Skip to main content
SpringerLink
Log in

Variational Calculation of the Period of Nonlinear Oscillators

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

The problem of calculating the period of second order nonlinear autonomous oscillators is formulated as an eigenvalue problem. We show that the period can be obtained from two integral variational principles dual to each other. Upper and lower bounds on the period can be obtained to any desired degree of accuracy. The results are illustrated by an application to the Duffing equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. Y. Biollay, Problème aux limites non-linéaire y''+λ 2 f(y)=0: Existence d'une solution et bornes pour λ 2(évaluations de la valeur critique), ZAMP 27:877–881 (1976).

    Google Scholar 

  2. R. D. Benguria and M. C. Depassier, A variational principle for eigenvalue problems of Hamiltonian systems, Phys. Rev. Lett. 77:2847–2850 (1996).

    Google Scholar 

  3. R. D. Benguria and M. C. Depassier, Phase space derivation of a variational principle for one-dimensional Hamiltonian systems, Phys. Lett. A 240:144–146 (1998).

    Google Scholar 

  4. R. D. Benguria and M. C. Depassier, A variational method for nonlinear eigenvalue problems, in Advances in Differential Equations and Mathematical Physics, E. Carlen, E. Harrell, and M. Loss, eds., Contemporary Mathematics, Vol. 217 (Amer. Math. Soc., 1998), pp. 1–17.

  5. I. Ekeland and R. Témam, Convex analysis and variational problems, Classics in Applied Mathematics, Vol. 28 (SIAM, Philadelphia, 1999).

    Google Scholar 

  6. D. D. Joseph, Nonlinear heat generation and stability of the temperature distribution in conducting solids, Int. J. Heat Mass Transfer 8:281–288 (1965).

    Google Scholar 

  7. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics Polymer Physics, and Finantial Markets, third extended edition (World Scientific, Singapore, 2003).

    Google Scholar 

  8. T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J. 20:1–13 (1970).

    Google Scholar 

  9. A. Pelster, H. Kleinert, and M. Schanz, High-order variational calculation for the frequency of time-periodic solutions, Phys. Rev. E 67:016604 (2003).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile

    Rafael Benguria & M. Cristina Depassier

Authors
  1. Rafael Benguria
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Cristina Depassier
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Benguria, R., Depassier, M.C. Variational Calculation of the Period of Nonlinear Oscillators. Journal of Statistical Physics 116, 923–931 (2004). https://doi.org/10.1023/B:JOSS.0000037219.42798.f7

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JOSS.0000037219.42798.f7

Search

Navigation