is studied. Explicit Gevrey estimates for the "-shadow expansion of the value of the parameter are deduced. It follows that summation \to the least term" of the formal series satisfying (1) yields a parameter value corresponding to a long canard and that this series is 1{summable except in the positive real direction. The article is written in the framework of nonstandard analysis. For the convenience of the reader, a short appendix explaining some notation in classical terms is provided.