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On Dynamics and Bifurcations of Nonlinear Evolution Equations Under Numerical Discretization

In: Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems

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  • Christian Lubich

    (Universität Tübingen, Mathematisches Institut)

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.

Suggested Citation

  • Christian Lubich, 2001. "On Dynamics and Bifurcations of Nonlinear Evolution Equations Under Numerical Discretization," Springer Books, in: Bernold Fiedler (ed.), Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pages 469-500, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-56589-2_21
    DOI: 10.1007/978-3-642-56589-2_21
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