Search

The coronavirus disease 2019 (COVID-19) pandemic has fatalized 216 countries across the world and has claimed the lives of millions of people globally. Researches are being carried out worldwide by scientists to understand the nature of this catastrophic virus and find a potential vaccine for it. The most possible efforts have been taken to present this paper as a form of contribution to the understanding of this lethal virus in the first and second wave. This paper presents a unique technique for the methodical comparison of disastrous virus dissemination in two waves amid five most infested countries and the death rate of the virus in order to attain a clear view on the behaviour of the spread of the disease. For this study, the data set of the number of deaths per day and the number of infected cases per day of the most affected countries, the USA, Brazil, Russia, India, and the UK, have been considered in the first and second waves. The correlation fractal dimension has been estimated for the prescribed data sets of COVID-19, and the rate of death has been compared based on the correlation fractal dimension estimate curve. The statistical tool, analysis of variance, has also been used to support the performance of the proposed method. Further, the prediction of the daily death rate has been demonstrated through the autoregressive moving average model. In addition, this study also emphasis a feasible reconstruction of the death rate based on the fractal interpolation function. Subsequently, the normal probability plot is portrayed for the original data and the predicted data, derived through the fractal interpolation function to estimate the accuracy of the prediction. Finally, this paper neatly summarized with the comparison and prediction of epidemic curve of the first and second waves of COVID-19 pandemic to visualize the transmission rate in the both times.

Initial applications of the ideas of chaos theory in hydrology were on rainfall data. Early studies essentially addressed the identification and prediction of chaotic behavior of rainfall data. Encouraging outcomes from these studies subsequently led to investigations on the chaotic nature of scaling relationships in rainfall and disaggregation of data, including development of a new chaotic approach for rainfall disaggregation. More recently, some studies have examined the spatial variability and classification of rainfall. In addition to these, a number of studies have also addressed the important methodological and data issues in the applications of chaos methods to rainfall data. This chapter presents a review of chaos studies on rainfall data. The presentation is organized into three parts to address three important problems associated with rainfall: identification and prediction of chaos, scaling and disaggregation, and spatial variability and classification. An example is presented for each of these to demonstrate the utility and effectiveness of chaos concepts and methods to study these different problems.

Despite the tremendous growth in the applications of chaos theory in hydrology, there have been lingering criticisms. These criticisms have been based on the fundamental assumptions involved in the development of methods for identification and prediction of chaos (e.g. infinite and noise-free time series, lack of clear-cut guidelines on the selection of parameters involved) and/or the limitations of hydrologic data (e.g. short and noisy data, presence of zeros). A number of issues have been raised in this regard, but some have attracted far more attention than the others. This chapter presents a review of studies that have addressed such issues in chaos studies in hydrology. The review mainly focuses on four major issues: selection of an optimum delay time for phase space reconstruction, minimum data size for correlation dimension estimation, effects of data noise, and influence of the presence of zeros in data. Examples are also provided to illustrate how these issues have been addressed to gain more confidence in the applications of the methods and in the interpretation of the outcomes.

Considerable interest in studying the chaotic behavior of natural, physical, and socio-economic systems have led to the development of many different methods for identification and prediction of chaos. An important commonality among almost all of these methods is the concept of phase space reconstruction. Other than this, the methods largely have different bases and approaches and often aim to identify different measures of chaos. All these methods have been successfully applied in many different scientific fields. This chapter describes some of the most popular methods for chaos identification and prediction, especially those that have found applications in hydrology. These methods include: phase space reconstruction, correlation dimension method, false nearest neighbor method, Lyapunov exponent method, Kolmogorov entropy method, surrogate data method, Poincare maps, close returns plot, and nonlinear local approximation prediction method. To put the utility of these methods in a proper perspective in the identification of chaos, the superiority of two of these methods (phase space reconstruction and correlation dimension) over two commonly used linear tools for system identification (autocorrelation function and power spectrum) is also demonstrated. Further, as the correlation dimension method has been the most widely used method for chaos identification, it is discussed in far more detail.

Chaos theory has found widespread applications in studies on river flow data. Indeed, river flow is the most studied data in the context of chaos studies in hydrology. Early applications mainly focused on identification and prediction of chaotic behavior in river flow dynamics. Later years witnessed studies on a wide range of problems associated with river flow, including scaling and disaggregation, missing data estimation, reconstruction of system equations, multivariable analysis, and spatial variability and classification. Many studies have also addressed the important issues in the applications of chaos theory to river flow data, including data size, data noise, and selection of parameters involved in chaos identification and prediction methods. This chapter presents a review of applications of chaos theory to river flow data. The studies are roughly grouped into three categories to represent the following aspects: identification and prediction, scaling and disaggregation, and spatial variability and classification. These applications are also illustrated through examples.

