Entropy in Dynamic Systems

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed

Color Image Encryption Algorithm Based on TD-ERCS System and Wavelet Neural Network

In order to solve the security problem of transmission image across public networks, a new image encryption algorithm based on TD-ERCS system and wavelet neural network is proposed in this paper. According to the permutation process and the binary XOR operation from the chaotic series by producing TD-ERCS system and wavelet neural network, it can achieve image encryption. This encryption algorithm is a reversible algorithm, and it can achieve original image in the rule inverse process of encryption algorithm. Finally, through computer simulation, the experiment results show that the new chaotic encryption algorithm based on TD-ERCS system and wavelet neural network is valid and has higher security

Doctor of Philosophy

dissertationThis dissertation consists of three independent essays on cyclical fluctuations of functional income distribution and effective demand in the Post-Keynesian tradition. The first essay investigates the longer run relationship between wage share and measurements of economic activity. Our main tools are wavelet covariance and multiresolution analysis. Results indicate that (1) Goodwin type cycles are observed even at longer run and (2) when considering smooth trends for periodicities longer than 32 years, a long Goodwin cycle seems to appear from the 1940s to mid-1990s that collapses afterwards. The second and third essays are related in the sense that they empirically investigate the possibility of strong internal dynamics in the business cycle model of effective demand and income distribution. Specifically, in the second essay, we study wage share and output gap in an univariate setting. Each time series is examined through chaos theory. The main tools are the nonlinear autoregressive neural network model, the dominant Lyapunov exponent, coefficient of determination, and local Lyapunov exponent. Results indicate that output gap might behave quasi-chaotically and wage share noisy-stable. Finally, the third essay inquires into the possibility of limit cycle in the two-dimensional model on wage share and output gap. For that, we use the multivariate nonlinear autoregressive neural network model. Our results indicate that limit cycle behavior describes well their dynamics and, furthermore, the instability is located on the wage share isocline. Chapters 1 through 3 open several questions that we hope further research will address. In Chapter 1, we conjecture that globalization plays a crucial role in the stagnation of the trends in the late 1990s. However, further research is required. Chapter 2 concludes that instability is rooted in the goods market dynamics rather than the distributive dynamics. Results on Chapter 3 indicate that the demand regime is stable and wage share locally unstable. This possibility remains largely unexplored both in the theoretical and empirical literature, and it creates a contradiction with the results found in Chapter 2. Further research is necessary on the robustness of the result, and possible mechanisms

A Fault Diagnosis System for Rotary Machinery Supported by Rolling Element Bearings

The failure of rolling element bearings is one of the foremost causes of breakdown in rotary machinery. So far, a variety of vibration-based techniques have been developed to monitor the condition of bearings; however, the role of vibration behavior is rarely considered in the proposed techniques. This thesis presents an analytical study of a healthy rotor-bearing system to gain an understanding of the different categories of bearing vibration. In this study, a two degree-of-freedom model is employed, where the contacts between the rolling elements and races are considered to be nonlinear springs. The analytical investigations confirm that the nature of the inner ring oscillation depends on the internal clearance. A fault-free bearing with a small backlash exhibits periodic behavior; however, bearings categorized as having normal clearance oscillate chaotically. The results from the numerical simulations agree with those from the experiments confirming bearing’s chaotic response at various rotational speeds. Bearing faults generate periodic impacts which affect the chaotic behavior. This effect manifests itself in the phase plane, Poincare map, and chaotic quantifiers such as the Lyapunov exponent, correlation dimension, and information entropy. These quantifiers serve as useful indices for detecting bearing defects. To compare the sensitivity and robustness of chaotic indices with those of well-accepted fault detection techniques, a comprehensive investigation is conducted. The test results demonstrate that the Correlation Dimension (CD), Normalized Information Entropy (NIE), and a proposed time-frequency index, the Maximum Approximate Coefficient of Wavelet transform (MACW), are the most reliable fault indicators. A neuro-fuzzy diagnosis system is then developed, where the strength of the aforementioned indices are integrated to provide a more robust assessment of a bearing’s health condition. Moreover, a prognosis scheme, based on the Adaptive Neuro Fuzzy Inference System (ANFIS), in combination with a set of logical rules, is proposed for estimating the next state of a bearing’s condition. Experimental results confirm the viability of forecasting health condition under different speeds and loads

