The solution structure of the Düffing oscillator’s transient response and general solution. (English) Zbl 1347.34056

Summary: A modification of homotopy analysis method (HAM) is developed in this paper. The solution structure of the Düffing oscillator’s free vibration at different damping levels is put forward based on this new modified HAM. Explicit expressions for the fundamental decaying rate and the fundamental frequency are derived. Numerical examples with different initial conditions are calculated to verify the proposed solution structures. The number of terms required to be considered in the modified HAM for yielding satisfactorily accurate solutions is analyzed. The structure of forced and damped Düffing oscillator’s general solution is also put forward and verified.


34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations


Full Text: DOI


[1] Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973) · Zbl 0265.35002
[2] Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981) · Zbl 0449.34001
[3] Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995) · Zbl 0848.34001
[4] Rao, S.S.: Mechanical Vibrations, 2nd edn. Addison-Wesley, Reading, MA (1990) · Zbl 0714.73050
[5] Wu, B; Li, P, A new approach to nonlinear oscillations, J. Appl. Mech., 68, 951-952, (2001) · Zbl 1110.74758
[6] Van Dyke, M.: Perturbation Methods in Fluid Mechanics. The Parabolic Press, Stanford (1975) · Zbl 0329.76002
[7] Liao, S.: Homotopy Analysis Method in Nonlinear Differential Equations. Springer & Higher Education Press, Berlin (2012) · Zbl 1253.35001
[8] Ergin, E. I.: Transient Response of Non-linear Spring-Mass Systems. Ph.D. thesis, California Institute of Technology (1954) · Zbl 1329.37051
[9] Elías-Zúñiga, ALEX, A general solution of the Düffing equation, Nonlinear Dyn., 45, 227-235, (2006) · Zbl 1121.70016
[10] Nourazar, S; Mirzabeigy, A, Approximate solution for nonlinear Düffing oscillator with damping effect using the modified differential transform method, Sci. Iran. B, 20, 364-368, (2013)
[11] Liao, S.: Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems. Ph.D. thesis, Shanghai Jiao Tong University (1992) · Zbl 1110.74758
[12] Zou, K; Nagarajaiah, S, An analytical method for analyzing symmetry-breaking bifurcation and period-doubling bifurcation, Commun. Nonlinear Sci. Numer. Simul., 22, 780-792, (2015) · Zbl 1329.37051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.