TY - JOUR
T1 - Residue harmonic balance solution procedure to nonlinear delay differential systems
AU - Guo, Zhongjin
AU - Ma, Xiaoyan
PY - 2014/6/15
Y1 - 2014/6/15
N2 - This paper develops the residue harmonic balance solution procedure to predict the bifurcated periodic solutions of some autonomous delay differential systems at and after Hopf bifurcation. In this solution procedure, the zeroth-order solution employs just one Fourier term. The unbalanced residues due to Fourier truncation are considered by solving linear equation iteratively to improve the accuracy. The number of Fourier terms is increased automatically. The well-known sunflower equation and van der Pol equation with unit delay are given as numerical examples. Their solutions are verified for a wide range of system parameters. Comparison with those available shows that the residue harmonic balance method is effective to solve the autonomous delay differential equations. Moreover, the present method works not only in determining the amplitude but also the frequency at bifurcation.
AB - This paper develops the residue harmonic balance solution procedure to predict the bifurcated periodic solutions of some autonomous delay differential systems at and after Hopf bifurcation. In this solution procedure, the zeroth-order solution employs just one Fourier term. The unbalanced residues due to Fourier truncation are considered by solving linear equation iteratively to improve the accuracy. The number of Fourier terms is increased automatically. The well-known sunflower equation and van der Pol equation with unit delay are given as numerical examples. Their solutions are verified for a wide range of system parameters. Comparison with those available shows that the residue harmonic balance method is effective to solve the autonomous delay differential equations. Moreover, the present method works not only in determining the amplitude but also the frequency at bifurcation.
KW - Residue harmonic balance
KW - Sunflower equation
KW - Van der Pol oscillator
KW - Hopf bifurcation
KW - Accurate periodic solution
KW - PSEUDO-OSCILLATOR ANALYSIS
KW - HOPF-BIFURCATION
KW - TIME-DELAY
KW - PERIODIC-SOLUTIONS
KW - ITERATION METHOD
KW - JERK EQUATIONS
KW - DYNAMICS
KW - FEEDBACK
KW - BIOLOGY
KW - MODEL
UR - http://www.scopus.com/inward/record.url?scp=84899012508&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84899012508&origin=recordpage
U2 - 10.1016/j.amc.2014.03.090
DO - 10.1016/j.amc.2014.03.090
M3 - 21_Publication in refereed journal
VL - 237
SP - 20
EP - 30
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
SN - 0096-3003
ER -