Nonlinear time-series analysis revisited

In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data---typically univariate---via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems

Forecasting high waters at Venice Lagoon using chaotic time series analisys and nonlinear neural netwoks

Time series analysis using nonlinear dynamics systems theory and multilayer neural networks models have been applied to the time sequence of water level data recorded every hour at 'Punta della Salute' from Venice Lagoon during the years 1980-1994. The first method is based on the reconstruction of the state space attractor using time delay embedding vectors and on the characterisation of invariant properties which define its dynamics. The results suggest the existence of a low dimensional chaotic attractor with a Lyapunov dimension, DL, of around 6.6 and a predictability between 8 and 13 hours ahead. Furthermore, once the attractor has been reconstructed it is possible to make predictions by mapping local-neighbourhood to local-neighbourhood in the reconstructed phase space. To compare the prediction results with another nonlinear method, two nonlinear autoregressive models (NAR) based on multilayer feedforward neural networks have been developed. From the study, it can be observed that nonlinear forecasting produces adequate results for the 'normal' dynamic behaviour of the water level of Venice Lagoon, outperforming linear algorithms, however, both methods fail to forecast the 'high water' phenomenon more than 2-3 hours ahead.Publicad

A dynamical systems approach to the discrimination of the modes of operation of cryptographic systems

Evidence of signatures associated with cryptographic modes of operation is established. Motivated by some analogies between cryptographic and dynamical systems, in particular with chaos theory, we propose an algorithm based on Lyapunov exponents of discrete dynamical systems to estimate the divergence among ciphertexts as the encryption algorithm is applied iteratively. The results allow to distinguish among six modes of operation, namely ECB, CBC, OFB, CFB, CTR and PCBC using DES, IDEA, TEA and XTEA block ciphers of 64 bits, as well as AES, RC6, Twofish, Seed, Serpent and Camellia block ciphers of 128 bits. Furthermore, the proposed methodology enables a classification of modes of operation of cryptographic systems according to their strength.Comment: 14 pages, 10 figure

Stochastic Effects in Physical Systems

A tutorial review is given of some developments and applications of stochastic processes from the point of view of the practicioner physicist. The index is the following: 1.- Introduction 2.- Stochastic Processes 3.- Transient Stochastic Dynamics 4.- Noise in Dynamical Systems 5.- Noise Effects in Spatially Extended Systems 6.- Fluctuations, Phase Transitions and Noise-Induced Transitions.Comment: 93 pages, 36 figures, LaTeX. To appear in Instabilities and Nonequilibrium Structures VI, E. Tirapegui and W. Zeller,eds. Kluwer Academi