PhD Thesis, University of Potsdam (2009)

Deciphering Dynamics Through Recurrences

C. Komalapriya

Since its introduction in 1890, by Henri Poincar´, the notion of recurrence of a system has become essential and very popular in the dynamical systems theory. Particularly, in the recent years, the concept of recurrence matrices and recurrence plots have proved to be highly useful for the analysis of a wide range of dynamical models and real world systems. The main part of the thesis is devoted to address one of the highly relevant problem of data analysis – the short or missing data sets. In such cases, deciphering the characteristics of the underlying system by using any conventional time series analysis techniques might not be possible, as many of these techniques, including recurrence plots, require temporally continuous data sets. In this thesis an automated algorithm is proposed based on the concept of recurrences, to overcome the important problem of short data sets or missing values. The method generates long artificial phase space trajectories – called Dynamically Reconstructed Trajectories (DRTs) – from a collection of short data sets. These dynamical replicants can then be easily characterised by the traditional time series analysis methods to extract the required dynamical details. More importantly, the DRTs generated by the algorithm reproduces the short and the long term dynamics of an underlying systems rather closely. The technique in its present form can be successfully applied to an ensemble of short trajectories, portions of which belong to different, but well separated, basins of attraction or to sharply divided phase space. Furthermore, the generality of the approach allows us to numerically characterise the properties of the chaotic saddles by generating an artificial long trajectory from a given aggregate of transiently chaotic trajectories. Moreover, the necessary procedures for the analysis of short scalar data sets are advocated and they are validated with a chemical oscillator data. The method might also be also useful to detect the existence of chaotic synchronisation, when only short trajectories/time series are available, from two or more interacting systems for analysis. Furthermore, in this work, the potential of the recurrence plots based approaches is illustrated by employing it to high-resolution EEG recordings and climate data sets.

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