An Incremental Harmonic Balance Method With a General Formula of Jacobian Matrix and a Direct Construction Method in Stability Analysis of Periodic Responses of General Nonlinear Delay Differential Equations

Abstract

A general formula of Jacobian matrix is derived in an incremental harmonic balance (IHB) method for general nonlinear delay differential equations (DDEs) with multiple discrete delays, where the fast Fourier transform is used to calculate Fourier coefficients of partial derivatives of residuals. It can be efficiently and automatically implemented in a computer program, and the only manual work is to derive the partial derivatives, which can be a much easier task than derivation of Jacobian matrix. An advantage of the IHB method in stability analysis is also revealed here. A direct construction method is developed for stability analysis of nonlinear differential equations with use of a relationship between Jacobian matrix in the IHB method and the system matrix of linearized equations. Toeplitz form of the system matrix can be directly constructed, and Hill’s method is used to calculate Floquet multipliers for stability analysis. Efficiency of stability analysis can be improved since no integration is needed to calculate the system matrix. Period-doubling bifurcations and period-p solutions of a delayed Mathieu–Duffing equation are studied to demonstrate use of the general formula of Jacobian matrix in the IHB method and the direct construction method in stability analysis. Its solution is the same as that from the numerical integration method using the spectral element method in the DDE toolbox in matlab, and it has a high convergence rate for solving a delayed Van der Pol equation.

Non-Reciprocal Wave Propagations in a One-Dimensional Periodic Structure Modified with a Linkage Mechanism

“…( 5), as shown in Appendix B. Meanwhile, the analytic formula of the Jacobian matrix of Q can be established by applying the method of the literature [37], as derived in Appendix C:…”

“…In order to avoid the complicated Galerkin integral calculation process, the analytical expression of JQ can be established by referring to the method in reference [37],…”

Periodic response of a nonlinear axially moving beam with a nonlinear energy sink and piezoelectric attachment

“…( 18) -Eq. ( 21), we will introduce the semi-analytical technique known as the incremental harmonic balance method (IHBM) [48][49][50]. One of the most important advantages of the IHBM lies in the fact that it can be easily combined with the continuation algorithms for finding the system response for different model parameters.…”

Steady-State Wave Responses of One-Dimensional Multi-Degree-of-Freedom Lattices with Strong Nonlinearities by an Improved Incremental Harmonic Balance Method

“…The steady-state solution of the DDE in Eq. ( 6) can be derived by modifying the method in [38]. First, a dimensionless time variable can be defined as…”

“…Using the new IHB method, Song [37] studied elastic wave propagations and control in 2-DoF systems. The key of this method is to transform the governing equation of a nonlinear wave problem into a delayed differential equation (DDE) with multiple delays and advances [38], and then the DDE can be solved by an improved IHB method [28]. It was proved that steady-state solutions of the original wave problem are the same as those of the transformed DDE.…”