Realization of Vector Fields and Normal Forms

Abstract

The meaning of realization comes from the following observation. Since the flow for a RFDE evolves in an infinite dimensional space, for any given integers N and n, it is perhaps to be expected that, for any system of ordinary differential equations of dimension N, there is an RFDE of dimension n such that the flow for the RFDE can be mapped onto the flow for the ODE. Of course, if n ≥ N, this is the case. If n < N, we will observe below that there are flows defined by an N dimensional ODE which cannot be realized by an RFDE in ℝⁿ. We verify this observation by considering the restrictions imposed on the ODE which determines the flow on a center manifold.