The numerical approximation of the ordinary differential equation (1) \({u_ t= f(u)}\), \(u(0)= u\), \(u\in \mathbb{C}(\mathbb{R}^ 1,\mathbb{R}^ p)\) is studied. A time discretization through the points \(t_ n= n\Delta t\) is introduced and the difference equation (2) \(U_{n+1}={\mathcal F}(U_ n,\Delta t)\), \(U_ 0= U\), \(U_ n\in \mathbb{R}^ p\) is considered as an approximation \(u(t_ n)\).
The study of a variety of sets invariant under the evolution generated by (1) is crucial. Standard error estimates comparing individual trajectories are of no direct use in this context. Instead of them, 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 studied are equilibrium points, together with their unstable manifolds and local phase portraits, periodic solutions, quasi-periodic solutions, and strange attractors.
The author introduces various notions from the theory of dynamical systems and pays particular attention to the development of a unified theory and to the development of an existence theory for invariant sets of the underlying differential equation that may be used directly to construct an analogous existence theory (and hence a simple approximation theory) for the numerical methods.
For the entire collection see [Zbl 0797.00003].