We consider digital feedback control systems with time-varying sampling period's consisting of an interconnection of a continuous-time nonlinear plant (described by systems of first-order ordinary differential equations), a nonlinear digital controller (described by systems of first-order ordinary difference equations), and appropriate interface elements between the plant and controller (A/D and D/A converters). For such systems we study the stability properties of an equilibrium (in the Lyapunov sense) and derive some results for local stability and instability via a linearization approach. These results are then used in the analysis of certain classes of switched systems and in the stabilization problem of nonlinear cascade control systems via hybrid feedback controllers.