On Dynamics and Bifurcations of Nonlinear Evolution Equations Under Numerical Discretization

  • Christian Lubich

Abstract

This article reviews recent results on long-time behaviour, invariant sets and bifurcations of evolution equations under discretization by numerical methods. The emphasis is on time discretization. Finite-time error bounds of low order for non-smooth data, of high order for smooth data, and attractive invariant manifolds are tools that pervade large parts of the article. To illustrate the mechanisms, the following combinations of dynamics/equations have been selected for a detailed discussion:
  • Shadowing near hyperbolic equilibria of singularly perturbed ODEs

  • Hyperbolic periodic orbits of delay differential equations

  • Hopf bifurcation of semilinear parabolic equations

  • Inertial manifolds of semilinear parabolic equations

  • Attractors of damped wave equations.

Keywords

Periodic Orbit Hopf Bifurcation Error Bound Invariant Manifold Delay Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Christian Lubich
    • 1
  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany

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