Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T03:22:30.341Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic solutions of high accuracy to the forced Duffing equation: Perturbation series in the forcing amplitude

Published online by Cambridge University Press:  17 February 2009

Lawrence K. Forbes
Lawrence K. Forbes
Affiliation:
Department of Mathematics, University of Queensland, St. Lucia, Queensland 4067, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

“Steady state” periodic solutions are sought to the forced Duffing equation. The solutions are expressed as formal Fourier series, giving rise to an infinite system of non-linear algebraic equations for the Fourier coefficients. This system is then solved using perturbation series in the amplitude of the forcing term. Solution profiles of high accuracy and phase-plane orbits are presented. The existence of limiting values of the forcing amplitude is discussed, and points of non-linear resonance are identified.

Type
Research Article
Information
The ANZIAM Journal , Volume 29 , Issue 1 , July 1987 , pp. 21 - 38
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Andersen, C. M. and Geer, J. F., “Power series expansions for the frequency and period of the limit cycle of the van der Pol equation”, SIAM J. Appl. Math. 42 (1982), 678–693.CrossRefGoogle Scholar
[2]Arnold, T. W. and Case, W., “Nonlinear effects in a simple mechanical system”, Amer. J. Phys. 50 (1982), 220–224.CrossRefGoogle Scholar
[3]Bazley, N. W. and Miletta, P., “Approximations to periodic solutions of a Duffing equation”, ZAMP 34 (1983), 301–309.Google Scholar
[4]Davis, H. T., Introduction to nonlinear differential and integral equations (Dover, New York, 1962).Google Scholar
[5]Domb, C. and Sykes, M. F., ‘On the susceptibility of a ferromagnetic above the Curie point”, Proc. Roy. Soc. London Ser. A 240 (1957), 214–228.Google Scholar
[6]Holmes, P., “A nonlinea oscillator with a strange attractor”, Philos. Trans. Roy. Soc. London Ser A. 292 (1979), 418–448.Google Scholar
[7]Parlitz, U. and Lauterborn, W., “Superstructure in the bifurcation set of the Duffing equation ”, Phys. Lett. A 107 (1985), 351–355.CrossRefGoogle Scholar
[8]Schmitt, B. V. “Sur la structure de ľequation de Duffing sans dissipation”, SIAM J. Appl. Math. 42 (1982), 868–894.CrossRefGoogle Scholar
[9]Shanks, D., “Non-linear transformations of divergent and slowly convergent sequences”, J. Math. Phys. 34 (1955), 1–42.CrossRefGoogle Scholar
[10]Urabe, M., “Galerkin's procedure for non-linear periodic systems”, Arch. Rational Mech. Anal. 20 (1965), 120–152.CrossRefGoogle Scholar
[11]van Dyke, M. D., “Analysis and improvement of perturbation series”, Quart. J. Mech. Appl. Math. 27 (1974), 423–450.CrossRefGoogle Scholar
[12]Wynn, P., “On the convergence and stability of the epsilon algorithm”, SIAM J. Numer. Anal. 3 (1966), 91–121.CrossRefGoogle Scholar