Modified Lindstedt-Poincaré methods for some strongly nonlinear oscillations. III: Double series expansion. (English) Zbl 1072.34507

Summary: We propose a new perturbation technique for strongly nonlinear oscillations with two parameters, which need not to be small in the present study. In this new method, the solution is expanded into a double series of the two parameters. In order to avoid the secular terms, a constant in the equation is also expressed in a double series expansion. The preliminary study shows that the obtained approximate solutions are uniformly valid on the whole solution domain.
For Part I: Expansion of a constant see Int. J. Non-Linear Mech. 37, No. 2, 309–314 (2002; Zbl 1116.34320); and Part II: A new transformation, ibid. 37, No. 2, 315–320 (2002; Zbl 1116.34321).


34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
Full Text: DOI


[1] He J.H, Int. J. Nonlinear Mech. 37 (2) pp 309– (2001)
[2] He J.H, Int. J. Nonlinear Mechanics 37 (2) pp 315– (2001)
[3] He J.H., International Journal of Nonlinear Sciences and Numerical Simulation 1 (1) pp 51– (2000)
[4] He J.H, Journal of Vibration and Control 7 (5) pp 631– (2001)
[5] He J.H, Int. J. Nonlinear Sei. & Numerical Simulation 2 (3) pp 203– (2001)
[6] He J.H, J. University of Shanghai Science & Technology 20 (4) pp 325– (1998)
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