An efficient and robust approach for continuation and bifurcation analysis of quasi-periodic solutions by multi-harmonic balance method

Abstract

To explore the stability and bifurcation of quasi-periodic solutions, it is crucial to obtain or track the full branches of the solutions during varying system parameters of nonlinear dynamical systems. In this aspect, the phase conditions related to the prior unknown frequencies of quasi-periodic solutions must be supplemented in order to ensure the uniqueness. However, the existing phase conditions cannot guarantee robust solution continuation because they are heavily dependent on the solution point of previous step or the accuracy of the predicted initial condition. In this paper, a simple phase condition is devised to secure a robust continuation, which utilizes the orthogonality in the sense of integral between the current solution point and its iterative variables at each step of continuation. Moreover, a formulation of two-dimensional fast Fourier transforms (2-FFT) for AFT method is introduced for the multi-harmonic balance method (MHB) in order to rapidly evaluate the nonlinear terms and their derivatives. Two examples of a 2DOFs nonlinear energy sink (NES) system and a 2DOFs rotor-stator rubbing system are studied by the developed approach to show the feasibility and efficiency. Both stable and unstable quasi-periodic solutions after Neimark-Sack and/or fold bifurcations are fully tracked.