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A semi-analytical procedure for studying，both qualitatively and quantitatively，stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous dynamical systems is developed. The procedure of analysis based mainly on the incremental harmonic balance（IHB) method. It is composed of three key steps，namely，the limit cycles approximation step，the monodromy matrix computation step and the step of selecting suitable initial conditions at the bifurcation points. The procedure can be used to analyze stability and successive bifurcations of limit cycles and calculate the series of bifurcation points. As for application，the procedure is used to investigate the dynamics of the limit cycle born in Hopf bifurcation in a three-dimensional nonlinear autonomous system. The symmetry-breaking bifurcation and the first and the second period-doubling bifurcations of the limit cycle are identified. The critical parameter values corresponding to these bifurcations are calculated. The validity of the obtained results is shown by the well consistence with those of direct numerical integrations using Runge-Kutta method.