BIT Numerical Mathematics

, Volume 28, Issue 1, pp 113–122 | Cite as

Invariant curves of one-step methods

  • timo Eirola
Part II Numerical Mathematics

Abstract

This paper considers the invariant sets of numerical one-step integration methods in a neighbourhood of a hyperbolic periodic solution of a nonlinear ODE. Using results from the dynamical systems theory it was possible to show that for the usual one-step methods the invariant sets areCk-circles (closed curves) for small enough stepsizeh. Here we give a direct proof for that and also show that they areO(hp)Ck-close to the true periodic trajectory, wherep is the order of the method.

Subject classification

AMS 65L 34C 

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References

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    W.-J. Beyn,On invariant closed curves for one-step methods, Numer. Math. (to appear).Google Scholar
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    M. Braun and J. Hershenov,Periodic solutions of finite difference equations, Quart. Appl. Math. 35 (1977), 139–147.Google Scholar
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    H. T. Doan,Invariant curves for numerical methods, Quart. Appl. Math. 43 (1985), 385–393.Google Scholar
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    T. J. Eirola,Two concepts for numerical periodic solutions of ODE's, Appl. Math. & Comp. (to appear).Google Scholar
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    M. Hirsh, C. Pugh and M. Shub,Invariant Manifolds, Lecture Notes in Math., No. 583, Springer-Verlag, Berlin, Heidelberg, 1977.Google Scholar

Copyright information

© BIT Foundations 1988

Authors and Affiliations

  • timo Eirola
    • 1
  1. 1.Institute of MathematicsHelsinki University of TechnologyEspooFinland

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