A Koopman-Hill framework for the bifurcation analysis of nonlinear dynamical systems in codimension-1 and -2

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, bifurcation test functions) and frequency-based (exploiting Hill’s method for extended systems) are combined optimally. In particular, the detection and localization of bifurcations in codimension-2 and higher is achieved through frequency-based, direct rank-one updates (Wielandt deflation) on the mon-odromy matrix. 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.