The Singularity of Expanded Jacobian Matrix in Incremental Harmonic Balance Method Directly Locates Bifurcation Points of Steady-State Responses

Abstract

Abstract As a semi-analytical approach, the incremental harmonic balance (IHB) method is widely implemented for solving steady-state (including both periodic and quasi-periodic) responses through an iteration process. The iteration is carried out through a Jacobian matrix (JM) and a residual vector, both updated in each iteration. Though the JM is known to be singular at certain bifurcation points, the singularity is still an open question and could play a pivotal role in real applications. In this study, we define and calculate an expanded JM (EJM) by applying an expanded solution expression in the IHB iteration. The singularity of the EJM at several different bifurcation points is proved in a general manner, according to the bifurcation theory for equilibria in nonlinear dynamical systems. Given the possible bifurcation type, furthermore, the singularity is applied to locate the corresponding bifurcation point directly and precisely. Considered are the cases of the period-doubling, symmetry breaking, and Neimark-Sacker bifurcations of periodic and/or quasi-periodic responses.