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A Direct Numerical Simulation (DNS) database of statistically planar flames ranging from the wrinkled flamelets to the thin reaction zones regime and DNS data for a Bunsen premixed flame representing the wrinkled flamelets regime have been utilised to evaluate the fractal dimensions of flame surfaces using the filtering dimension method, the box-counting algorithm and the correlation dimension approach. The fractal dimension evaluated based on the fully resolved three-dimensional data has been found to be reasonably approximated by adding unity to the equivalent fractal dimension evaluated based on two-dimensional projections irrespective of the methodology of extracting fractal dimension. This indicates that the flame surface can be approximated as a self-similar fractal surface for the range of Karlovitz and Damköhler numbers considered here. While all methods, provide results identical to each other for benchmark problems, it has been found that the fractal dimension evaluation based on box-counting method provides almost identical results as that obtained using the filtering dimension method for both three and two dimensions, while the fractal dimensions based on the correlation dimension tend to be slightly smaller. The findings of the current analysis have the potential to be used to reliably estimate the actual fractal dimension in 3D based on experimentally obtained 2D binarised reaction progress variable field. The inner cut-off scales estimated based on all three methodologies yield comparable results in terms of order of magnitude with the box-counting method predicting a smaller value of inner cut-off scale in comparison to other methods. The execution times for fractal dimension extraction based on filtering dimension and box-counting methodologies are found to be comparable but the correlation dimension method is found to be considerably faster than the two alternative approaches and provides results consistent with theoretical bounds in all cases. [ABSTRACT FROM AUTHOR]

Combining biomechanics and motor control, the aim of this study was to investigate the limit cycle dynamics during the high bar longswing across the UK elite gymnastics pathway age groupings. Senior, junior and development gymnasts (N = 30) performed three sets of eight consecutive longswings on the high bar. The centre of mass motion was examined through Poincaré plots and recurrence quantification analysis exploring the limit cycle dynamics of the longswing. Close to one-dimensional limit cycles were displayed for the senior (correlation dimension (CD) = 1.17 ±.08), junior (CD = 1.26 ±.08) and development gymnasts (CD = 1.33 ±.14). Senior elite gymnasts displayed increased recurrence characteristics in addition to longer longswing duration (p <.01) and lower radial angular velocity of the mass centre (p <.01). All groups of gymnasts had highly recurrent and predictable limit cycle characteristics. The findings of this research support the postulation that the further practice, experience and individual development associated with the senior gymnasts contribute to the refinement of the longswing from a nonlinear dynamics perspective. These findings support the idea of functional task decomposition informing the understanding of skill and influencing coaches' decisions around skill development and physical preparation. [ABSTRACT FROM AUTHOR]

This paper explores the fractal properties of token embedding spaces in GPT-2 language models by analyzing the stability of the correlation dimension, a measure of geometric complexity. Token embeddings represent words or subwords as vectors in a high-dimensional space. We hypothesize that the correlation dimension D 2 remains consistent across different vocabulary subsets, revealing fundamental structural characteristics of language representation in GPT-2. Our main objective is to quantify and analyze the stability of D 2 in these embedding subspaces, addressing the challenges posed by their high dimensionality. We introduce a new theorem formalizing this stability, stating that for any two sufficiently large random subsets S 1 , S 2 ⊂ E , the difference in their correlation dimensions is less than a small constant ε. We validate this theorem using the Grassberger–Procaccia algorithm for estimating D 2 , coupled with bootstrap sampling for statistical consistency. Our experiments on GPT-2 models of varying sizes demonstrate remarkable stability in D 2 across different subsets, with consistent mean values and small standard errors. We further investigate how the model size, embedding dimension, and network depth impact D 2 . Our findings reveal distinct patterns of D 2 progression through the network layers, contributing to a deeper understanding of the geometric properties of language model representations and informing new approaches in natural language processing. [ABSTRACT FROM AUTHOR]

The quantitative relationship between material microstructures, such as grain distributions, and the nonlinear strain-hardening behaviors of polycrystalline metals has not yet been completely understood. This study finds that the grain correlation dimension of polycrystals D is universally equal to the reciprocal of the strain-hardening exponent by experimental research and fractal geometry analysis. From a geometric perspective, the correlation dimension of grains is consistent with that of the equivalent plastic strain field, which represents the correlation dimension of the material manifold. According to the definition of the Hausdorff measure and Ludwik constitutive model, the strain-hardening exponent represents the exponent derived from the Dth root of the measure relationship. This universal law indicates that the strain-hardening behaviors are fractal geometrized and that the strain-hardening exponent represents a geometrical parameter reflecting the multifractal characteristics of grain structures. This conclusion can enhance the comprehension of the relationship between microstructure and mechanical properties of materials and highlights the importance of designing materials with non-uniform grain distributions to achieve desired hardening properties. [ABSTRACT FROM AUTHOR]

Wind energy is a clean, sustainable, and renewable source. It is receiving a large amount of attention from governments and energy companies worldwide as it plays a significant role as an alternative source of energy in reducing carbon emissions. However, due to long-term operation in reduced and difficult weather conditions, wind turbine blades are always seriously damaged. Hence, damage detection in blade structure is essential to evaluate its operational condition and ensure its structural integrity and safety. We aim to use fractal, entropy, and chaos concepts as descriptors for the diagnosis of wind turbine blade condition. They are, respectively, estimated by the correlation dimension, approximate entropy, and the Lyapunov exponent. Formal statistical tests are performed to check how they are different across wind turbine blade conditions. The experimental results follow. First, the correlation dimension is not able to distinguish between all conditions of wind turbine blades. Second, approximate entropy is suitable to distinguish between healthy and erosion conditions and between healthy and mass imbalance conditions. Third, chaos is not a discriminative feature to distinguish between wind turbine blade conditions. Fourth, wind turbine blades with either erosion or mass imbalance exhibit less irregularity in their respective signals than healthy wind turbine blades. [ABSTRACT FROM AUTHOR]

