ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 62, 2 (1993)
pp. 249-275

DISCRETIZATION AND SOME QUALITATIVE PROPERTIES OF ORDINARY DIFFERENTIAL EQUATIONS ABOUT EQUILIBRIA
B. M. GARAY

Abstract. Discretizations and Grobman-Hartman Lemma, discretizations and the hierarchy of invariant manifolds about equilibria are considered. For \hboxone-step methods, it is proved that the linearizing conjugacy for ordinary differential equations in \hboxGrobman-Hartman Lemma is, with decreasing stepsize, the limit of the linearizing conjugacies of the discrete systems obtained via \hboxtime-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 Ref. 6, Beyn and Lorenz \cite7 and Feckan Ref. 17, Ref. 19.

AMS subject classification.
Keywords. Discretization, invariant manifolds, Grobman-Hartman lemma

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