Field Of Nonlinear Time Series Analysis Research Articles

Signatures of chaotic and stochastic dynamics uncovered with ε -recurrence networks

An old and important problem in the field of nonlinear time-series analysis entails the distinction between chaotic and stochastic dynamics. Recently, ε -recurrence networks have been proposed as a tool to analyse the structural properties of a time series. In this paper, we propose the applicability of local and global ε -recurrence network measures to distinguish between chaotic and stochastic dynamics using paradigmatic model systems such as the Lorenz system, and the chaotic and hyper-chaotic Rössler system. We also demonstrate the effect of increasing levels of noise on these network measures and provide a real-world application of analysing electroencephalographic data comprising epileptic seizures. Our results show that both local and global ε -recurrence network measures are sensitive to the presence of unstable periodic orbits and other structural features associated with chaotic dynamics that are otherwise absent in stochastic dynamics. These network measures are still robust at high noise levels and short data lengths. Furthermore, ε -recurrence network analysis of the real-world epileptic data revealed the capability of these network measures in capturing dynamical transitions using short window sizes. ε -recurrence network analysis is a powerful method in uncovering the signatures of chaotic and stochastic dynamics based on the geometrical properties of time series.

A new algorithm for packet routing problems using chaotic neurodynamics and its surrogate analysis

We propose a new algorithm for routing packets which effectively avoids packet congestion in computer networks. The algorithm involves chaotic neurodynamics. To realize effective packet routing, we first composed a basic method by a neural network, which routes packets with shortest path information between two nodes in the computer network. When the computer network has an irregular topology, the basic routing method does not work well, because most of packets cannot be transmitted to their destinations due to packet congestion in the computer network. To avoid such an undesirable problem, we employed chaotic neurodynamics to extend the basic method. Numerical experiments show that our proposed method exhibits good performance for scale-free networks. We also analyze why the proposed routing method is effective, comparing the proposed method with several stochastic methods. We introduced the method of surrogate data, a statistical hypothesis testing which is often used in the field of nonlinear time-series analysis. Analysis of the proposed method by the method of surrogate data reveals that the chaotic neurodynamics is most effective to decentralize the packet congestion in the computer network.

ON THE EVALUATION OF NOISE LEVELS FOR QUANTIZED DATA

In recent years, quantitative methods for evaluating chaotic properties have been developed in the field of nonlinear time-series analysis. The embedding theorem, which is a mathematical background for the methods, assumes an ideal situation in which noiseless time series are observed with infinite resolution and an infinite amount of data points. However, under real situations we cannot ignore two classes of noises which are included in really observed data. The first one is observational and dynamical noises which depend on internal nonlinear systems and performance of observational instruments. The second one is quantization error included in the time series since we usually use digital computers for applying the methods. In the present paper, we derive formulae for evaluating the levels of observational and quantization noises in the case of embedding observed time series in reconstructed state spaces. By measuring a distance between noiseless and noisy attractors, we also confirm that the derived formulae are appropriate for quantifying the noise included in the reconstructed attractor.

Structural Health Monitoring Through Chaotic Interrogation

The field of vibration based structural health monitoring involves extracting a ‘feature’ which robustly quantifies damage induced changes to the structure in the presence of ambient variation, that is, changes in ambient temperature, varying moisture levels, etc. In this paper, we present an attractor-based feature derived from the field of nonlinear time-series analysis. Emphasis is placed on the use of chaos for the purposes of system interrogation. The structure is excited with the output of a chaotic oscillator providing a deterministic (low-dimensional) input. Use is made of the Kaplan–Yorke conjecture in order to ‘tune’ the Lyapunov exponents of the driving signal so that varying degrees of damage in the structure will alter the state space properties of the response attractor. The average local attractor variance ratio (ALAVR) is suggested as one possible means of quantifying the state space changes. Finite element results are presented for a thin aluminum cantilever beam subject to increasing damage, as specified by weld line separation, at the clamped end. Comparisons of the ALAVR to two modal features are evaluated through the use of a performance metric.