The “Duck Survival” Problem in Three-Dimensional Singularly Perturbed Systems with Two Slow Variables

Abstract

We consider the system of ordinary differential equations x = f(x,y), εy = g(x,y), where x ∈ℝ², y ∈ ℝ, 0 <ε ≪ 1, and f,g ∈C∞. It is assumed that the equation g = 0 determines two different smooth surfaces y = ϕ(x) and y = ψ(x) intersecting generically along a curve l. It is further assumed that the trajectories of the corresponding degenerate system lying on the surface y = ϕ(x) are ducks, i.e., as time increases, they intersect the curve l generically and pass from the stable part {y =ϕ(x),g'_y < 0} of this surface to the unstable part {y =ϕ(x),g'_y > 0}. We seek a solution of the so-called “duck survival” problem, i.e., give an answer to the following question: what trajectories from the one-parameter family of duck trajectories for ɛ= 0 are the limits as ɛ → 0 of some trajectories of the original system.