Multifractal Dynamics Research Articles

Lifelong longitudinal assessment of the contribution of multi-fractal fluctuations to heart rate and heart rate variability in aging mice: role of the sinoatrial node and autonomic nervous system.

Aging is a major risk factor for sinoatrial node (SAN) dysfunction, which can impair heart rate (HR) control and heart rate variability (HRV). HR and HRV are determined by intrinsic SAN function and its regulation by the autonomic nervous system (ANS). The purpose of this study wasto use multi-scale multi-fractal detrended fluctuation analysis (MSMFDFA; a complexity-based approach to analyze multi-fractal dynamics) to longitudinally assess changes in multi-fractal HRV properties and SAN function in ECG time series recorded repeatedly across the full adult lifespan in mice. ECGs were recorded in anesthetized mice in baseline conditions and after autonomic nervous system blockade every three months beginning at 6months of age until the end of life. MSMFDFA was used to assess HRV and SAN function every three months between 6 and 27months of age. Intrinsic HR (i.e. HR during ANS blockade) remained relatively stable until 15months of age, and then progressively declined until study endpoint at 27months of age. MSMFDFA revealed sudden and rapid changes in multi-fractal properties of the ECG RR interval time series in aging mice. In particular, multi-fractal spectrum width (MFSW, a measure of multi-fractality) was relatively stable between 6months and 15months of age and then progressively increased at 27months of age. These changes in MFSW were evident in baseline conditions and during ANS blockade. Thus,intrinsic SAN function declines progressively during aging and is manifested by age-associated changes in multi-fractal HRV across the lifespan in mice, which can be accurately quantified by MSMFDFA.

Inner Multifractal Dynamics in the Jumps of Cryptocurrency and Forex Markets

Jump dynamics in financial markets exhibit significant complexity, often resulting in increased probabilities of subsequent jumps, akin to earthquake aftershocks. This study aims to understand these complexities within a multifractal framework. To do this, we employed the high-frequency intraday data from six major cryptocurrencies (Bitcoin, Ethereum, Litecoin, Dashcoin, EOS, and Ripple) and six major forex markets (Euro, British pound, Canadian dollar, Australian dollar, Swiss franc, and Japanese yen) between 4 August 2019 and 4 October 2023, at 5 min intervals. We began by extracting daily jumps from realized volatility using a MinRV-based approach and then applying Multifractal Detrended Fluctuation Analysis (MFDFA) to those jumps to explore their multifractal characteristics. The results of the MFDFA—especially the fluctuation function, the varying Hurst exponent, and the Renyi exponent—confirm that all of these jump series exhibit significant multifractal properties. However, the range of the Hurst exponent values indicates that Dashcoin has the highest and Litecoin has the lowest multifractal strength. Moreover, all of the jump series show significant persistent behavior and a positive autocorrelation, indicating a higher probability of a positive/negative jump being followed by another positive/negative jump. Additionally, the findings of rolling-window MFDFA with a window length of 250 days reveal persistent behavior most of the time. These findings are useful for market participants, investors, and policymakers in developing portfolio diversification strategies and making important investment decisions, and they could enhance market efficiency and stability.

Interplay Multifractal Dynamics Among Carbon Trading Market, Geopolitical Risk and Economic Policy Uncertainty

This paper explores the variations in cross-correlations among the carbon trading market, geopolitical risk (GPR) and economic policy uncertainty (EPU), focusing on their multifractality and asymmetric properties. Therefore, the study employs the multifractal detrended cross-correlation analysis (MF-DCCA) and multifractal asymmetric detrended cross-correlation analysis (MF-ADCCA) approaches to examine these relationships from a three-dimensional perspective. Also, the study looks into cross-correlations at different phases of carbon trading. Our findings reveal that the multifractal cross-correlations between the carbon trading market and GPR or EPU exhibit anti-persistence and asymmetry. The anti-persistent cross-correlations are notably stronger when GPR experiences a downtrend or when EPU undergoes an uptrend. Furthermore, distinct fluctuation patterns emerge across various trends and scales on Hurst surfaces. In the short term, the carbon trading market exhibits heightened sensitivity to changes in GPR or EPU when they rise or fall. The cross-correlations remain multifractal, anti-persistent and asymmetric across different stages of carbon trading. Notably, the multifractality of cross-correlations is most pronounced for the series pair EUA/GPR in Phase II and for the series pair EUA/UK EPU in Phase III and Phase IV. Except for the series pair EUA/UK EPU in Phase II, the anti-persistent cross-correlations are more pronounced during uptrend than downtrend for the other series pairs. Moreover, the series pair EUA/UK EPU exhibits the highest degree of asymmetry at all stages.

Multifractal Multiscale Analysis of Human Movements during Cognitive Tasks.

