REALIZATION OF ORDINARY DIFFERENTIAL-EQUATIONS BY RETARDED FUNCTIONAL-DIFFERENTIAL EQUATIONS IN NEIGHBORHOODS OF EQUILIBRIUM POINTS

This paper addresses the realisation of ordinary differential equation s (ODEs) by retarded functional differential equations (FDEs) in finit e-dimensional invariant manifolds, locally around equilibrium points. A necessary and sufficient condition for realisability of C-1 vector f ields is established in terms of their linearisations at the equilibri um. It is also shown that any arbitrary finite jet of vector fields of ODEs can be realised without any further restrictions than those impo sed by the realisability of its linear term, a fact of relevance for d iscussing the flows defined by FDEs around singularities, and their bi furcations. Besides, it is proved that such a realisation can always b e achieved with FDEs whose nonlinearities are defined in terms of a fi nite number of delayed values of the solutions.