An initial value theorem for nonlinear singularly perturbed systems (Q795979)

Let \((x_{\epsilon},y_{\epsilon})\) be the solution (which is assumed to exist and to be unique) of the singularly perturbed system \(\dot x=f(x,y,\epsilon)\), \(x(0)=x_ 0\), \(\epsilon\) \(\dot y=g(x,y,\epsilon)\), \(y(0)=y_ 0\). Introducing Lyapunov functions for the reduced and for the boundary systems the authors give new conditions under which \((x_{\epsilon},y_{\epsilon})\) converges to the solution \((x_ 0,y_ 0)\) of the reduced system, associated with \(\epsilon =0\), uniformly on the semi-infinite time-interval \([0,+\infty)\). A comparison with other results is presented.