Skip to main content
SpringerLink
Log in

Periodic solutions of asymptotically linear Hamiltonian systems

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We prove existence and multiplicity results for periodic solutions of time dependent and time independent Hamiltonian equations, which are assumed to be asymptotically linear. The periodic solutions are found as critical points of a variational problem in a real Hilbert space. By means of a saddle point reduction this problem is reduced to the problem of finding critical points of a function defined on a finite dimensional subspace. The critical points are then found using generalized Morse theory and minimax arguments.

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. H. AMANN and E. ZEHNDER: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Annali Scuola Norm. Sup. Pisa, in press.

  2. H. AMANN and E. ZEHNDER: Multiple periodic solutions for a class of nonlinear autonomous wave equations. Houston J. Math., in press.

  3. G.D. BIRKHOFF: An extension of Poincaré's last geometric theorem. Acta. Math.,47, (1925), 297–311.

    Google Scholar 

  4. D.C. CLARK: A variant of the Lusternik-Schnirelman theory, Indiana Univ. Math. J.,22, (1972), 65–74.

    Google Scholar 

  5. C. CONLEY: Isolated invariant sets and the Morse index. A.M.S. Regional Conference Series in Mathem., Nr. 38., 1978

  6. E.R. FADELL and P.H. Rabinowitz: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Inventiones math.,45, (1978), 139–174.

    Google Scholar 

  7. J. MOSER: New aspects in the theory of stability of Hamiltonian systems. Comm. Pure and Appl. Math.,11, (1958), 81–114.

    Google Scholar 

  8. —: A fixed point theorem in symplectic geometry. Acta mathematica,141, (1978), 17–34.

    Google Scholar 

  9. P. RABINOWITZ: Periodic solutions of Hamiltonian systems. Comm. Pure and Appl. Math.,31, (1978), 157–184.

    Google Scholar 

  10. A. WEINSTEIN: Periodic orbits for convex Hamiltonian systems. Ann. of Math.,108, (1978), 507–518

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut der Universität Zürich, CH-8032, Zürich, Switzerland

    Herbert Amann

  2. Mathematisches Institut der Ruhr-Universität Bochum, D-4630, Bochum, Germany

    Eduard Zehnder

Authors
  1. Herbert Amann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eduard Zehnder
    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

Amann, H., Zehnder, E. Periodic solutions of asymptotically linear Hamiltonian systems. Manuscripta Math 32, 149–189 (1980). https://doi.org/10.1007/BF01298187

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01298187

Keywords

Search

Navigation