Properties of an optimal linearization transformation

摘要

In this article, we develop properties of a transformation that was introduced by the authors in an earlier paper. This transformation enabled us to construct iterates which give, to any desired level of detail, the asymptotic behaviour of solutions of a certain class of nonlinear vector differential equations which approach a sink as time tends to infinity. First, we present smoothness results. Second, we show that this transformation is the optimal linearization transformation and satisfies the Hartman-Grobman conjugacy condition. Finally, we amend the aforementioned iterates to yield iterates which not only approximate solutions but are flows and yield higher-order conjugacy conditions. Examples are provided.