Some researchers have suggested that the hydro-climatic process is a complex system, with nonlinearity as its basic characteristic. But there is still a lack of effective means available to thoroughly discover the dynamics of hydro-climatic process at different time scales. Therefore, more studies are required to explore the nonlinearity of hydro-climatic process from different perspectives and using different methods. Based on the hydrologic and meteorological data in the areas of the Tarim headwaters, this chapter investigated the nonlinear hydro-climatic process by a comprehensive method including correlation dimension, R/S analysis, wavelet analysis, regression and artificial neural network modeling. The main findings are as follows

Multifractal analysis is the well developed theory in the non-linear analysis of chaotic signals. Quantification of chaotic nature and complexity of the waveforms requires estimation of the Generalized Fractal Dimensions (GFD) spectrum where the complexity means higher variability in general fractal dimension spectrum. The focal theme of this paper is to develop a fuzzy multifractal theory to define the Fuzzy Generalized Fractal Dimensions (F-GFD) by introducing fuzzy membership function in classical Generalized Fractal Dimensions method. It was shown that, the designed Fuzzy GFD method accurately classifies the complexity of the chaotic waveforms such as Weierstrass functions by comparing graphically with the classical GFD method. Hence the fuzzy multifractal analysis performs significantly than the classical multifractal analysis and also the proposed Fuzzy GFD is a generalized method of the classical GFD.

We have recently proposed an algorithmic scheme [22] for the non-subjective computation of correlation dimension from time series data. Here it is extended for the computation of generalized dimensions and the multifractal spectrum and applied to a number of EEG and ECG data sets from normal as well as certain pathological states of the brain and the cardiac system. Comparisons are drawn using a standard low dimensional chaotic system. Our method has the advantage that the analysis is done under identical prescriptions built into the algorithm and hence the comparison of resulting indices becomes non-subjective. This also enables a quantitative characterization of the relative complexity between practical time series such as, those corresponding to the changes in the physiological states of the same system, from the view point of underlying dynamics.

There were great expectations in the 1980s in connection with the practical applications of mathematical processes which were built mainly upon fractal dimension mathematical basis. Results were achieved in the first times in several fields: examination of material structure, simulation of chaotic phenomena (earthquake, tornado), modelling real processes with the hilp of information technology equipment, the definition of length of revers or riverbanks. Significant results were achieved in practical applications later in the fields of informationtechnology, certain image processing areas. data compression, and computer classification. In the present publication the so far well known algorithms calculating fractal dimension in a simple way will be introduced as well sa the new mathematical concept named by th author ’ spectral fractal dimension’ [8] the algorithm derived from this concept and the possiblities of their practical usage.

This paper presents a new method to determine the fractal dimension of plane curves. Two main problems are described, namely the direct and the inverse problem. It is shown that the periods of harmonic oscillators obtained by folding wire like structures with the same geometry as the terms of a fractal sequence provide the necessary information to determine the fractal char- acteristics of that sequence. More interesting is the determination of the fractal properties of a given curve from a sequence of samples cut off the original curve. Building up harmonic oscillators with these samples it is shown that fractal characteristics can be identified. The dynamical response depends on the initial conditions providing complementary information about the fractal characteristics of the object. The last section deals with a simple random fractal and confirms the use of different initial conditions to characterize completely the fractal sequence.

Noisy signals obtained during a tonic-clonic epileptic seizure, are usually neglected for visual inspection by the physicians due to the presence of muscle artifacts. Although noise obscures completely the recording, information about the underlying brain activity can be obtained by filtering, through the Orthogonal Wavelet Transforms, those frequencies bands associated with muscle activity. After generating a “noise free” signal by removing the muscle artifacts with wavelets, a dynamical analysis of the brain behavior will be performed by using nonlinear dynamics methods. The values for nonlinear metric invariants, like the correlation dimension and the maximum Lyapunov exponent, confirm that the brain dynamical behavior is more ordered during the epileptic seizure than pre-seizure stage.

It is now well established that the irregularities in the light intensities of some variables are chaotic in origin due to the inherent nonlinearities in the underlying dynamics. By taking typical stars, we present an analysis of their light intensities using time series analysis emphasizing aspects of nonlinear dynamics. The fractal dimensions and other indices are calculated to establish the strength of chaos and the minimum modes involved in their pulsations. In CY Aqr we have found high dimensional chaos

The time dimension (temporal dimension) of computation models is studied. We argue that there is no time dimension associated with the traditional computation models such as Church’s /arada-calculus, Turing machines, and partial recursive functions. We define Iterative Turing Machines that have the time dimension. We also study the structures of time and their relation to computation models.

Local and global methods for the determination of fractal dimensions are applied to astrophysical time series. The analysis reveals that local dimension methods are better suitable for such kind of time series, which are non-stationary and which represent real-world systems. It is shown that local dimension methods can provide physical insights into the system even in cases in which pure determinism cannot be established.

This article reports nonlinear analysis of ECG R-R interval time-series obtained from healthy individuals and some cardiac patients. The R-R interval time-series data from 6 healthy individuals and 3 cardiac patients were transformed into multidimensional phase-space vectors by time-delay embedding. The largest Lyapunov exponent and correlation dimension (CD) were calculated. Nonlinearity was tested by comparing the CDs obtained from the original data with those obtained from surrogate data sets. Results are discussed with reference to results obtained in previous studies.

The correlation dimension D 2 is used to develop a method of classification for phase space orbits. D 2 depends only on the mutual distances of the orbit’s points, therefore the time development of the orbit is reflected in the time development of the correlation dimension approximants. It is shown, that this technique allows to investigate the dynamical properties of a phase space orbit, a classification of chaotic and regular orbits and a detection of sticky orbits.