Skip to main content
SpringerLink
Log in

Transversal homoclinic points and hyperbolic sets for non-autonomous maps II

  • Original Papers
  • Published:
Zeitschrift für angewandte Mathematik und Physik ZAMP Aims and scope Submit manuscript

Abstract

The method of Melnikov is generalized to non-autonomous maps. If the Melnikov function has infinitely many zeros with derivatives bounded away from zero then the system admits a generalized hyperbolic set as it was introduced in part I. The developed theory is applied to almost periodically perturbed differential equations.

Zusammenfassung

Die Methode von Melnikov wird verallgemeinert für nichtautonome Abbildungen. Falls die Melnikov-Funktion unendlich viele Nullstellen mit von Null weg beschränkten Ableitungen hat, dann enthält das System eine verallgemeinerte hyberbolische Menge, wie sie in Teil I eingeführt wurde. Die entwickelte Theorie wird auf fast periodisch gestörte Systeme angewandt.

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. N. N. Bogoljubov and Y. A. Mitropolski,Asymptotische Methoden in der Theorie der nichtlinearen Schwingungen, VEB Leipzig 1965.

  2. H. Bohr,Almost Periodic Functions, Chelsea, New York 1947.

    Google Scholar 

  3. U. Kirchgraber,Erratische Lösungen der periodisch gestörten Pendelgleichung, preprint No. 88, Universität Würzburg 1982.

  4. H. W. Knobloch and F. Kappel,Gewöhnliche Differentialgleichungen, Teubner, Stuttgart 1974.

    Google Scholar 

  5. T. Y. Li and J. A. Yorke,Period three implies chaos, Am. Math. Monthly, Vol.82 (1975).

  6. V. K. Melnikov,On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc. Vol.12 (1963).

  7. J. Moser,Stable and Random Motions in Dynamical Systems, Princeton Univ. Press, New Jersey 1973.

    Google Scholar 

  8. K. J. Palmer,Exponential dichotomies and transversal homoclinic points, J. Diff. Equations, Vol.55 (1984).

  9. K. J. Palmer,Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported, Vol.1 (1988).

  10. G. Preuss,Allgemeine Topologie, Springer, Berlin 1975.

    Google Scholar 

  11. J. Scheurle,Chaotic solutions of systems with almost periodic forcing, J. Appl. Math. Phys. ZAMP, Vol.37 (1986).

  12. S. Smale,Differentiable dynamical systems, Bull. Amer. Math. Soc. Vol.73 (1967).

  13. D. Stoffer,Some geometric and numerical methods for perturbed integrable systems, Diss. ETH8456 (1988).

  14. D. Stoffer,Transversal homoclinic points and hyperbolic sets for non-autonomous maps I, J. Appl. Math. Phys. ZAMP, Vol.39 (1988).

  15. K. R. Meyer and G. R. Sell,Melnikov Transforms, Bernoulli-Bundles, and almost periodic perturbations, IMA, Preprint Series385 (1987).

Download references

Author information

Authors and Affiliations

  1. Seminar für Angewandte Mathematik, ETH-Zürich, Zürich

    Daniel Stoffer

Authors
  1. Daniel Stoffer
    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

Stoffer, D. Transversal homoclinic points and hyperbolic sets for non-autonomous maps II. Z. angew. Math. Phys. 39, 783–812 (1988). https://doi.org/10.1007/BF00945119

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00945119

Keywords

Search

Navigation