Entropy Measures in Machine Fault Diagnosis: Insights and Applications

Entropy, as a complexity measure, has been widely applied for time series analysis. One preeminent example is the design of machine condition monitoring and industrial fault diagnostic systems. The occurrence of failures in a machine will typically lead to non-linear characteristics in the measurements, caused by instantaneous variations, which can increase the complexity in the system response. Entropy measures are suitable to quantify such dynamic changes in the underlying process, distinguishing between different system conditions. However, notions of entropy are defined differently in various contexts (e.g., information theory and dynamical systems theory), which may confound researchers in the applied sciences. In this paper, we have systematically reviewed the theoretical development of some fundamental entropy measures and clarified the relations among them. Then, typical entropy-based applications of machine fault diagnostic systems are summarized. Further, insights into possible applications of the entropy measures are explained, as to where and how these measures can be useful towards future data-driven fault diagnosis methodologies. Finally, potential research trends in this area are discussed, with the intent of improving online entropy estimation and expanding its applicability to a wider range of intelligent fault diagnostic systems

Fractal Analysis and Chaos in Geosciences

The fractal analysis is becoming a very useful tool to process obtained data from chaotic systems in geosciences. It can be used to resolve many ambiguities in this domain. This book contains eight chapters showing the recent applications of the fractal/mutifractal analysis in geosciences. Two chapters are devoted to applications of the fractal analysis in climatology, two of them to data of cosmic and solar geomagnetic data from observatories. Four chapters of the book contain some applications of the (multi-) fractal analysis in exploration geophysics. I believe that the current book is an important source for researchers and students from universities

Dynamic non-linear system modelling using wavelet-based soft computing techniques

The enormous number of complex systems results in the necessity of high-level and cost-efficient modelling structures for the operators and system designers. Model-based approaches offer a very challenging way to integrate a priori knowledge into the procedure. Soft computing based models in particular, can successfully be applied in cases of highly nonlinear problems. A further reason for dealing with so called soft computational model based techniques is that in real-world cases, many times only partial, uncertain and/or inaccurate data is available. Wavelet-Based soft computing techniques are considered, as one of the latest trends in system identification/modelling. This thesis provides a comprehensive synopsis of the main wavelet-based approaches to model the non-linear dynamical systems in real world problems in conjunction with possible twists and novelties aiming for more accurate and less complex modelling structure. Initially, an on-line structure and parameter design has been considered in an adaptive Neuro- Fuzzy (NF) scheme. The problem of redundant membership functions and consequently fuzzy rules is circumvented by applying an adaptive structure. The growth of a special type of Fungus (Monascus ruber van Tieghem) is examined against several other approaches for further justification of the proposed methodology. By extending the line of research, two Morlet Wavelet Neural Network (WNN) structures have been introduced. Increasing the accuracy and decreasing the computational cost are both the primary targets of proposed novelties. Modifying the synoptic weights by replacing them with Linear Combination Weights (LCW) and also imposing a Hybrid Learning Algorithm (HLA) comprising of Gradient Descent (GD) and Recursive Least Square (RLS), are the tools utilised for the above challenges. These two models differ from the point of view of structure while they share the same HLA scheme. The second approach contains an additional Multiplication layer, plus its hidden layer contains several sub-WNNs for each input dimension. The practical superiority of these extensions is demonstrated by simulation and experimental results on real non-linear dynamic system; Listeria Monocytogenes survival curves in Ultra-High Temperature (UHT) whole milk, and consolidated with comprehensive comparison with other suggested schemes. At the next stage, the extended clustering-based fuzzy version of the proposed WNN schemes, is presented as the ultimate structure in this thesis. The proposed Fuzzy Wavelet Neural network (FWNN) benefitted from Gaussian Mixture Models (GMMs) clustering feature, updated by a modified Expectation-Maximization (EM) algorithm. One of the main aims of this thesis is to illustrate how the GMM-EM scheme could be used not only for detecting useful knowledge from the data by building accurate regression, but also for the identification of complex systems. The structure of FWNN is based on the basis of fuzzy rules including wavelet functions in the consequent parts of rules. In order to improve the function approximation accuracy and general capability of the FWNN system, an efficient hybrid learning approach is used to adjust the parameters of dilation, translation, weights, and membership. Extended Kalman Filter (EKF) is employed for wavelet parameters adjustment together with Weighted Least Square (WLS) which is dedicated for the Linear Combination Weights fine-tuning. The results of a real-world application of Short Time Load Forecasting (STLF) further re-enforced the plausibility of the above technique

