Twice Harmonic Balance Method for Stability and Bifurcation Analysis of Quasi-Periodic Responses

A twice harmonic balance (THB) method is proposed to compute and analyze quasi-periodic (QP) responses of nonlinear dynamical systems, with emphasis on the stability and bifurcation of QP responses. In the first harmonic balancing, the original system is transformed into a truncated system via harmonic balance method with variable-coefficients. The truncated system is further solved via the second harmonic balancing, more specifically the incremental harmonic balance (IHB) method. The equivalence is addressed between the periodic solutions of the truncated system and the QP responses of the original system. According to the relationship, the presented method is in essence to convert the problem of solving the original system for QP responses into a truncated system for periodic solutions. Numerical examples show that the semi-analytical QP solutions obtained by the THB method are in well consistence with the solutions obtained by the Runge–Kutta (RK) method and the IHB method with two time scales, respectively. More importantly, the stability of the attained QP solutions can be analyzed by just applying the Floquet theory to the periodic response of the truncated system. The continuation of the QP responses is generated by the presented method, on which the possible bifurcations resulted from the stability reversal are analyzed in detail. In addition, the evolution of QP responses can also be tracked from periodic solutions, such as that due to the onset of a Neimark–Sacker bifurcation.