<p>This paper proposes a full numerical bifurcation analysis framework built on the harmonic balance method. For Floquet stability analysis, the framework leverages the Koopman-Hill projection method, enabling a “best of both worlds” computational strategy wherein results from typically time-based (monodromy matrix as obtained by Koopman-Hill projection, bifurcation test functions) and frequency-based (retaining Hill’s method for extended systems) approaches are combined optimally. In particular, the detection of bifurcations in codimension-2 is achieved through frequency-based, direct rank-one updates on the monodromy matrix thanks to the Koopman-Hill projection, while their exact localization is then performed using insights from classical Hill theory. Furthermore, this work details the application of the Koopman-Hill projection to different formulations of dynamical systems, in such a way that the proposed techniques are straightforwardly applicable to cases of wide practical interest, namely: second order ODEs and dynamical systems involving a state-dependent mass matrix. The robustness and performance of the novel framework are tested on three benchmark examples and compared to the traditional sorting-based Hill method, showing clear evidence in favor of using the Koopman-Hill projection in terms of reduced computation times and more precise results.</p>