Land use/land cover (LULC) conversion and droughts impact river water quality. Since the river water quality is significant, it was attempted to investigate the effect of hydrological drought (HD) and LULC changes on the water quality parameters of the Helle river in Iran. All the time series (1969–2020) were analyzed using dynamic analysis applications. The results indicated that due to the alteration of LULC and HD patterns, water hardness and salinity were increased by 13.7% and 16.1%, respectively. Moreover, the water quality parameters concentration was intensified. However, the river water potential of hydrogen (pH) decreased by 0.4%. The number of random elements in the time series of the pH, bicarbonate, sulfate, magnesium, and sodium was increased noticeably, and their predictability and long-term memory were reduced. The total dissolved solids (TDS), electrical conductivity (EC), chloride, potassium, and calcium's time series became more chaotic, and their predictability and long-term memory were improved. All the time series faced an increase in the number of noises. The alteration of HD patterns induced a reduction of correlation between HD and sulfate, sodium, magnesium, bicarbonate, pH, and TDS concentration and an augmentation of correlation between HD and chloride, calcium, potassium, and salinity concentration. The Helle river water quality deteriorated due to the alteration of HD patterns and LULC. Future studies can address the strategies to improve the water quality of the Helle river. [ABSTRACT FROM AUTHOR]

Depression, a debilitating mental illness, profoundly impacts an individual's cognition, behaviour, and emotions. Despite efforts to quantify depression rates through electro-encephalo-gram (EEG) signal analysis, classification remains challenging due to inherent noise. This paper introduces a Bat based UNET signal analysis, aimed at accurately classifying depression rates using a normal EEG dataset. It comprises of pre-processing, feature extraction, feature selection, and classification stages. The framework excels at noise reduction during pre-processing, enhancing dataset integrity. Feature extraction leverages band power and correlation dimension to extract crucial features. Furthermore, feature selection optimizes classification accuracy by refining the fitness function of bats in the classification layer. Utilizing a standardized EEG dataset implemented in Matrix Laboratory (MATLAB), the proposed technique demonstrates superior performance compared to existing methods, as evidenced by metrics such as accuracy, area under the curve, precision, and recall (or sensitivity). This innovative framework represents a significant advancement in the classification of depression rates. [ABSTRACT FROM AUTHOR]

Sea surface temperature (SST), with its complex and dynamic behavior, is a major driver of ocean–atmosphere interactions. The purpose of this study is to investigate the behavior of SST and its prediction using a chaotic approach. Average mutual information (AMI) and Cao methods were used to reconstruct the phase space. The Lyapunov exponent and correlation dimension were used to investigate chaos. The Lyapunov exponent index was used to predict SST with a 5-year average prediction horizon using the local prediction method between 2023 and 2027. The results showed a 3-month delay time for the Pacific and Antarctic Oceans, and a 2-month delay time for the Atlantic, Indian, and Arctic Oceans. The optimal embedding dimension for all oceans is between 6 and 7. Our analysis reveals that the dynamics of SST in all oceans exhibit varying degrees of chaos, as indicated by the correlation dimension. The local prediction method achieves relatively accurate short-term SST predictions due to the clustering of SST points around specific attractors in the phase space. However, in the long term, the accuracy of this method decreases as the points in the phase space of SST can spread randomly. The model performance ranking with a Percent Mean Relative Absolute Error shows that the Indian Ocean has the best performance compared to other oceans, while the Atlantic, Pacific, and Antarctic and Arctic Oceans are in the next ranks. This study contributes to understanding the dynamics of SST and has practical value for use in the development of climate models. [ABSTRACT FROM AUTHOR]

The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based on fractal dimension (FD). The existing model selection methods lack interpretability and exhibit arbitrariness in choosing mode dimension truncation levels. To address these issues, this paper analyzes the geometric features of modes for the dimensional characteristics of dynamical systems. By calculating the box counting dimension (BCD) of modes and the correlation dimension (CD) and embedding dimension (ED) of the original dynamical system, it achieves guidance on the importance ranking of modes and the truncation order of modes in DMD. To validate the practicality of this method, it is applied to the reduction applications on the reconstruction of the velocity field of cylinder wake flow and the force field of compressor blades. Theoretical results demonstrate that the proposed selection technique can effectively characterize the primary dynamic features of the original dynamical systems. By employing a loss function to measure the accuracy of the reconstruction models, the computed results show that the overall errors of the reconstruction models are below 5%. These results indicate that this method, based on fractal theory, ensures the model's accuracy and significantly reduces the complexity of subsequent computations, exhibiting strong interpretability and practicality. [ABSTRACT FROM AUTHOR]

Automatic voice pathology detection enables an objective assessment of pathologies that influence the voice production strategy. By utilizing the conventional pipeline model as well as the modern deep learning-centric end-to-end methodology, numerous pathological voice analyzing techniques have been developed. The conventional methodology is still a valid choice owing to the lack of enormous amounts of training data in the study region of pathological voice. In the meantime, obtaining higher precision, higher accuracy, and stability is still a complicated task. Therefore, by amalgamating the nonlinear measure, the pathological voices are analyzed to abate such risks. The viability of six nonlinear discriminating measures derived from the phase space realm, involving healthy and pathological voice signals, is studied in this work. The analyzed parameters are Singularity spectrum coefficients ( α min , α max , γ 1 and γ 2 ). Correlation entropy at optimum embedding dimension ( K 2 m ) and correlation dimension at optimum embedding dimension ( D 2 m ). Analyzing the pathological voices with better accuracy rates is the major objective of the proposed methodology. Here, the Support Vector Machine (SVM) was utilized as the classifier. Experimentations were performed on VOiceICarfEDerico (VOICED) databases subsuming 208 healthy, as well as pathological voices, amongst these 50 samples, were utilized. Here, the model obtained 97% of accuracy with 99% as of the classifier with Gaussian kernel function. Therefore, to differentiate normal as well as pathological subjects, the six proposed characteristics are highly beneficial; in addition, they will be supportive in pathology diagnosis. [ABSTRACT FROM AUTHOR]

