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

We consider the system of ordinary differential equations $\dot x = f(x,y)$, $\varepsilon\dot y=g(x,y)$, where $x\in\mathbb R^2$, $y\in\mathbb R$, $0<\varepsilon \ll 1$ and $f,g\in C^\infty$. It is assumed that the equation $g = 0$ determines two different smooth surfaces $y=\varphi(x)$ and $y=\psi(x)$ intersecting generically along a curve $l$. It is further assumed that the trajectories of the corresponding degenerate system lying on the surface $y=\varphi(x)$ are ducks, i.e., as time increases, they intersect the curve $l$ generically and pass from the stable part $\{y=\varphi(x), g'_y<0\}$ of this surface to the unstable part $\{y=\varphi(x), g'_y>0\}$. 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 $\varepsilon=0$ are the limits as $\varepsilon\to 0$ of some trajectories of the original system.