Study on a multi-frequency homotopy analysis method for period-doubling solutions of nonlinear systems. (English) Zbl 1391.34034

Summary: In this paper, a modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system. Firstly, the process of calculating the two-degree-of-freedom coupled Duffing system is presented. Secondly, the single periodic solutions and double periodic solutions are obtained by solving the constructed nonlinear algebraic equations. Finally, comparing the periodic solutions obtained by the multi-frequency homotopy analysis method (MFHAM) and the fourth-order Runge-Kutta method, it is found that the approximate solution agrees well with the numerical solution.


34A45 Theoretical approximation of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
34C25 Periodic solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations


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[1] Akbarzade, M.; Ganji, D. D., Coupled method of homotopy perturbation method and variational approach for solution to nonlinear cubic-quintic Duffing oscillator, Adv. Th. Appl. Mech., 3, 329-337, (2010)
[2] Beléndez, A.; Pascual, C.; Fernández, E., Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified he’s homotopy methods, Physica Scripta, 77, 025004, (2008) · Zbl 1175.70023
[3] Cui, J.; Liang, J.; Lin, Z., Stability analysis for periodic solutions of the van der Pol-Duffing forced oscillator, Physica Scripta, 91, 015201, (2016)
[4] Ganjiani, M.; Ganjiani, H., Solution of coupled system of nonlinear differential equations using homotopy analysis method, Nonlin. Dyn., 56, 159-167, (2009) · Zbl 1172.65379
[5] Harat, S. M. H.; Babolian, E.; Heydari, M., An efficient method for solving strongly nonlinear oscillators: combination of the multi-step homotopy analysis and spectral method, Iranian J. Sci. Technol. Trans. A Sci., 39, 455-462, (2015)
[6] He, J. H., Modified Lindstedt-Poincaré methods for some strongly nonlinear oscillations: part II: A new transformation, Int. J. Non-Lin. Mech., 37, 309-314, (2002) · Zbl 1116.34320
[7] Jafarian, A.; Ghaderi, P.; Golmankhaneh, A. K., Homotopy analysis method for solving coupled Ramani equations, Romanian J. Phys., 59, 26-35, (2014)
[8] Jiang, H. P.; Zhang, T. H.; Song, Y. L., Delay-induced double Hopf bifurcations in a system of two delay-coupled van der Pol-Duffing oscillators, Int. J. Bifurcation and Chaos, 25, 1550058-1-18, (2015) · Zbl 1314.34143
[9] Jiang, T.; Yang, Z.; Jing, Z., Bifurcations and chaos in the Duffing equation with parametric excitation and single external forcing, Int. J. Bifurcation and Chaos, 27, 1750125-1-31, (2017) · Zbl 1377.34047
[10] Khan, Y.; Vázquez-Leal, H.; Faraz, N., An efficient new iterative method for oscillator differential equation, Scientia Iranica, 19, 1473-1477, (2012)
[11] Kimiaeifar, A.; Saidi, A. R.; Bagheri, G. H., Analytical solution for van der Pol-Duffing oscillators, Chaos Solit. Fract., 42, 2660-2666, (2009) · Zbl 1198.65145
[12] Kimiaeifar, A., An analytical approach to investigate the response and stability of van der Pol-Mathieu-Duffing oscillators under different excitation functions, Math. Meth. Appl. Sci., 33, 1571-1577, (2010) · Zbl 1304.34098
[13] Liao, S. J. [1992] “The proposed homotopy analysis techniques for the solution of nonlinear problems,” PhD dissertation, Shanghai Jiao Tong University, Shanghai.
[14] Liao, S., Homotopy Analysis Method in Nonlinear Differential Equations, (2012), Springer, Berlin-Heidelberg · Zbl 1253.35001
[15] Liu, H. M., Approximate period of nonlinear oscillators with discontinuities by modified Lindstedt-Poincaré method, Chaos Solit. Fract., 23, 577-579, (2005) · Zbl 1078.34509