Copyright of Science Technology & Engineering is the property of Chinese Society of Technology Economics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This study examines the market efficiency in the prices and volumes of transactions of 41 cryptocurrencies. Specifically, the correlation dimension (CD), Lyapunov Exponent (LE), and approximate entropy (AE) were estimated before and during the COVID-19 pandemic. Then, we applied Student's t-test and F-test to check whether the estimated nonlinear features differ across periods. The empirical results show that (i) the COVID-19 pandemic has not affected the means of CD, LE, and AE in prices, (ii) the variances of CD, LE, and AE estimated from prices are different across pre-pandemic and during pandemic periods, and specifically (iii) the variance of CD decreased during the pandemic; however, the variance of LE and the variance of AE increased during the pandemic period. Furthermore, the pandemic has not affected all three features estimated from the volume series. Our findings suggest that investing in cryptocurrencies is advantageous during a pandemic because their prices become more regular and stable, and the latter has not affected the volume of transactions. [ABSTRACT FROM AUTHOR]

Copyright of Environmental & Social Management Journal / Revista de Gestão Social e Ambiental is the property of Environmental & Social Management Journal and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

These days, physical health concerns are gradually elevated to the same degree of significance as essential issues such as education, health, and protection as national living standards improve. Enhancing the physical health of university students and encouraging healthy growth are significant moments for China, due to which each university controls a physical examination every year. Utilizing wireless sensors for data capture in university students' physical fitness evaluation systems has played an essential role in various areas. Therefore, this paper presents a body domain network multimodal sensor and decision algorithm-based physical health assessment system for university students. The proposed system uses multiple body domain network sensors to collect physiological indicators data by using decision algorithms to combine multiple assessment indicators for analysis and prediction, ultimately generating a health assessment report and providing customized health advice and health plans for each user. In addition, to improve the physical health evaluation of university students in the context of sports medicine integration, this paper also proposes a physical health evaluation method based on the ID3 decision tree algorithm. It constructs a physical health information mining and feature extraction model for university students in the context of sports medicine integration. It uses a decision tree optimization method for classification detection. The proposed method also uses an associative multidimensional feature detection algorithm to evaluate university students' physical fitness and health status. It establishes a decision indicator function to investigate information fusion and constrained feature decomposition through the significant difference and balanced training fusion methods. In the ID3 decision tree, the branching system of college students' physical fitness health information in the context of sports medicine fusion is designed. It examines the entropy weight index parameters for managing test data management to assess university students' physical fitness in fusing sports medicine. The experimental results show that the system can accurately assess the physical health status of university students and provide corresponding recommendations and plans, which is essential for improving the health status of university students. In addition, these results proved that the proposed method has good output stability and strong optimization ability, which improves the classification management and information integration of college students' physical fitness and health information. [ABSTRACT FROM AUTHOR]

Given a dynamical system, we prove that the shortest distance between two n -orbits scales like n to a power even when the system has slow mixing properties, thus building and improving on results of Barros, Liao and the first author [On the shortest distance between orbits and the longest common substring problem. Adv. Math. 344 (2019), 311–339]. We also extend these results to flows. Finally, we give an example for which the shortest distance between two orbits has no scaling limit. [ABSTRACT FROM AUTHOR]

The acoustic characteristics of the rock failure process can reflect the evolution of crack development and expansion. To study the evolution of different types of internal cracks during the rock failure process, the acoustic emission (AE) signal of the yellow sandstone under uniaxial compression was collected with AE monitoring technology, and the acoustic characteristics, crack types and evolution of the correlation dimension of the rock failure process were studied. The parameter analysis of RA (rise time/amplitude) and AF (counts/duration) for classifying different cracking modes during loading revealed that tensile cracks are dominated in the crack closure stage and linear elastic deformation stage, and the proportion of shear cracks increased in the plastic deformation stage and the post-peak failure stage. High-energy events were more widely distributed in the areas of shear cracks. The average AE energy of shear cracks is higher than that of tensile cracks, and the AE energy ratio for shear cracks and tensile cracks is within the range of 4.29–4.78. The correlation between stress and AE data was calculated using fractal theory methods and Grassberger–Procaccia (G–P) algorithms to study the evolution of different types of internal cracks. The correlation dimension displayed a stable stage, a descending stage and a fluctuating low value stage. The descending stage and the fluctuating low value stage appeared during the plastic deformation stage and the post-peak failure stage, respectively, indicating that the type of rock internal micro-crack changed greatly in the above two stages. The findings in this study contribute to better understanding of the rock failure mechanism. Highlights: The evolution of proportion of tensile and shear cracks during the rock failure process were analyzed. The average AE energy of shear cracks is higher than that of tensile cracks, and the AE energy ratio for within the range of 4.29–4.78. The evolution of the type of micro-cracks in rock failure process is analyzed by Grassberger–Procaccia (G–P) algorithms. [ABSTRACT FROM AUTHOR]

Introduction: It is proved that there are differences between gait performed by females and males, which appear in movements of selected body parts. Despite numerous state-of-the-art studies related to the discriminative analysis of motion capture data, the question of whether measures of signal complexity and uncertainty can extract valuable features for the problem of sex distinction still remains open. It is the subject of the paper. Methods: Correlation dimension, as well as approximate and sample entropies, are selected to describe motion data. In the numerical experiments, the collected dataset with 884 samples of 25 females and 30 males was used. The measurements took place in the Human Motion Laboratory (HML), equipped with a highly precise motion capture system. Two variants of data representation were investigated-time series that contain joint rotations of taken skeleton model as well as positions of the markers attached to the human body. Finally, a comparative analysis between the populations of females and males using descriptive statistics, non-parametric estimation, and statistical hypotheses verification was carried out. Results: There are statistically significant sex differences extracted by the taken measures. In general, the movements of lower limbs result in greater values of correlation dimension and entropies for females, while selected upper body parts play a similar role for males. The dissimilarities are mainly observed in hip, ankle, shoulder, and head movements. Discussion: Correlation dimension and entropy measures provide robust and explainable features of motion capture data with a valuable description of the human locomotion system. Thus, beyond the importance of discovered differences between females and males, their interpretation and understanding are also known. [ABSTRACT FROM AUTHOR]

