Embedding in higher dimensions causes ambiguity indetermining equations from data

A linear system of dimension D can be embedded by D time-delay vectors. However, a nonlinear system often requires more vectors, almost always not more than 2D+1. This paper is regarding the consequence of such an embedding. We encountered this problem in an attempt to obtain a set of equations from the observed data. Our goal was to generate a system which is topologically conjugate to the original system. We found that many a times our efforts led to highly unstable systems. Now, we have an explanation. Embedding in higher dimensions causes a topological foliation and this foliation has a D-dimensional “leaf”. All the data were on this leaf and therefore we were unable to distinguish this foliated system from those where the foliation had collapsed. We illustrate the phenomenon by examples from numerically generated data and suggest possible remedies.