Step-like contrast structure for a quasilinear system of singularly perturbed differential equations with a zero characteristic number (Q299125)

Consider the singularly perturbed system \[ \begin{aligned} \varepsilon{dz\over dt} &= A(u,t)y+\varepsilon B(u,t),\\ \varepsilon{dy\over dt} &= z,\\ \varepsilon {du\over dt} &= C(u,t)y+\varepsilon D(u,t),\end{aligned}\tag{1} \] where \(\varepsilon\) is a small positive parameter and \(A\), \(B\), \(C\), \(D\) are sufficiently smooth functions, with the boundary conditions \[ z(0,\varepsilon)= z^0,\quad z(1,\varepsilon)= z^1,\quad u(0,\varepsilon)= u^0,\tag{2} \] where \(z\), \(y\) and \(u\) are scalar variables. The authors derive conditions on the functions \(A\), \(B\), \(C\) such that the boundary value problem (1), (2) has for sufficiently small \(\varepsilon\) a solution with a step at some transition point \(t^*(\varepsilon)\in (0,1)\). They present a procedure for the construction of a uniform asymptotic expansion of the solution including the transition point.