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Residue harmonic balance solution procedure to nonlinear delay differential systems. (English) Zbl 1334.34145
Summary: 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.

34K07 Theoretical approximation of solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
Full Text: DOI
[1] Chen, G. R.; Lu, J. L.; Nicholas, B.; Ranganathan, S. M., Bifurcation dynamics in discrete time delayed-feedback control systems, Int. J. Bifurcation Chaos, 9, 287-293, (1999) · Zbl 0967.93511
[2] Heil, T., Delay dynamics of semiconductor laser with short external cavities: bifurcation scenarios and mechanism, Phys. Rev. E, 67, 066214-1, (2003)
[3] Huang, Y., Delay differential equations with application to population dynamics, (1993), Academic Press New York
[4] Nayfeh, A. H.; Chin, C. M.; Pratt, J., Perturbation methods in nonlinear dynamics: application to machining dynamics, J. Manuf. Sci. Eng., 119, 485-493, (1997)
[5] Iooss, G.; Josseph, D. D., Elementary stability and bifurcation theory, (1989), Springer New York
[6] Diekmann, O.; Gils, S. A.V.; Lunel, S. M.V.; Walther, H. O., Delay equations, functional-, complex-, and nonlinear analysis, (1995), Springer New York
[7] Kuznestov, Y. A., Elements of applied bifurcation theory, (1998), Springer New York
[8] Xu, J.; Chung, K. W., Effects of time delayed position feedback on a van der Pol Duffing oscillator, Physica D, 180, 17-39, (2003) · Zbl 1024.37028
[9] Lakrib, M., The method of averaging and functional differential equations with delay, Int. J. Math. Math. Sci., 26, 8, 497-511, (2001) · Zbl 0994.34054
[10] Hu, H. Y.; Wang, Z. H., Singular perturbation methods for nonlinear dynamic systems with time delays, Chaos Solitons Fractals, 40, 1, 13-27, (2009) · Zbl 1197.34113
[11] Das, S. L.; Chatterjee, A., Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations, Nonlinear Dyn., 30, 323-335, (2002) · Zbl 1038.34075
[12] Wang, H. L.; Hu, H. Y., Remarks on the perturbation methods in solving the second-order delay differential equations, Nonlinear Dyn., 33, 379-398, (2003) · Zbl 1049.70013
[13] Casal, A.; Freedman, M., A Poincaré-Lindstedt approach to bifurcation problems for differential-delay equations, IEEE Trans. Autom. Control, 25, 967-973, (1980) · Zbl 0437.34058
[14] Bojadziev, G.; Chan, S., Asymptotic solutions of differential equations with time delay in population dynamic, Bull. Math. Biol., 41, 3, 325-342, (1979) · Zbl 0402.92024
[15] Shakeri, F.; Dehghan, M., Solution of delay differential equations via a homotopy perturbation method, Math. Comput. Model., 48, 486-498, (2008) · Zbl 1145.34353
[16] You, X. C.; Xu, H., Analytical approximations for the periodic motion of the Duffing system with delayed feedback, Numer. Algorithms, 56, 4, (2011), 561-57 · Zbl 1211.65094
[17] Khan, H.; Liao, S. J.; Mohapatra, R. N.; Vajravelu, K., An analytical solution for a nonlinear time-delay model in biology, Commun. Nonlinear Sci. Numer. Simul., 14, 3141-3148, (2009) · Zbl 1221.65204
[18] Ali, K. E.; Mohammad, K., A homotopy analysis method for limit cycle of the van der Pol oscillator with delayed amplitude limiting, Appl. Math. Comput., 217, 2, 9404-9411, (2011) · Zbl 1215.65122
[19] Chung, K. W.; Chan, C. L.; Xu, J., A perturbation-incremental method for delay differential equations, Int. J. Bifurcation Chaos, 16, 9, 2529-2544, (2006) · Zbl 1142.34388
