SCALAR OBSERVATIONS FROM A CLASS OF HIGH-DIMENSIONAL CHAOTIC SYSTEMS - LIMITATIONS OF THE TIME-DELAY EMBEDDING

The time delay embedding for the reconstruction of a state space from scalar data introduces strong folding of the smooth manifold in which a chaotic attractor is embedded, which is absent in some more natural state space. In order to observe the deterministic nature of data, the typical length scale related to this folding has to be resolved. Abov e this length scale the data appear to be random. For a particular mod el class we prove these statements and we derive analytically the depe ndence of this length scale on the complexity of the system. We show t hat the number of scalar observations required to observe determinism increases exponentially in the product of the system's entropy and dim ension. (C) 1997 American Institute of Physics.