The "duck survival" problem in three-dimensional singularly perturbed systems with two slow variables

We consider the system of ordinary differential equations x (x, y), epsilony = g(x, y), where x is an element of R-2, y is an element of R, 0 < epsilon much less than 1, and f, g is an element of C-infinity. It is assumed that the equation g = 0 determines two different smooth surfaces y = phi(x) and y = psi(x) intersecting generically along a curve 1. It is further assumed that the trajectories of the corresponding degenerate system lying on the surface y = phi(x) are ducks, i.e., as time increases, they intersect the curve l generically and pass from the stable part {y = phi(x), g'(y) < 0} of this surface to the unstable part {y = phi(x), g'(y) > 0}. We seek a solution of the so-called "duck survival" problem, i.e., y give an answer to the following question: what trajectories from the one-parameter family of duck trajectories for epsilon = 0 are the limits as epsilon--> 0 of some trajectories of the original system.