Copyright of Boletín de Ciencias de la Tierra is the property of Universidad Nacional de Colombia and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Fractals are widely used in different fields of science. The main characteristic of a fractal set is its fractal dimensions. The presented work addresses the accuracy of estimation of the correlation fractal dimension D2 and corresponding confidence interval. For example, in geosciences D2 is applied for characterizing surfaces or the spatial distribution of fracture systems. Two qualitatively different problems are studied. The first problem is related to the analysis of the fractal sets assigned on regular grids. The fractals consisting of a family of irregular points are considered in the second problem. The Monte Carlo method is proposed to estimate the confidence intervals of correlation fractal dimension for the first problem. This approach is based on the statistical generation of the fractal sets realizations with a given D2. Numerical testing using synthetic models showed the reliability of this method. A comparison with PBM and BCA bootstrap methods is performed. The paper also investigates the accuracy of the D2 and the corresponding confidence intervals estimate for the second studied problem. An algorithm employing the jackknife method is suggested. Its accuracy has been studied and proved numerically. [ABSTRACT FROM AUTHOR]

Bubble columns (BCs) are widely used in the chemical industry. In many industrial applications, these important gas-liquid contactors operate in a churn-turbulent flow regime. In principle, it is essential to determine the operating conditions in every BC reactor, in which local isotropic turbulence is established. In this work, it was demonstrated that several different parameters (Kolmogorov entropy, correlation dimension and novel hybrid index) follow a monotonic decreasing trend. This finding could be explained by the constantly increasing coalesced bubble size, which brings more order into the gas-liquid system and thus any entropic or chaotic parameter should decrease with the increase in the superficial gas velocity Ug. The profiles of the new parameters in various gas-liquid systems were studied. They were extracted from different pressure signals (gauge or absolute). In this research, BCs of different diameter and equipped with different gas distributors were used. It was demonstrated that the studied parameters could be successfully correlated with the length scale of the micro eddies and thus the Ug range of applicability of the local isotropic turbulence theory under various operating conditions was indirectly determined. The overall gas holdup profiles were analyzed and, based on the exponent of the Ug value, it was found that in the aqueous solutions of alcohols studied, the conditions in the bubble bed (BB) are homogeneous, whereas in the air-tap water system aerated in different BCs, the conditions in the BB are heterogeneous. This result implies that the local isotropic turbulence conditions predominate mainly around the corresponding measurement positions. [ABSTRACT FROM AUTHOR]

Dynamic failure of surrounding rock often causes many casualties and financial losses. Predicting the precursory characteristics of rock failure is of great significance in preventing and controlling the dynamic failure of surrounding rock. In this paper, a triaxial test of granite is carried out, and the acoustic emission events are monitored during the test. The fractal characteristics of acoustic emission events' energy distribution and time sequence are analyzed. The correlation dimension and the b value are used to study the size distribution and sequential characteristics. Furthermore, a rock failure prediction method is proposed. The correlation dimension is chosen as the main index and the b value is chosen as a secondary index for the precursor of granite failure. The study shows that: (1) The failure process can be divided into an initial stage, active stage, quiet stage, and failure stage. (2) The b value and correlation dimension both can describe the process of rock failure. There is a continuous decline before failure. Because of the complexity of the field, it is difficult to accurately estimate the stability of surrounding rock using a single index. (3) The combination of the b value and correlation dimension to establish a new method, which can accurately represent the stability of the surrounding rock. When the correlation dimension is increasing, the surrounding rock is stable with stress adjusting. When the correlation dimension is decreasing and the b value remains unchanged after briefly rising, the surrounding rock is stable, and stress is finished adjusting. When the correlation dimension and b value are both decreasing, the surrounding rock will be destroyed. [ABSTRACT FROM AUTHOR]

Copyright of Przegląd Elektrotechniczny is the property of Przeglad Elektrotechniczny and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The purpose of this study was to assess the effects of transverse friction massage (TFM) on the electromechanical delay components and complexity of the surface electromechanical activity in the rectus femoris (RF) and vastus medialis (VM) muscles and to identify possible mechanisms behind TFM-induced alterations in the dynamics of RF and VM activity. Seven female and five male healthy subjects participated in this study. The subjects generated five maximal voluntary isometric contractions (MVICs) consecutively before and after TFM. Meanwhile, electromyography (EMG), mechanomyography (MMG), and force were recorded. The onset times of the recorded signals were detected offline by setting the threshold to three times the SD of the baseline. The delays between EMG and MMG (Δt(EMG–MMG)), MMG and force (Δt(MMG–Force)), and EMG and force (Δt(EMG–Force)) were computed from the detected onsets. The fractal dimension (FD) of the EMG time series was computed using the correlation dimension method. TFM increased Δt(MMG–Force) and Δt(EMG–Force) significantly in the RF but decreased Δt(EMG–MMG) and increased Δt(MMG–Force) in the VM. TFM decreased the FD in the RF and increased it in the VM. The results imply that TFM decreased the stiffness of both the RF and VM and decreased the duration of the electrochemical processes in the VM. It is proposed that the decrease in EMG complexity in the RF may be associated with the decreased stiffness of the RF, and the increase in EMG complexity in the VM may be associated with the decreased electrochemical processes in this muscle. It is also suggested that the opposite changes in EMG complexity in the RF and VM can be used as a discriminating parameter to search for the effects of an intervention in the quadriceps muscles. The present study also demonstrates how to discriminate the nonlinear dynamics of a complex muscle system from a noisy time series. [ABSTRACT FROM AUTHOR]

