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Abstract
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.
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Acknowledgements
Most of the material in this article can be found in M.S. Calder's PhD thesis Citation25 written under the supervision of D. Siegel. Moreover, we would like to thank the reviewers for their helpful comments and suggestions and also Pei Yu for a helpful discussion. Research by M.S. Calder was partially supported by an Ontario Graduate Scholarship. Research by D. Siegel was supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.