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Published 1994 | metadata_only
Journal Article

Numerical analysis of dynamical systems


This article reviews the application of various notions from the theory of dynamical systems to the analysis of numerical approximation of initial value problems over long-time intervals. Standard error estimates comparing individual trajectories are of no direct use in this context since the error constant typically grows like the exponential of the time interval under consideration. Instead of comparing trajectories, the effect of discretization on various sets which are invariant under the evolution of the underlying differential equation is studied. Such invariant sets are crucial in determining long-time dynamics. The particular invariant sets which are studied are equilibrium points, together with their unstable manifolds and local phase portraits, periodic solutions, quasi-periodic solutions and strange attractors. Particular attention is paid to the development of a unified theory and to the development of an existence theory for invariant sets of the underlying differential equation which may be used directly to construct an analogous existence theory (and hence a simple approximation theory) for the numerical method.

Additional Information

© 1994 Cambridge University Press. This work was supported by the Office of Naval Research, contract number N00014-92-J-1876 and by the National Science Foundation, contract number DMS-9201727. I am greatly indebted to the people who helped in the task of checking this article: Fengshan Bai, Chris Budd, Adrian Hill, Arieh Iserles, Yunkang Liu, Gerald Moore and Alastair Spence.

Additional details

August 20, 2023
August 20, 2023