To verify whether fractal characteristics are measured by infrared radiation (IR) experiments, IR and AE monitoring of sandstone samples under uniaxial compression was carried out. The relationship between the infrared radiation count correlation dimension (DIR) based on the G-P algorithm and fractal theory was established by these tests. The fractal characteristics and precursors of IR during sandstone fracture were obtained by comparing the correlation between the IR and AE characteristics. The following relationships were established: (1) infrared radiation counts (IRC) and acoustic emission ringing-down count (AEC) have simultaneous and sudden changes in all sandstones before final failure. (2) during the compression process, the DIR shows a trend of fluctuating increase → rising fluctuation amplitude → peak → sudden drop, and this trend is similar to the development of the AEC count correlation dimension (DAE). This indicates that the IR on the sandstone surface has fractal characteristics and is consistent with the internal fracture process, which is an orderly process of dimensionality reduction. (3) The average stress at the DIR sudden drops is 90.3% of the peak value, while the time leading to the peak stress and final failure is 30.5 s and 80.8 s on average, respectively, suggesting that the sudden drop of DIR after reaching the peak can be regarded as the precursor information of sandstone fracture and stability loss. These research results provide a theoretical basis and data support for the use of IR to monitor and predict sudden and brittle rock mass instability and damage. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of the time-series of water temperature of the Skokomish River (2019–2020) at hourly time scale by employing well-known nonlinear methods of chaotic data analysis including average mutual information, false nearest neighbors, correlation exponent, and local divergence rates. The delay time and the embedding dimension were calculated as 1400 and 9, respectively. The results indicated that the thermal regime in this river is chaotic due to the correlation dimension (1.38) and the positive largest Lyapunov exponent (0.045). Furthermore, complex networks have been applied to study the periodicity of thermal time-series throughout a year. A special algorithm is then used to find the so-called communities of the nodes. The algorithm found three communities which have been called Cold, Intermediate, and Warm. The temperatures in these three communities are, respectively, in the intervals (0.8, 5.8), (5.8, 11.63), and (11.63, 15.8). This analysis indicates that highest variations in water temperature occur between warm and cold seasons, and complex networks are highly capable to analyze hydrothermal fluctuations and classify their time-series. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Nanjing University of Information Science & Technology (Natural Science Edition) / Nanjing Xinxi Gongcheng Daxue Xuebao (ziran kexue ban) is the property of Editorial Office of Journal of Nanjing University of Information Science & Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Copyright of Science Technology & Engineering is the property of Chinese Society of Technology Economics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Estimation of actual evapotranspiration (ETa) is challenging due to its complex interactions with surrounding environments; thus, understanding the complexity and predictability of ETa along with its influencing factors is essential to improve ETa estimation. Based on the FLUXNET and ChinaFLUX datasets, we first examined whether daily ETa exhibited chaotic behaviors, and then investigated how daily ETa complexity and predictability varied across climate regimes and ecosystems using the chaos theory. The results of recurrence plot and correlation dimension (CD) analysis suggested the existence of chaotic behaviors in daily ETa, implying the system controlling ETa processes was deterministic with a limited number of controlling variables. The recurrence quantification analysis further revealed a varying degree of complexity for ETa processes (e.g., determinism‐DET with a range from 0.01% to 40.7%) across the sites. Specifically, daily ETa featured stronger deterministic properties (e.g., higher DET and lower CD values) and thus higher predictability in humid and arid regions; whereas, higher degrees of daily ETa complexity emerged in sub‐humid and semi‐arid regions due to more complex interactions with environmental factors. Moreover, primary environmental controls on daily ETa complexity varied with climatic and land surface conditions. Results showed that energy and water supplies along with average daily temperature and relative humidity were the primary controls on the ETa complexity across the sites, while leaf area index also played an important role at drier sites. As a first attempt, this study provides additional avenues to understand the complexity of ETa processes from a global perspective using the chaos theory. Key Points: Chaotic behaviors of daily actual evapotranspiration (ETa) across 81 sites were investigated using the chaos theoryHigher degrees of daily ETa complexity emerged at sub‐humid and semi‐arid sites than those at humid and arid sitesEnvironmental controls on daily ETa complexity varied with climatic and land surface conditions [ABSTRACT FROM AUTHOR]

Neurodegenerative diseases are caused by progressive degeneration of the central nervous system (CNS)'s neuronal structure. Well-known diseases in this category include Alzheimer's disease (AD), Parkinson's disease (PD), and multiple sclerosis (MS), which are also addressed in this study. The CNS appears to be a complex dynamic system, whose parameters change during the disease due to neuronal damage, resulting in various symptoms. Since the change in dynamic behavior is due to the neurons' death and change in neurons' connectivity, brain images of the affected areas appear to provide a good understanding of this change. This work attempts to focus on brain magnetic resonance images (MRI) and examine the effect of neuronal loss on the images. To this end, the complex features of these images, including 2D and Higuchi's fractal dimensions (HFD), correlation dimension (CD), largest Lyapunov exponent (LLE), and approximate entropy (ApEn) were calculated. Despite small differences in numerical values (0.01–0.35), these values differ significantly. This shows that the brain dynamic system behaves and functions differently in the disease state, which is clear in the behavior of global features. These three diseases have the same functional pattern, and this study seems to have captured the roots of these seemingly variant diseases. [ABSTRACT FROM AUTHOR]

Copyright of Machine Tool & Hydraulics is the property of Guangzhou Mechanical Engineering Research Institute (GMERI) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We investigated the relationship between the mechanical degrees of freedom (DoF) and its postural dynamics. The joint DoF was fixed to constrain the mechanical DoF. Nine participants were required to perform a single-leg stance task. The center of pressure trajectory data was analyzed. Ankle fixation induced a larger amount of variability in the anteroposterior direction, and less dimensionality and complexity in the mediolateral direction. These results suggest that the ankle joint fixation caused limited postural sway in the mediolateral direction; therefore, functional DoF and complexity decreased. In contrast, it increased the amount of postural sway variability in the anteroposterior direction. Our findings imply a direct relationship between the mechanical DoF of the human movement system and its postural dynamics. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Railway Science & Engineering is the property of Journal of Railway Science & Engineering Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Chronoamperometry (CA), phase space reconstruction, correlation dimension, multifractal detrended fluctuation analysis (MFDFA) combining linear sweep voltammetry (LSV), electrochemical impedance spectroscopy (EIS), and surface morphology observation were used to systematically analyze the nonlinear dynamics behavior of electrochemical dissolution of Ti-48Al-2Cr-2Nb alloy at high applied potentials. The relationships of applied potential E with nonlinear dynamic characteristic (saturation correlation dimension D 2 ′ , Hurst exponent H, the area of the spectrum S) and electrochemical corrosion parameters (capacitance ratio (Ca/Cb)) were analyzed, respectively. The results show that non-uniform corrosion at low potentials (7–13 V) results in rough surfaces; a higher potential (16 V) contributes to obtaining smoother surface. The growth of saturated correlation dimension reveals that the corrosion behavior of Ti-48Al-2Cr-2Nb alloy changes from non-uniform corrosion to uniform corrosion. The number of major variables in the electrochemical system is 6 for 7–13 V and is 7 for 16 V. The inversely proportional relationships of the area of the spectrum S and Hurst exponent H between the applied potential, respectively, testify that high potential is favorable for uniform corrosion of Ti-48Al-2Cr-2Nb alloy. [ABSTRACT FROM AUTHOR]