Thalamocortical Oscillations in Sleep and Anaesthesia

The last 20 years have seen a substantial advancement in the understanding of the molecular targets of general anaesthetics however the neural mechanisms involved in causing loss of consciousness remain poorly understood. Thalamocortical oscillations are present in natural sleep and are induced by many general anaesthetics suggesting that modulation of this reciprocal system may be involved in the regulation of consciousness. Dynamic changes of thalamocortical oscillations in natural sleep and anaesthesia were investigated in rats chronically implanted with skull screw and depth electrodes in the cortex and thalamus. The hypothesis that discrete areas within the thalamus are responsible for regulation of arousal was tested. The anaesthetics propofol and dexmedetomidine but not midazolam produced switches in delta frequency at loss of righting reflex (LORR). This switch in frequency mirrored that seen within non-rapid eye movement sleep (NREM), whereas the onset of NREM was characterized by a switch from theta to delta in the EEG. Depth recordings during NREM indicated that the switch into a NREM state occurred in the central medial thalamus (CMT) significantly before the cingulate, barrel cortex and ventrobasal nucleus (VB), and that the CMT switch corresponded to the switch seen in the global EEG. Dexmedetomidine hypnosis showed a delta frequency shift that occurred simultaneously within the thalamus and cortex, and furthermore that the thalamus exhibited phase advancement over the cortex at the point of LORR. In conclusion, globalised changes within the thalamocortical system occur for propofol and dexmedetomidine LORR in the rat. This change represents a transition within drug free NREM and may implicate a common pathway responsible for a decrease in arousal. Furthermore, the phase advancement of the intralaminar thalamus over the cortex at LORR suggests a crucial role for this part of the thalamocortical system for regulating consciousness

Imaging the spatial-temporal neuronal dynamics using dynamic causal modelling

Oscillatory brain activity is a ubiquitous feature of neuronal dynamics and the synchronous discharge of neurons is believed to facilitate integration both within functionally segregated brain areas and between areas engaged by the same task. There is growing interest in investigating the neural oscillatory networks in vivo. The aims of this thesis are to (1) develop an advanced method, Dynamic Causal Modelling for Induced Responses (DCM for IR), for modelling the brain network functions and (2) apply it to exploit the nonlinear coupling in the motor system during hand grips and the functional asymmetries during face perception. DCM for IR models the time-varying power over a range of frequencies of coupled electromagnetic sources. The model parameters encode coupling strength among areas and allows the differentiations between linear (within frequency) and nonlinear (between-frequency) coupling. I applied DCM for IR to show that, during hand grips, the nonlinear interactions among neuronal sources in motor system are essential while intrinsic coupling (within source) is very likely to be linear. Furthermore, the normal aging process alters both the network architecture and the frequency contents in the motor network. I then use the bilinear form of DCM for IR to model the experimental manipulations as the modulatory effects. I use MEG data to demonstrate functional asymmetries between forward and backward connections during face perception: Specifically, high (gamma) frequencies in higher cortical areas suppressed low (alpha) frequencies in lower areas. This finding provides direct evidence for functional asymmetries that is consistent with anatomical and physiological evidence from animal studies. Lastly, I generalize the bilinear form of DCM for IR to dissociate the induced responses from evoked ones in terms of their functional role. The backward modulatory effect is expressed as induced, but not evoked responses