Statistical Mechanics and Information-Theoretic Perspectives on Complexity in the Earth System

Peer reviewedPublisher PD

Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects

We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation in $R^{3}$ phase space. We demonstrate that it accommodates the phase space dynamics of low dimensional dissipative systems such as the much studied Lorenz and R\"{o}ssler Strange attractors, as well as the more recent constructions of Chen and Leipnik-Newton. The rotational, volume preserving part of the flow preserves in time a family of two intersecting surfaces, the so called {\em Nambu Hamiltonians}. They foliate the entire phase space and are, in turn, deformed in time by Dissipation which represents their irrotational part of the flow. It is given by the gradient of a scalar function and is responsible for the emergence of the Strange Attractors. Based on our recent work on Quantum Nambu Mechanics, we provide an explicit quantization of the Lorenz attractor through the introduction of Non-commutative phase space coordinates as Hermitian $N \times N$ matrices in $R^{3}$. They satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction it violates the quantum commutation relations. We demonstrate that the Heisenberg-Nambu evolution equations for the Quantum Lorenz system give rise to an attracting ellipsoid in the $3 N^{2}$ dimensional phase space.Comment: 35 pages, 4 figures, LaTe

Readings in the 'New Science': a selective annotated bilbiography

Die vorliegende kommentierte Bibliographie will hauptsächlich Historikern eine Orientierungshilfe für die Literaturfülle zum Thema 'New Science' geben. Die knapp besprochenen Arbeiten sind nach folgenden Themenkomplexen gruppiert: Unentscheidbarkeit, Ungewißheit und Komplexität; Makrostrukturen: Systeme und die humane Dimension; Dynamische Systeme (Spieltheorie, Katastrophentheorie, Chaos, Fraktale Geometrie, Antizipatorische Systeme, Lebende Systeme); Computer (Informationstheorie, Kognitionswisssenschaft und Künstliche Intelligenz); Die Mikro- und die Makrodimensionen; Zeit; Kultur und Erkenntnistheorie. (pmb

Super Unification of Physics and Mathematics

This short letter proposes to convert physics and mathematics not to classical mathematical physics but to a far more radically different entity. We call this new logical product "wild topology" which is also known  as a general kind of topology ramifying into a Cantor set. Thus the work goes far more beyond our older P-Adicunification of physics and mathematics. In the course of this process, we enhance both understanding as well as computation of not only classical physics but more importantly also quantum physics and cosmology. In particular, we free quantum mechanics from several of the paradoxes and counter intuitive features which has historically plagued it since its very inception

Forecasting chaotic systems : the role of local Lyapunov exponents

We propose a novel methodology for forecasting chaotic systems which is based on the nearest-neighbor predictor and improves upon it by incorporating local Lyapunov exponents to correct for its inevitable bias. Using simulated data, we show that gains in prediction accuracy can be substantial. The general intuition behind the proposed method can readily be applied to other non-parametric predictors.Chaos theory, Lyapunov exponent, logistic map, Monte Carlo simulations.