The hydrological regime is the main factor governing the functioning of floodplain rivers. A full comprehension of its dynamic leads to a better understanding of the system's behaviour. We analysed the daily water level of the Paraná River during the last century at three gauge stations using linear and non‐linear tools to characterise the hydrological dynamic and to analyse to what extent chaotic behaviour prevails. The three water level time series were characterised as non‐linear and non‐stationary by power spectrum, autocorrelation function, and surrogate test analyses. A strange attractor was developed when the phase space was reconstructed, having a low dimensional chaos supported by correlation dimension, positive maximum Lyapunov exponents, and recurrence quantification analysis. The sample entropy analysis shows that the system resulted unpredictable beyond a temporal threshold, and with an intermediate hydrological complexity, while Hurst exponent characterised it as persistent and with sensitive dependence on initial conditions. In a general overview, all the evidence obtained indicates that the Paraná River's behaviour is at the edge of chaos. A latitudinal gradient of decreasing chaoticity was observed as the floodplain extent increased, whereas complexity was highest at the intermediate river station due to the inflow of tributaries with different hydrology. This paper attempts to offer some additional insights for understanding the hydrological behaviour of floodplain rivers and the most appropriate methods to understand their complexity. [ABSTRACT FROM AUTHOR]

Few studies have evaluated the effect of a secondary motor task on the standing posture based on nonlinear analysis. However, it is helpful to extract information related to the complexity, stability, and adaptability to the environment of the human postural system. This study aimed to analyze the effect of two motor tasks with different difficulty levels in motor performance complexity on the static standing posture in healthy young adults. Thirty-five healthy participants (23.08 ± 3.92 years) performed a postural single task (ST: keep a quiet standing posture) and two motor dual tasks (DT). i.e., mot-DT(A)—perform the ST while performing simultaneously an easy motor task (taking a smartphone out of a bag, bringing it to the ear, and putting it back in the bag)—and mot-DT(T)—perform the ST while performing a concurrent difficult motor task (typing on the smartphone keyboard). The approximate entropy (ApEn), Lyapunov exponent (LyE), correlation dimension (CoDim), and fractal dimension (detrending fluctuation analysis, DFA) for the mediolateral (ML) and anterior-posterior (AP) center-of-pressure (CoP) displacement were measured with a force plate while performing the tasks. A significant difference was found between the two motor dual tasks in ApEn, DFA, and CoDim-AP (p < 0.05). For the ML CoP direction, all nonlinear variables in the study were significantly different (p < 0.05) between ST and mot-DT(T), showing impairment in postural control during mot-DT(T) compared to ST. Differences were found across ST and mot-DT(A) in ApEn-AP and DFA (p < 0.05). The mot-DT(T) was associated with less effectiveness in postural control, a lower number of degrees of freedom, less complexity and adaptability of the dynamic system than the postural single task and the mot-DT(A). [ABSTRACT FROM AUTHOR]

Burnishing experiments with different burnishing parameters were performed on a computer numerical control milling machine to characterize the surface roughness of an aluminum alloy during burnishing. The chaos theory was employed to investigate the nonlinear features of the burnishing system. The experimental results show that the power spectrum is broadband and continuous, and the Lyapunov exponent λ is positive, proving that burnishing has chaotic characteristics. The chaotic characteristic parameter, the correlation dimension D, is sensitive to the time behavior of the system and is used to establish the corresponding relationship with the surface roughness. The correlation dimension was the largest, when the surface roughness was the smallest. Furthermore, when the correlation dimension curve decreases, the roughness curve increases. The correlation dimension and surface roughness exhibit opposite variation trends. The higher the correlation dimension, the lower the surface roughness. The surface roughness of the aluminum alloy can be characterized online by calculating the correlation dimension during burnishing. [ABSTRACT FROM AUTHOR]

The potential earthquake and tsunami hazard along the Sumatra subduction margin, especially around the coast of West Sumatra-Bengkulu, was investigated based on the availability of pre-seismic surface displacement data and shallow crustal earthquake catalogue data from 1907 to 2016. The pre-seismic surface displacement data is based on the displacement data prior to and corrected displacement data after major earthquakes. Using the results of our previous study on the local covariance function and the relationship of Correlation Dimension (DC) with the b-value of Gutenberg-Richter (GR) Law, we estimated the maximum horizontal crustal strain rate (SHmax) and DC around the study area. Least squares prediction based on horizontal displacement data using the local covariance function is used to estimate the displacement model in the entire gridding study area with a 10 km x 10 km size. Furthermore, DC is calculated based on the b-value using the maximum likelihood method based on the input of a constant number of earthquake samples, assuming the regional b-value of GR Law equals 1. Furthermore, the spatial correlation of SHmax and DC can define the area of possible earthquake hazard potential. The identification results are then linked with previous stress reconstruction results for seismic hazard study and analysis. Based on the finding, we then estimate the Seismic Hazard Function (SHF) and Tsunami Height simulation to estimate the possible hazard risk at several observation points. We suggest that the result of this study could be beneficial to understand better the potential seismic and tsunami hazard in the future, mainly to support mitigation purposes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the discrete-time fractional systems from the following aspects. First, the discrete-time fractional unified system in Caputo sense is established with the help of Euler's discretization method. Furthermore, the dynamic behaviors of the discrete-time fractional Lü system (DFLS) which is deemed as a representative for unified system are observed. Then, the correlation dimension ( D 2 ) and Kaplan–Yorke dimension ( D K Y ) of the DFLS are evaluated by the aid of Grassberger–Procaccia algorithm and the Lyapunov exponent spectrum, respectively. Finally, the intrinsic connections between D 2 and D K Y are analyzed by the statistical modeling idea when the DFLS is in chaotic vibrations. The main results show that D 2 shares a positive correlation with D K Y for the chaotic DFLS, while the differences between D 2 and D K Y are not only related to the ratio of the largest and smallest Lyapunov exponents, but also closely tied up with the fractional order v itself. [ABSTRACT FROM AUTHOR]

