Topological considerations of an attractor based on temporal locality along its phase trajectories

Abstract

The nonlinear method of topological analysis of an attractor reconstructed from a chaotic time series (TS) is developed. Using temporal localization along phase trajectories (in contrast to spatial localization that occurs in most conventional methods), we show that the analysis of topological dynamics with respect to phase trajectories of the attractor at enlarging dimensionality can be reduced to successive statistical processing of the TS by means of its partition into segments with maximum overlap. This approach allows us to obtain the asymptotic measure of topological instability at changing an embedding dimension of the attractor that can be considered as the averaged characteristic of temporal evolution. The developed method provides a possibility to estimate a number of freedom degrees of the chaotic system and implement minimization of the embedding dimension m for its attractor. The algorithm allows to achieve the essential reduction of computation time and required experimental data in comparison with the most conventional algorithms of fractal analysis that allows the algorithm to be realized even for higher-dimensional cases ( m > 20 in the present paper).