Twist Character of the Fourth Order Resonant Periodic Solution

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Abstract

In this paper, we will give, for the periodic solution of the scalar Newtonian equation, some twist criteria which can deal with the fourth order resonant case. These are established by developing some new estimates for the periodic solution of the Ermakov–Pinney equation, for which the associated Hill equation may across the fourth order resonances. As a concrete example, the least amplitude periodic solution of the forced pendulum is proved to be twist even when the frequency acroses the fourth order resonances. This improves the results in Lei et al. (2003). Twist character of the least amplitude periodic solution of the forced pendulm. SIAM J. Math. Anal. 35, 844–867.

Keywords

Twist character periodic solution fourth order resonance third order approximation forced pendulum 
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References

  1. 1.
    Chow, S.-N., Li, C.Z., Wang, D. 1994Normal Forms and Bifurcation of Planar Vector FieldsCambridge University PressCambridgeGoogle Scholar
  2. 2.
    Lei, J., Li, X., Yan, P., Zhang, M. 2003Twist character of the least amplitude periodic solution of the forced pendulemSIAM J. Math. Anal35844867CrossRefGoogle Scholar
  3. 3.
    Lei, J., Zhang, M. 2002Twist property of periodic motion of an atom near a charged wireLett. Math. Phys60917CrossRefGoogle Scholar
  4. 4.
    Mawhin, J. 1998Nonlinear complex-valued differential equations with periodic, Floquet or nonlinear boundary conditions. International Conference on Differential Equations (Lisboa, 1995)World ScientificRiver Edge, NJ154164Google Scholar
  5. 5.
    Moser J. (1962). On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. II, 1–20.Google Scholar
  6. 6.
    Núñez, D. 2002The method of lower and upper solutions and the stability of periodic oscillationsNonlinear Anal5112071222CrossRefGoogle Scholar
  7. 7.
    Núñez, D., Ortega, R. 2000Parabolic fixed points and stability criteria for nonlinear Hill’s equationsZ. Angew. Math. Phys51890911Google Scholar
  8. 8.
    Núñez, D., Torres, P.J. 2001Periodic solutions of twist type of an earth satellite equationDiscr. Cont. Dynam. Syst7303306Google Scholar
  9. 9.
    Núñez, D., Torres, P.J. 2003Stable odd solutions of some periodic equations modeling satellite motionJ. Math. Anal. Appl279700709CrossRefGoogle Scholar
  10. 10.
    Ortega, R. 1992The twist coefficient of periodic solutions of a time-dependent Newton’s equationJ. Dynam. Differential Equations4651665CrossRefGoogle Scholar
  11. 11.
    Ortega, R. 1994The stability of the equilibrium of a nonlinear Hill’s equationSIAM J. Math. Anal2513931401CrossRefGoogle Scholar
  12. 12.
    Ortega, R. 1996Periodic solution of a Newtonian equation: Stability by the third approximationJ. Differential Equations128491518CrossRefGoogle Scholar
  13. 13.
    Ortega, R. (2003). The stability of the equilibrium: a search for the right approximation, Preprint: Available at http://www.ugr.es/ ecuadif/fuentenueva.htmGoogle Scholar
  14. 14.
    Siegel, C.L., Moser, J.K. 1971Lectures on Celestial MechanicsSpringer-VerlagBerlinGoogle Scholar
  15. 15.
    Torres, P.J., Zhang, M. 2004Twist periodic solutions of repulsive singular equationsNonlinear Anal56591599CrossRefGoogle Scholar
  16. 16.
    Yan, P., Zhang, M. 2003Higher order nonresonance for differential equations with singularitiesMath. Methods Appl. Sci2610671074CrossRefGoogle Scholar
  17. 17.
    Zhang, M. 2001The rotation number approach to eigenvalues of the one-dimensional p-Laplacian with periodic potentialsJ. London Math. Soc64125143Google Scholar
  18. 18.
    Zhang, M. 2003The best bound on the rotations in the stability of periodic solutions of a Newtonian equationJ. London Math. Soc67137148CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Zhou Pei-Yuan Center for Applied MathematicsTsinghua UniversityBeijingChina
  2. 2.Departamento de Matemática AplicadaUniversidad de GranadaGranadaSpain
  3. 3.Department of Mathematical SciencesTsinghua UniversityBeijingChina

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