[16] Liu, Q. X.; Liu, J. K.; Chen, Y. M., Asymptotic limit cycle of fractional van der Pol oscillator by homotopy analysis method and memory-free principle, Appl. Math. Model., 40, 3211-3220, (2016)
[17] Nayfeh, A. H., Dynamic General Equilibrium Modeling, Perturbation methods, 75-173, (1973), Springer-Verlag, Berlin-Heidelberg
[18] Pathak, S.; Singh, T., An analytic solution of mathematical model of boussinqs equation in homogeneous porous media during infiltration of groundwater flow, J. Geograph. Environ. Earth Sci. Int., 3, 1-8, (2015)
[19] Pathak, S.; Singh, T., Optimal homotopy analysis methods for solving the linear and nonlinear Fokker-Planck equations, British J. Math. Comput. Sci., 7, 965-967, (2015)
[20] Pathak, S.; Singh, T., Solution of coupled non-linear system by optimal homotopy analysis method, Int. J. Concept. Comput. Inform. Technol., (2015)
[21] Pathak, S.; Singh, T., A mathematical modeling of imbibition phenomenon in inclined homogenous porous media during oil recovery process, Perspect. Sci., 8, 183-186, (2016)
[22] Pathak, S.; Singh, T., Study on fingero imbibition phenomena in inclined porous media by optimal homotopy analysis method, Ain Shams Engineering J, (2016)
[23] Pathak, S.; Singh, T., The solution of non-linear problem arising in infiltration phenomenon in unsaturated soil by optimal homotopy analysis method, Int. J. Adv. Appl. Math. Mech., 4, 21-28, (2016) · Zbl 1370.76177
[24] Qian, Y.; Chen, S., Accurate approximate analytical solutions for multi-degree-of-freedom coupled van der Pol-Duffing oscillators by homotopy analysis method, Commun. Nonlin. Sci. Numer. Simul., 15, 3113-3130, (2010) · Zbl 1222.65092
[25] Qian, Y. H.; Zhang, Y. F., Optimal extended homotopy analysis method for multi-degree-of-freedom nonlinear dynamical systems and its application, Struct. Engin. Mech., 61, 107-118, (2017)
[26] Rajasekar, S.; Murali, K., Resonance behaviour and jump phenomenon in a two coupled Duffing-van der Pol oscillators, Chaos Solit. Fract., 19, 925-934, (2004) · Zbl 1058.34048
[27] Sanliturk, K. Y.; Ewins, D. J., Modelling two-dimensional friction contact and its application using harmonic balance method, J. Sound Vibr., 193, 511-523, (1996) · Zbl 1232.74077
[28] Sayevand, K.; Baleanu, D.; Fardi, M., A perturbative analysis of nonlinear cubic-quintic Duffing oscillators, Proc. Romanian Acad. — Series A: Math. Phys. Techn. Sci. Inform. Sci., 15, 228-234, (2014)
[29] Shukla, A. K.; Ramamohan, T. R.; Srinivas, S., A new analytical approach for limit cycles and quasi-periodic solutions of nonlinear oscillators: the example of the forced van der Pol-Duffing oscillator, Physica Scripta, 89, 075202, (2014)
[30] Turkyilmazoglu, M., An effective approach for approximate analytical solutions of the damped Duffing equation, Physica Scripta, 86, 15301-15306, (2012)
[31] van Dyke, M., Perturbation methods in fluid mechanics, Nature, 206, 226-227, (1965)
[32] Wen, J.; You, B.; Zhao, J., Superharmonic resonances of parametricly excited gear system solved by homotopy analysis method, Int. J. Hybrid Inform. Technol., 7, 147-154, (2014)
[33] Yamapi, R.; Filatrella, G., Strange attractors and synchronization dynamics of coupled van der Pol-Duffing oscillators, Commun. Nonlin. Sci. Numer. Simul., 13, 1121-1130, (2008) · Zbl 1221.37070
[34] Yuan, P. X.; Li, Y. Q., Approximate solutions of primary resonance for forced Duffing equation by means of the homotopy analysis method, Chinese J. Mech. Engin., 24, 501-506, (2011)
[35] Zou, K.; Nagarajaiah, S., An analytical method for analyzing symmetry-breaking bifurcation and period-doubling bifurcation, Commun. Nonlin. Sci. Numer. Simul., 22, 780-792, (2015) · Zbl 1329.37051
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