Discretization and Some Qualitative Properties of Ordinary Differential Equations about Equilibria

Discretizations and Grobman-Hartman Lemma, discretizations and the hierarchy of invariant manifolds about equilibria are considered. For one-step methods, it is proved that the linearizing conjugacy for ordinary differential 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 differential 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 Fečkan [17], [19].