The north-east sector of the Himalaya is one of the most active tectonic belts, with complex geological and tectonic features. The b-value and spatial correlation dimension (Dc) of earthquake distribution in the north-east Himalaya and its adjacent regions (20–32°N and 88–98°E) are estimated in the present study. Based on seismicity and faulting pattern, the region is divided into five active regions, namely the (i) South-Tibet, (ii) Eastern-Syntaxis, (iii) Himalayan-Frontal Arc, (iv) Arakan-Yoma belt and (v) Shillong-Plateau. A homogeneous catalogue of 1,416 earthquakes (mb ≥ 4.5) has been prepared from a revised catalogue of the ISC (International Seismological Centre). The b-value has been appraised by the maximum likelihood estimation method, while Dc values have been calculated by the correlation integral method; b-values of 1.08 ± 0.09, 1.13 ± 0.05, 0.92 ± 0.05, 1.00 ± 0.03 and 0.98 ± 0.08 have been computed for the South-Tibet, Eastern-Syntaxis, Himalayan-Frontal Arc, Arakan-Yoma belt and Shillong-Plateau region, respectively. The Dc values computed for the respective regions are 1.36 ± 0.02, 1.74 ± 0.04, 1.57 ± 0.01, 1.8 ± 0.01, and 1.83 ± 0.02. These values are > 1.5, except for the South-Tibet (1.36 ± 0.02). The b-values around the global average value (1.0) reflect the stress level and seismic activity of the regions, while high Dc values refer to the heterogeneity of the seismogenic sources. [ABSTRACT FROM AUTHOR]

SEM microfractographies of Zircaloy-4 are studied by fractal analysis and the time-series method. We first develop a computer application that associates the fractal dimension and lacunarity to each SEM micrograph picture, and produce a nonlinear analysis of the data acquired from the quantitatively evaluated time series. Utilizing the phase space-embedding technique to reconstruct the attractor and to compute the autocorrelation dimension, the fracture surface of the Zircaloy-4 samples is investigated. The fractal analysis method manages to highlight damage complications and provide a description of morphological parameters of various fractures by calculating the fractal dimension and lacunarity. [ABSTRACT FROM AUTHOR]

This study intends to evaluate the possible correlation between the correlation dimension (DC) and the seismic moment rate for different late Quaternary active fault data, shallow crustal earthquakes, and GPS on the island of Sumatra Probabilistic Seismic Hazard Analysis (PSHA). The seismicity smoothing was applied to estimate the DC of active faults (DF) and earthquake data (DE) and then to correlate that with the b-value, which will be used to identify seismic hazard functions (SHF) along with the Sumatra Fault Zone (SFZ). The seismicity based on GPS data was calculated by the seismic moment rate that is estimated based on pre-seismic horizontal surface displacement data. The correlation between DF, DE, and the b-value was analyzed, and a reasonable correlation between the two seismotectonic parameters, DF-b, and DE-b, respectively, could be found. The relatively high DC coincides with the high seismic moment rate model derived from the pre-seismic GPS data. Furthermore, the SHF curve of total probability of exceedance versus the mean of each observation point's peak ground acceleration (PGA) shows that the relatively high correlation dimension coincides with the high SHF. The results of this study might be very beneficial for seismic mitigation in the future. [ABSTRACT FROM AUTHOR]

The issue of monitoring and early warning of rock instability has received increasing critical attention in the study of rock engineering. To investigate the damage evolution process of granite under triaxial compression tests, acoustic emission (AE) tests were performed simultaneously. This study firstly introduced two novel parameters, i.e., the coefficient of variation (CoV) of the information entropy and correlation dimension of the amplitude data from the AE tests, to identify the precursor of the failure of granite. Then the relationship between the changes in these parameters and the stress-time curve was compared and analyzed. The results of this study show that: (1) There is a strong correlation between the CoV of the information entropy and the failure process of granite. The granite failed when the CoV curve raised to a plateau, which could be used as an indicator of rock instability. (2) The fluctuation of the correlation dimension indicates the different stages during the loading process, i.e., the initial compaction stage, the linear elastic stage, the yield stage, and the failure stage. Each stage contains a descending and a rising process in the correlation dimension curve, and the exhibited starting point or the bottom point at the correlation dimension curve could be selected as the indicator point for the rock instability. (3) The combined analysis of the Information entropy and Correlation dimension can improve the accuracy of rock instability prediction. This study provides new insights into the prediction of rock instability, which has theoretical implications for the stability of subsurface engineering rock masses. [ABSTRACT FROM AUTHOR]

In this study, the characterization method and evolution law of phase-point saturation based on the capacity dimension in fractal theory are studied to investigate the running-in process and state identification of a system. Tests of the running-in process under different working conditions were conducted on a ring-disc tribometer, and the friction signals were collected. The phase trajectories in different stages were then constructed and the fractal dimensions were calculated. The phase-point saturation parameter is proposed to reflect the degree of saturation of a phase space filled with phase points, and its reliability, noise-resistance, correlation with the fractal dimension, and effectiveness of the characterization of the running-in process were thoroughly investigated. The results demonstrate that the proposed phase-point saturation parameter is reliable and resistant to noise and can be used to identify changes in the system state. Furthermore, there is a negative correlation between the phase-point saturation parameter and the fractal dimension, and the phase-point saturation parameter can effectively characterize the state characteristics and describe the running-in process. During the running-in process, the phase-point saturation changes from high to low and tends to be stable when it reaches the running-in state. When entering the stable wear stage, the value of the parameter is maintained near the minimum, it fluctuates little as compared with the correlation dimension, and it exhibits good stability. Therefore, the proposed phase-point saturation parameter has important theoretical significance and application value for the quantitative characterization of the running-in process and state identification. [ABSTRACT FROM AUTHOR]