Non-linear dynamical analysis of biosignals

Biosignals are physiological signals that are recorded from various parts of the body. Some of the major biosignals are electromyograms (EMG), electroencephalograms (EEG) and electrocardiograms (ECG). These signals are of great clinical and diagnostic importance, and are analysed to understand their behaviour and to extract maximum information from them. However, they tend to be random and unpredictable in nature (non-linear). Conventional linear methods of analysis are insufficient. Hence, analysis using non-linear and dynamical system theory, chaos theory and fractal dimensions, is proving to be very beneficial. In this project, ECG signals are of interest. Changes in the normal rhythm of a human heart may result in different cardiac arrhythmias, which may be fatal or cause irreparable damage to the heart when sustained over long periods of time. Hence the ability to identify arrhythmias from ECG recordings is of importance for clinical diagnosis and treatment and also for understanding the electrophysiological mechanism of arrhythmias. To achieve this aim, algorithms were developed with the help of MATLAB® software. The classical logic of correlation was used in the development of algorithms to place signals into the various categories of cardiac arrhythmias. A sample set of 35 known ECG signals were obtained from the Physionet website for testing purposes. Later, 5 unknown ECG signals were used to determine the efficiency of the algorithms. A peak detection algorithm was written to detect the QRS complex. This complex is the most prominent waveform within an ECG signal and its shape, duration and time of occurrence provides valuable information about the current state of the heart. The peak detection algorithm gave excellent results with very good accuracy for all the downloaded ECG signals, and was developed using classical linear techniques. Later, a peak detection algorithm using the discrete wavelet transform (DWT) was implemented. This code was developed using nonlinear techniques and was amenable for implementation. Also, the time required for execution was reduced, making this code ideal for real-time processing. Finally, algorithms were developed to calculate the Kolmogorov complexity and Lyapunov exponent, which are nonlinear descriptors and enable the randomness and chaotic nature of ECG signals to be estimated. These measures of randomness and chaotic nature enable us to apply correct interrogative methods to the signal to extract maximum information. The codes developed gave fair results. It was possible to differentiate between normal ECGs and ECGs with ventricular fibrillation. The results show that the Kolmogorov complexity measure increases with an increase in pathology, approximately 12.90 for normal ECGs and increasing to 13.87 to 14.39 for ECGs with ventricular fibrillation and ventricular tachycardia. Similar results were obtained for Lyapunov exponent measures with a notable difference between normal ECG (0 – 0.0095) and ECG with ventricular fibrillation (0.1114 – 0.1799). However, it was difficult to differentiate between different types of arrhythmias.Biosignals are physiological signals that are recorded from various parts of the body. Some of the major biosignals are electromyograms (EMG), electroencephalograms (EEG) and electrocardiograms (ECG). These signals are of great clinical and diagnostic importance, and are analysed to understand their behaviour and to extract maximum information from them. However, they tend to be random and unpredictable in nature (non-linear). Conventional linear methods of analysis are insufficient. Hence, analysis using non-linear and dynamical system theory, chaos theory and fractal dimensions, is proving to be very beneficial. In this project, ECG signals are of interest. Changes in the normal rhythm of a human heart may result in different cardiac arrhythmias, which may be fatal or cause irreparable damage to the heart when sustained over long periods of time. Hence the ability to identify arrhythmias from ECG recordings is of importance for clinical diagnosis and treatment and also for understanding the electrophysiological mechanism of arrhythmias. To achieve this aim, algorithms were developed with the help of MATLAB® software. The classical logic of correlation was used in the development of algorithms to place signals into the various categories of cardiac arrhythmias. A sample set of 35 known ECG signals were obtained from the Physionet website for testing purposes. Later, 5 unknown ECG signals were used to determine the efficiency of the algorithms. A peak detection algorithm was written to detect the QRS complex. This complex is the most prominent waveform within an ECG signal and its shape, duration and time of occurrence provides valuable information about the current state of the heart. The peak detection algorithm gave excellent results with very good accuracy for all the downloaded ECG signals, and was developed using classical linear techniques. Later, a peak detection algorithm using the discrete wavelet transform (DWT) was implemented. This code was developed using nonlinear techniques and was amenable for implementation. Also, the time required for execution was reduced, making this code ideal for real-time processing. Finally, algorithms were developed to calculate the Kolmogorov complexity and Lyapunov exponent, which are nonlinear descriptors and enable the randomness and chaotic nature of ECG signals to be estimated. These measures of randomness and chaotic nature enable us to apply correct interrogative methods to the signal to extract maximum information. The codes developed gave fair results. It was possible to differentiate between normal ECGs and ECGs with ventricular fibrillation. The results show that the Kolmogorov complexity measure increases with an increase in pathology, approximately 12.90 for normal ECGs and increasing to 13.87 to 14.39 for ECGs with ventricular fibrillation and ventricular tachycardia. Similar results were obtained for Lyapunov exponent measures with a notable difference between normal ECG (0 – 0.0095) and ECG with ventricular fibrillation (0.1114 – 0.1799). However, it was difficult to differentiate between different types of arrhythmias