Fleuves oscillants

We present in this article some results about the asymptotic of scalar ordinary differential equations. We are interested in a family of equations, for which asymptotic concentrations of oscillatory solutions occur. With the following theorems, the presence of these trajectories can be deduct from the structure of the fields. The expression of their asymptotic expansion, and the algorithm to use for computing the coefficients (which are periodic functions) are given. Some studies for non oscillatory cases have already be done. There are related to what is called the “river phenomenon”.