Investigating the behavior of water surface evaporation is one of the basic issues in design, operation, and studies related to water engineering. Therefore, the application of new methods such as chaos theory in hydrology and water resources has recently been considered due to its innovation and capabilities. Since the fluctuations of evaporation from free water surfaces are dynamic and non-linear in nature, the aim of this study was to investigate the possibility of chaotic behavior in evaporation from the free water surface in the Semnan synoptic station on daily and monthly time scales in 1995-2018 using the concepts of chaos theory. The daily, monthly, and annual evaporation rates of this synoptic station were calculated to be 68.8, 200, and 2600 mm, respectively. To reconstruct the state space, two parameters of delay time and embedding dimension are needed. The mean of mutual information and false nearest neighborhood has been used to estimate these two parameters. The first step to study a process with chaos theory is to investigate the chaotic nature of the correlation dimension method as one of the most common methods. First, the embedded dimension was calculated by the nearest neighborhood method equal to 3. To calculate the delay time, cross-evaporation diagrams were drawn at Semnan station at different time scales. According to this method, the first local minimum in the diagram is considered the latency, which was obtained for evaporation at daily and monthly scales of 30 and 3, respectively. Unlike complicated and conventional computational methods, these results are obtained by observation and in the least amount of time, as follows: monthly data are more chaotic than daily data. The enclosed dimension and the slope of the correlation dimension diagram were obtained at 8.8 and 9.8, respectively, after calculating the latency and reconstruction of the state space. [ABSTRACT FROM AUTHOR]

Copyright of Advanced Engineering Science / Gongcheng Kexue Yu Jishu is the property of Advanced Engineering Science Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Cyber-physical systems (CPSs) realize the automatic control of entities through computing systems and networks. Stability is an important factor in CPS for system upgrading and troubleshooting. Traditional analysis methods focus on simulation and formal analysis, which have two major limitations: first, the current state information of CPS is difficult to obtain; second, most CPS face the state space explosion problem. These problems can be avoided and a good analysis can be provided based on empirical data. The main work of this article is summarized as follows: first, a phase space reconstruction method is designed to divide the dataset into several subsequences with the same shape; second, we propose a stability analysis method based on correlation dimensions. Results indicate that the proposed approach can obtain a stable correlation dimension. CPS perform better if the correlation dimension is maintained within a certain range; otherwise, a destabilizing factor exists. The proposed stability analysis has less complexity and running time. [ABSTRACT FROM AUTHOR]

Complex systems such as river flow do not obey the law of linearity but most often than not behave in a nonlinear manner. In this study, the dynamics of the River Niger discharge along two stations located in Nigeria were investigated using the methods of phase space reconstruction, correlation dimension and Lyapunov exponent. The analysis was carried out by using the daily data for three different periods — 1914–1939 (period 1), 1940–1964 (period 2), and 1966–1991 (period 3) spanning before and after the Kainji dam construction. Time delay embedding values at Baro and Lokoja for the three time periods were used to reconstruct the phase space of the river discharge at both locations. The embedding dimension represents the number of variables to completely describe the system. The presence of chaos was confirmed with positive values of Lyapunov exponents in both locations at the three different periods considered. The Lyapunov exponents at Baro were in the range 0.0014–0.0150 while the range at Lokoja was 0.007–0.0145. Significant correlation dimension values obtained at Baro (2.11–2.81) and Lokoja (2.28–4.51) are indicative of low dimensional chaos. A possible explanation for this could be the temporal sequence in the dam installation. The forecast horizon, the inverse of the largest Lyapunov exponents, gives a prediction boundary on a chaotic time series. In this study, the forecast horizon for both locations is expected to be in the range of 40–58 days into the future. [ABSTRACT FROM AUTHOR]

This study assessed the characteristics of water-level time series of a tidal river by decomposing it into tide, wave, rainfall-runoff, and noise components. Especially, the analysis for chaotic behavior of each component was done by estimating the correlation dimension with phase-space reconstruction of time series and by using a close returns plot (CRP). Among the time series, the tide component showed chaotic characteristics to have a correlation dimension of 1.3. It was found out that the water level has stochastic characteristics showing the increasing trend of the correlation exponent in the embedding dimension. Other components also showed the stochastic characteristics. Then, the CRP was used to examine the characteristics of each component. The tide component showed the chaotic characteristics in its CRP. The CRP of water level showed an aperiodic characteristic which slightly strayed away from its periodicity, and this might be related to the tide component. This study showed that a low water level is mainly affected by a chaotic tide component through entropy information. Even though the water level did not show chaotic characteristics in the correlation dimension, it showed stochastic chaos characteristics in the CRP. Other components showed stochastic characteristics in the CRP. It was confirmed that the water level showed chaotic characteristics when it was not affected by rainfall and stochastic characteristics deviating from the bounded trajectory when water level rises due to rainfall. Therefore, we have shown that the water level related to the chaotic tide component can also have chaotic properties because water level is influenced by chaotic tide and rainfall shock, thus it showed stochastic chaos characteristics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The measured vibrational responses of the pumping station pipeline in the irrigation site were chosen to confirm the chaotic characteristics of the pumping station pipeline vibration and to determine the vibrational excitation that makes it chaotic. First, the chaotic properties of the pipeline vibration responses were investigated using a saturation correlation dimension and the maximum Lyapunov exponent. The vibration excitation with chaotic features was obtained using an improved variational mode decomposition (IVMD) method to examine the multi-time-scale chaotic characteristics of the pipeline vibration responses. The results show that the vibrational responses of each measuring point of the pipeline under different operating conditions have clear chaotic characteristics, where the chaotic characteristics of the axial points and bifurcated pipe points are relatively strong. The vibration of the operating conditions and measurement points affected by the unit's operation and flow state change is further complicated. The intrinsic mode function (IMF) produces a low-dimensional chaotic attractor after the IVMD disrupts the vibration response. Still, the vibration excitation of the remaining components on behalf of the units does not have chaotic properties, implying that water pulsation excitation makes the pumping station pipeline vibrations chaotic. The vibration excitation caused by the unit's operation covers the chaotic characteristics of the pipeline vibration and increases its uncertainty. The outcomes of this study provide a theoretical basis for further exploration of the vibration characteristics of pumping station pipelines, and a new method of chaos analysis is proposed. [ABSTRACT FROM AUTHOR]