Averaging techniques without requiring a fast time-varying differential equation

An averaging result is presented for local uniform asymptotic stability of nonlinear differential equations without requiring a fast time-varying vectorfield. The nonlinearity plays a crucial role: close to the origin, the trajectories vary slowly compared to the time dependence of the vectorfield. The result generalises averaging results which prove stability properties for systems having a homogeneous vectorfield with positive order. The result is illustrated with several examples.