Research Papers

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

[+] Author and Article Information
Xuefeng Wang

Department of Mechanical Engineering,
The University of Alabama,
Tuscaloosa, AL 35487
e-mail: xwang201@eng.ua.edu

Weidong Zhu

Fellow ASME
Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle, Baltimore, MD 21250
e-mail: wzhu@umbc.edu

Xi Zhao

Department of Mathematics and Computer Science,
West Virginia State University,
Institute, WV 25112
e-mail: xi.zhao@wvstateu.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the Journal of Applied Mechanics. Manuscript received October 6, 2018; final manuscript received February 11, 2019; published online April 1, 2019. Assoc. Editor: Ahmet S. Yigit.

J. Appl. Mech 86(6), 061011 (Apr 01, 2019) (11 pages) Paper No: JAM-18-1558; doi: 10.1115/1.4042836 History: Received October 06, 2018; Accepted February 11, 2019

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.

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Grahic Jump Location
Fig. 1

(a) Transient solution of the delayed Mathieu–Duffing equation with ϵ = 0.45 and (b) state-space plots of period-1 solutions from the numerical integration method in the DDE toolbox in matlab and the IHB method

Grahic Jump Location
Fig. 2

(a) Transient solution of the delayed Mathieu–Duffing equation with ϵ = 0.59 and (b) state-space plots of period-2 solutions from the numerical integration method in the DDE toolbox in matlab and the IHB method

Grahic Jump Location
Fig. 3

(a) Transient solution of the delayed Mathieu–Duffing equation with ϵ = 0.664 and (b) state-space plots of period-4 solutions from the numerical integration method in the DDE toolbox in matlab and the IHB method

Grahic Jump Location
Fig. 4

State-space plot for a transient solution of the delayed Mathieu–Duffing equation with ϵ = 0.7

Grahic Jump Location
Fig. 5

Poincare maps for (a) period-1, (b) period-2, (c) period-4, and (d) chaotic solutions

Grahic Jump Location
Fig. 6

L2 norms of solution errors for different orders of Fourier series

Grahic Jump Location
Fig. 7

Periodic solution of the delayed Van der Pol equation from the IHB method

Grahic Jump Location
Fig. 8

L2 norms of solution errors of the delayed Van der Pol equation from the IHB method



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