Discretization and Some Qualitativeproperties of Ordinary

Discretizationsand Grobman-HartmanLemma, discretizations and the hierarchy of invariant manifolds about equilibria are considered. For one-step methods , it is proved that the linearizing conjugacy for ordinary diierential equations in Grobman-Hartman Lemma is, with decreasing stepsize, the limit of the linearizing conjugacies of the discrete systems obtained via time-discretizations. Similar results are proved for all types of invariant manifolds about equilibria. The estimates are given in terms of the degree of smoothness of the original ordinary diierential equation as well as in terms of the stepsize and of the order of the discretization method chosen. The results sharpen and unify those of Beyn 6], Beyn and Lorenz 7] and Fe ckan 17], 19]. 0. Introduction In recent years, several papers were devoted to studying the qualitative properties of discrete-time dynamical systems obtained via discretization methods. The basic question was/is whether the qualitative properties of continuous-time systems are preserved under discretization. Various concepts of diierentiable dynamics were investigated. Without claiming completeness, we mention here results on stability and attraction properties 17], 23], bifurcations 8], periodic orbits 5], 15], 29], 36], invariant tori 13], 14], the saddle-point structure about equilib-ria 6], 1], invariant manifolds about equilibria 6], 7], 17], algebraic-topological invariants 23], 27], averaging 19]. Most of these results relate one-step methods for ordinary diierential equations. Several authors remark that their results are also true for multistep methods (but the details are usually not presented. Such remarks do not seem to be entirely justiied. The starting point is a result for the Euler method. Though somewhat more technical skill is required, generalizations for other one-step methods are, from the conceptual point of view, easily given.