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Realisation of ordinary differential equations by retarded functional differential equations in neighbourhoods of equilibrium points*

Published online by Cambridge University Press:  14 November 2011

Teresa Faria  and
Luis T. Magalhães
Teresa Faria
Affiliation:
Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1700 Lisboa, Portugal; CAPS, Complexo Interdisciplinar I do INIC, 1096 Lisboa Codex, Portugal
Luis T. Magalhães
Affiliation:
Departamento de Matemática, IST, Universidade Técnica de Lisboa, 1096 Lisboa Codex, Portugal; Complexo Interdisciplinar I do INIC, 1096 Lisboa Codex, Portugal
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Abstract

This paper addresses the realisation of ordinary differential equations (ODEs) by retarded functional differential equations (FDEs) in finite-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C1 vector fields is established in terms of their linearisations at the equilibrium.

It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those imposed by the realisability of its linear term, a fact of relevance for discussing the flows defined by FDEs around singularities, and their bifurcations. Besides, it is proved that such a realisation can always be achieved with FDEs whose nonlinearities are defined in terms of a finite number of delayed values of the solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 4 , 1995 , pp. 759 - 776
Copyright
Copyright © Royal Society of Edinburgh 1995

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