Dynamics from time series: Iteration maps in correlation space

The attractor for the dynamics of a complex system can be constructed from the time series measurement of a single variable. A recently proposed procedure is to construct a covariance matrix using an embedding window on the time series. An analysis of the meaning of the eigenvalues of the covariance matrix of the time series is undertaken here. It is argued that each principal eigenvalue can be decomposed into components which describe the time evolution of the correlations of the system along the given principal direction. A one-dimensional iterative map of these components can be constructed in correlation space. Such a map displays the regular or chaotic nature of the dynamics for each principal direction of the attractor. Illustrative examples of such maps are constructed for regular and random time series and for the Lorenz attractor.