Bifurcation for non linear ordinary differential equations with singular perturbation

We study a family of singularly perturbed ODEs with one parameter and compare their solutions to the ones of the corresponding reduced equations. The interesting characteristic here is that the reduced equations have more than one solution for a given set of initial conditions. Then we consider how those solutions are organized for different values of the parameter. The bifurcation associated to this situation is studied using a minimal set of tools from non standard analysis