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On the generalized averaging method of a class of strongly nonlinear forced oscillators

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  • Mahmoud, Gamal M.

Abstract

We present a modified version of the generalized averaging method for studying the periodic solutions of a class of strongly nonlinear forced oscillators of the form ẍ + ω2x + εf(x) P(Ωt) = 0, where f(x) is a nonlinear function of x, P(Ωt) is a periodic function of t, ε need not be small, and ω is a constant parameter. This equation can be used to describe, e.g., a pendulum with a vibrating length or the displacements of colliding particle beams in high energy accelerators. The new version is based on defining a new parameter α = α(ε) and a linear transformation of the time. This version is applied for the cases f(x) = x3 and f(x) = x4 with P(Ωt) = cos t and excellent agreement is found with the results of numerical experiments, for large values of ε.

Suggested Citation

  • Mahmoud, Gamal M., 1993. "On the generalized averaging method of a class of strongly nonlinear forced oscillators," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 199(1), pages 87-95.
    • Handle: RePEc:eee:phsmap:v:199:y:1993:i:1:p:87-95
    DOI: 10.1016/0378-4371(93)90099-P
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    Cited by:

    1. Mahmoud, Gamal M., 1998. "Approximate solutions of a class of complex nonlinear dynamical systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 253(1), pages 211-222.
    2. A. M. A. El-Sayed & S. M. Salman, 2019. "Dynamical analysis of a complex logistic-type map," Indian Journal of Pure and Applied Mathematics, Springer, vol. 50(2), pages 427-450, June.
    3. Mohd Taib Shatnawi & Adel Ouannas & Ghenaiet Bahia & Iqbal M. Batiha & Giuseppe Grassi, 2021. "The Optimal Homotopy Asymptotic Method for Solving Two Strongly Fractional-Order Nonlinear Benchmark Oscillatory Problems," Mathematics, MDPI, vol. 9(18), pages 1-13, September.

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