Development of new analytical techniques for chaotic time series

In the present thesis two main results are presented. The first is a study of the statistical properties of the finite-time maximum Lyapunov exponent determined out of a time series by using the divergent rate method. To reach this goal, we developed a new, completely automatic algorithm based on the method developed by Gao and Zheng. A main achievement of this part of the work is the interpretation of the uncertainty in the light of the work by Grassberger, Badii e Politi of 1988 on the theoretical distribution of maximum Lyapunov exponents. We showed that the analysis and identification of clusters in diagrams representing uncertainty vs. maximum Lyapunov exponent can provide useful information about the optimal choice of the embedding parameters. In addition, our results allow us to identify systems that can provide suitable benchmarks for the comparison and ranking of different embedding methods. The second main result concerns the development of a new method for the assessment of the optimal embedding parameters. Our method is based on two assumptions: a potential-like quantity is defined on the lattice of points that characterize the embedding; the optimal embedding choice coincides with local extrema (maxima or minima) of this potential. Throughout the work, we used "synthetic" time series generated by numerically integrating the difference and differential equations that describe the following dynamical systems: the Hénon map, the Lorenz attractor, the Rössler attractor and the Mackey-Glass attractor. These four systems are widely used as references in the scientific literature. In the last part of the work, we have started to examine EEG recordings by using the techniques developed in the main part of the work. The EEG recordings are sampled on healthy subjects in resting-state. These investigations are still at a starting phase.