Über die Konstruktion Invarianter Tori, welche von Einer Stationären Grundlösung Eines Reversiblen Dynamischen Systems Abzweigen

  • Jürgen Scheurle
Part of the International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Serie Internationale D’Analyse Numerique book series (ISNM, volume 48)


An existence theorem for invariant tori near equilibria of reversible dynamical systems is presented. It is assumed that the linearized vector field has exactly n pairs of simple conjugate eigenvalues moving along the imaginary axis with non-vanishing velocities corresponding to certain external parameters. The remaining part of the spectrum may be arbitrary. The motion on the tori turns out to be quasiperiodic. Finally a generalized Newton method is described which enables one to construct the tori inspite of the arising difficulty with small divisors.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Kirchgässner K. und Scheurle J.: On the bounded solutions of a semilinear elliptic equation in a strip, erscheint in J. Diff. Equat.Google Scholar
  2. [2]
    Kolmogorov A.: On conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk. SSSR 98 (1954), 527–530.Google Scholar
  3. [3]
    Moser J.: Stable and random motions in dynamical systems. Ann. Math. Studies 77, Princeton New Jersey, Princeton Univ. Press 1973.Google Scholar
  4. [4]
    Moser J.: Convergent series expansions for quasiperiodic motions. Math. Ann. 169 (1967), 136–176.CrossRefGoogle Scholar
  5. [5]
    Scheurle J.: Bifurcation of a stationary solution of a dynamical system into n-dimensional tori of quasiperiodic solutions. Proceedings der FDE-und AFP-Konferenz Bonn 1978, erscheint in Springer lecture notes.Google Scholar
  6. [6]
    Scheurle J.: Newton iterations without inverting the derivative. Praeprint 1978.Google Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Jürgen Scheurle
    • 1
  1. 1.Math. Institut AUniversität StuttgartStuttgart 80Germany

Personalised recommendations