where vector notation is used and ,u is a small parameter. This topological method was developed in previous papers [2-61 for the case in which A(t) is a constant matrix and the large forcing term g(t) is zero. Here we generalize the method to study the equation above and show that the generalized method provides a unified treatment of certain topics (somewhat diverse from the physical viewpoint) which arise in nonlinear theory: subharmonic oscillations, entrainment of frequency, asynchronous oscillations, and parametric excitation. These topics have been included under the name nonlinear resonance by Minorsky [7]. The relation of this topological method to the work of other writers, in particular Friedrichs [I], Lefschetz [lo, 111 and Bass [12] has been described in [a]. For references on nonlinear resonance, see Minorsky [7-91.