[20] Xu, J.; Chung, K. W., A perturbation-incremental scheme for studying Hopf bifurcation in delayed differential systems, Sci. China Ser. E, 52, 3, 698-708, (2009) · Zbl 1201.37088
[21] Dehghan, M.; Salehi, R., Solution of a nonlinear time-delay model in biology via semi-analytical approaches, Comput. Phys. Commun., 181, 1255-1265, (2010) · Zbl 1219.65062
[22] Yang, S. P.; Xiao, A. G., Convergence of the variational iteration method for solving multi-delay differential equations, Comput. Math. Appl., 61, 8, 2148-2151, (2011) · Zbl 1219.65086
[23] He, J. H., Periodic solutions and bifurcations of delay-differential equations, Phys. Lett. A, 347, 228-230, (2005) · Zbl 1195.34116
[24] Wang, Z. H.; Hu, H. Y., Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation, Int. J. Bifurcation Chaos, 17, 8, 2805-2814, (2007) · Zbl 1298.34126
[25] Gao, F.; Wang, H. L.; Wang, Z. H., Hopf bifurcation of a nonlinear delayed system of machine tool vibration via pseudo-oscillator analysis, Nonlinear Anal. Real, 8, 1561-1568, (2007) · Zbl 1146.70013
[26] Zhang, L. L.; Huang, L. H.; Zhang, Z. Z., Hopf bifurcation of the maglev time-delay feedback system via pseudo-oscillator analysis, Math. Comput. Model., 52, 5-6, 667-673, (2010) · Zbl 1202.34124
[27] Koen, E.; Tatyana, L.; Dirk, R., Numerical bifurcation analysis of delay differential equations, J. Comput. Appl. Math., 125, 1-2, 265-275, (2000) · Zbl 0970.65090
[28] Fudziah, I.; Mohammed, B. S., Solving delay differential equations using intervalwise partitioning by Runge-Kutta method, Appl. Math. Comput., 121, 1, 37-53, (2001) · Zbl 1024.65056
[29] Urabe, M., Galerkin’s procedure for nonlinear periodic systems, Arch. Ration. Mech. Anal., 20, 120-152, (1965) · Zbl 0133.35502
[30] S.L. Lau, Incremental harmonic balance method for nonlinear structural vibration (Ph.D. thesis), University of Hong Kong, 1982.
[31] Lai, S. K.; Lim, C. W., Newton-harmonic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators, Appl. Math. Model., 33, 852-866, (2009) · Zbl 1168.34321
[32] Wu, B. S.; Sun, W. P.; Lim, C. W., An analytical approximate technique for a class of strongly nonlinear oscillators, Int. J. Nonlinear Mech., 41, 766-774, (2006) · Zbl 1160.70340
[33] Wu, B. S.; Lim, C. W.; Sun, W. P., Improved harmonic balance approach to periodic solutions of non-linear Jerk equations, Phys. Lett. A, 354, 95-100, (2006)
[34] Blaquiére, A., Nonlinear systems analysis, (1966), Academic press New York
[35] Macdonald, N., Harmonic balance in delay-differential equations, J. Sound Vibr., 186, 4, 649-656, (1995) · Zbl 0963.34068
[36] Leung, A. Y.T.; Guo, Z. J., Residue harmonic balance approach to limit cycles of non-linear Jerk equations, Int. J. Nonlinear Mech., 46, 898-906, (2011)
[37] Leung, A. Y.T.; Guo, Z. J., Forward residue harmonic balance for autonomous and non-autonomous systems with fractional derivative damping, Commun. Nonlinear Sci. Numer. Simul., 16, 2169-2183, (2011) · Zbl 1221.34014
[38] Schanz, M.; Pelster, A., Analytical and numerical investigations of the phase-locked loop with time delay, Phys. Rev. E, 67, (2003), 056205-1
[39] Wang, Z. H., An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation, Nonlinear Dyn., 53, 1-11, (2008) · Zbl 1179.34071
[40] Li, J. Y., Hopf bifurcation of the sunflower equation, Nonlinear Anal. Real, 10, 2574-2580, (2009) · Zbl 1163.34347
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