Choosing the Dimension of Reconstructed Phase Space

Abstract

We wish to determine the integer global dimension where we have the necessary number of coordinates to unfold observed orbits from self overlaps arising from projection of the attractor to a lower dimensional space. For this we go into the data set and ask when the unwanted overlaps occur. The lowest dimension which unfolds the attractor so that none of these overlaps remains is called the embedding dimension d _E . d _E is an integer. If we measure two quantities s _A (n) and s _B (n) from the same system, there is no guarantee that the global dimension d _E for from each of these is the same. Each measurement along with its timelags provides a different nonlinear combination of the original dynamical variables and can provide a different global nonlinear mapping of the true space x(n) into a reconstructed space of dimension d _E where smoothness and uniqueness of the trajectories is preserved. Recall that d _E is a global dimension and may well be different from the local dimension of the underlying dynamics.