Predicting Chaos Most of the Time from Embeddings with Self-Intersections

Embedding techniques for predicting chaotic time series from experimental data may fail if the reconstructed attractor self-intersects, and such intersections often occur unless the embedding dimension exceeds twice the attractor close-quote s box counting dimension. Here we consider embeddings with self-intersection. When the dimension M of the measurement space exceeds the information dimension D1 of the attractor, reliable prediction is found to be still possible from most orbit points. In particular, the fraction of state space measure from which prediction fails typically scales as var-epsilon M-D1 for small var-epsilon where var-epsilon is the diameter of the current state close-quote s neighborhood used for prediction. copyright 1998 The American Physical Society