Singular initial value problem for a system of ordinary differential equations with a small parameter (Q1048505)

The paper is devoted to the study of asymptotic behavior of the solution to the initial value problem \[ \varepsilon\;\frac{dU}{dt}=f(t,U,V), \quad \frac{dV}{dt} = g(t,U,V), \quad U(0)=A, V(0)=0, \] with a small positive parameter \(\varepsilon\). It is assumed that the reduced problem has a solution \((U_0(t),V_0(t))\). This problem is well studied under the assumption that for \(t\geq0\) \[ \frac{\partial f}{\partial U} (t,U_0(t),V_0(t)) <0. \] In the present study this assumption is violated at the initial point. In this case, the asymptotic behavior of the solution is complicated and it is shown that the correct asymptotic expansion is not a series in power of \(\varepsilon\) even far away from the initial point. The algorithm for a construction of the uniformly valid asymptotic expansions is presented.