``Buridan's ass'' problem in relaxation systems with one slow variable (Q1966219)

Consider the singularly perturbed scalar differential equation \((*)\) \(\varepsilon dy/dx = f(x,y)\), \(0 < \varepsilon \ll 1\) where \(f\) is \(C^\infty\). It is assumed that \(f(x,y) =0\) has two solutions \(y = \varphi (x)\) and \(x = \psi (y)\) intersecting transversally at \((x_0 , y_0)\) and forming a pitchfork such that \(f^{-1} (0)\) has three branches for \(x > x_0\). The authors are interested in the asymptotic behavior of the solution \(y(x,\varepsilon)\) to \((*)\) satisfying \(y(x_0 - a, \varepsilon) = \alpha\) \((a > 0)\) as \(\varepsilon\) tends to zero, especially, they ask for the branch of \(f^{-1} (0)\) approximated by the solution \(y (x_0 - a,\varepsilon)\) for \(x> x_0\) and for conditions implying a delayed approximation. For proofs the authors refer to other papers.