The oscillatory instability and the family of limit cycles associated with a general autonomous dynamical system described by n nonlinear first order differential equations and an independently assignable scalar parameter are examined via an intrinsic method of harmonic analysis. The method is essentially a variation of the classical method of “harmonic balancing”, and is designed to eliminate the drawbacks and shortcomings associated with the latter. Indeed, the new approach yields consistent approximations for the nonlinear dynamical bifurcation problem under consideration through a systematic perturbation procedure. It has thus been possible to derive explicit, formula-type expressions for the post-critical family of the periodic solutions, frequency of oscillations and the path which represents the family bifurcating from a flutter-critical point on an initially stable equilibrium path. The results are available to be used directly in the analysis of specific problems which fall within the scope of the formulation, without actually performing much analysis. Two illustrative examples are provided.