The harmonic balance approach is applied to a class of periodic dynamical systems with a specific (but quite general) feedback structure, in order to analyse approximately tangent (saddle-node) and flip (period-doubling) bifurcations of periodic solutions. By exploiting the system structure, an algebraic system of equations, whose unknowns are the coefficients of a truncated Fourier series, is defined and the relationship between the bifurcations of the roots of this algebraic system and the tangent and flip bifurcations of the periodic solutions is pointed out. An example concerning an adaptive control system illustrates the proposed methodology.