On the structures and quantification of recurrence plots

Recurrence plots (RPs) often have fascinating structures, especially when the embedding dimension is 1. We identify four basic patterns of a RP, namely, patterns along the main (45 degrees) diagonal, patterns along the 135 degrees diagonal, block-like structures, and square-like textures. We also study how the structures of and quantification statistics for RPs vary with the embedding parameters. By considering the distribution of the main diagonal line segments for chaotic systems, we relate some of the known statistics for the quantification of a RP to the Lyapunov exponent. This consideration enables us to introduce new ways of quantifying the diagonal line segments. Furthermore, we categorize recurrence points into two classes. A number of new quantities are identified which may be useful for the detection of nonstationarity in a time series, especially for the detection of a bifurcation sequence. A noisy transient Lorenz system is studied, to demonstrate how to identify a true bifurcation sequence, to interpret false bifurcation points, and to choose the embedding dimension. (C) 2000 Published by Elsevier Science B.V. All rights reserved.