New periodic solutions of nonautonomous systems of nonlinear differential equations with derivatives multiplied by a small parameter and their stability (Q416967)

Consider the singularly perturbed system \[ \varepsilon\dot{x} = f(x,y,\beta_1(t))\;,\;\dot{y} = g(x,y,\beta_2(t)) \] with \(\dim x=\dim\beta_1=k\), \(\dim y=\dim\beta_2=l,\) and \(\beta_1\), \(\beta_2\) vector valued \(T_0\)-periodic synchronizing signals. The autonomous system \[ \varepsilon\dot{x} = f(x,y,0)\;,\;\dot{y} = g(x,y,0) \] is assumed to have a solution such that \(x(t+T_0)\equiv x(t)\). There are given conditions in order to ensure stability of the overall periodic motions of the system subject to the synchronizing signals.