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A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables isdoi:10.1090/s0025-5718-08-02139-x fatcat:jcd5exhbifcijb2bf2mj4r7ely