On structural stability of ordinary differential equations with respect to discretization methods (Q1923317)

Consider the Morse-Smale differential system with no periodic orbits \((*)\) \(x'=f(x)\), \(x\in{\mathcal N}\), where \(\mathcal N\) is an open neighbourhood of the unit disc \(D^m\) in \(\mathbb R^m\). It is shown that such systems are correctly reproduced by one-step integration schemes in the following sense: for \(p\)th order methods with stepsize \(h\) the induced local solution flow for \((*)\) and the corresponding discretized system are joined by a homeomorphism \({\mathcal H}_h\), defined on \(D^m\), which is \(O(h^p)\) close to the identity. A similar result had been proved by the author for \(m=2\) by a geometric argument which does not apply to higher dimensions. In this paper, he uses a differential-topological approach constructing the homeomorphism \({\mathcal H}_h\) has the (unique) solution of an appropriate functional equation. The paper is a contribution to the problem which may be formulated as ``When are hyperbolic/transversal structures preserved under discretization schemes?''.