Continuous adaptations of the movement system to changing environments or task demands rely on superposed fractal processes exhibiting power laws, that is, multifractality. The estimators of the multifractal spectrum potentially reflect the adaptive use of perception, cognition, and action. To observe time-specific behavior in multifractal dynamics, a multiscale multifractal analysis based on DFA (MFMS-DFA) has been recently proposed and applied to cardiovascular dynamics. Here we aimed at evaluating whether MFMS-DFA allows identifying multiscale structures in the dynamics of human movements. Thirty-six (12 females) participants pedaled freely, after a metronomic initiation of the cadence at 60 rpm, against a light workload for 10 min: in reference to cycling (C), cycling while playing "Tetris" on a computer, alone (CT) or collaboratively (CTC) with another pedaling participant. Pedal revolution periods (PRP) series were examined with MFMS-DFA and compared to linearized surrogates, which attested to a presence of multifractality at almost all scales. A marked alteration in multifractality when playing Tetris was evidenced at two scales, τ ≈ 16 and τ ≈ 64 s, yet less marked at τ ≈ 16 s when playing collaboratively. Playing Tetris in collaboration attenuated these alterations, especially in the best Tetris players. This observation suggests the high sensitivity to cognitive demand of MFMS-DFA estimators, extending to the assessment of skill/demand interplay from individual behavior. So, by identifying scale-dependent multifractal structures in movement dynamics, MFMS-DFA has obvious potential for examining brain-movement coordinative structures, likely with sufficient sensitivity to find echo in diagnosing disorders and monitoring the progress of diseases that affect cognition and movement control.

Interplay of multifractal dynamics between shadow policy rates and energy markets

This study examines the interconnections between short shadow rates (SSR) and energy markets. We conduct multifractal detrended cross-correlation analysis (MF-DXA) on the daily SSR from four major economies (USA, Eurozone, Japan, and UK) and the daily prices of four energy markets (WTI, Brent, Natural Gas, and Heating Oil) spanning January 1995 to March 2022. Our analysis shows that all energy market-short shadow rate pairs display significant multifractal characteristics. The presence of power law cross-correlation suggests that substantial changes in short shadow rates are associated with significant price changes in energy markets. The highest multifractal cross-correlation is observed between energy prices and shadow short rates of Japan and the US. Energy markets demonstrate persistent cross-correlations for small fluctuations compared to large fluctuations. However, the rolling windows analysis reveal a shift from persistent to anti-persistence cross-correlations during global financial crises (GFC), the 2014–2016 energy crises and the COVID-19 pandemic. Lastly, the cross-correlation exhibit positive skewness. We suggest that integrating these long-term dependency features into financial analysis has the potential to support decision-making for market participants.

THE (IN)EFFICIENCY OF USA EDUCATION GROUP STOCKS: BEFORE, DURING AND AFTER COVID-19

This paper represents a pioneering effort to investigate multifractal dynamics that exclusively encompass the return time series of USA Education Group Stocks concerning two non-overlapping periods (before, during, and after COVID-19). Given this, we employ the Multifractal Detrended Fluctuations Analysis (MF-DFA). In this sense, we investigate the generalized Hurst exponent [Formula: see text] and the Rényi exponent [Formula: see text] for each asset and quantify their statistical properties, which allowed us to observe separately the contributing small scale (primarily via the negative moments [Formula: see text]) and the large scale (via the positive moments [Formula: see text]). We perform a fourth-degree polynomial regression fit to estimate the complexity parameters that describe the degree of multifractality of the underlying process. Also, we shall apply the inefficiency multifractal index to assess the COVID-19 shock for both periods. Our findings show that for both periods, the majority of these assets are marked by multifractal dynamics associated with persistent behavior [Formula: see text], a higher degree of multifractality and the dominance of large fluctuations. At the same time, most of these assets show asymmetry parameter [Formula: see text] for both periods, indicating that large fluctuations contributed more to multifractality in the time series of returns.

STATISTICAL ANALYSIS BY WAVELET LEADERS REVEALS DIFFERENCES IN MULTI-FRACTAL CHARACTERISTICS OF STOCK PRICE AND RETURN SERIES IN TURKISH HIGH FREQUENCY DATA

The price and return time series are two distinct features of any financial asset. Hence, examining the evolution of multiscale characteristics of price and returns sequential data in time domain would be helpful in gaining a better understanding of the dynamical evolution mechanism of the financial asset as a complex system. In fact, this is important to understand their respective dynamics and to design their appropriate predictive models. The main purpose of the current work is to investigate the multiscale fractals of price and return high frequency data in Turkish stock market. In this regard, the wavelet leaders computational method is applied to each high frequency data to reveal its multi-fractal behavior. In particular, the method is applied to a large set of Turkish stocks and statistical results are performed to check for (i) presence of multi-fractals in price and return series and (ii) differences between prices and returns in terms of multi-fractals. Our statistical results show strong evidence that high frequency price and return data exhibit multi-fractal dynamics. In addition, they show evidence of distinct fractal characteristics on different scales between price and return series. Furthermore, our statistical results show evidence of differences in local fluctuation characteristics of price and return time series. Therefore, differences in local characteristics are useful to build specific predictive models for each type of data for better modeling and prediction to generate profits. Besides, we found evidence that both long-range correlations and fat-tail distributions contribute to the multifractality in Turkish stocks. This finding can be attributed to the major role played by international investors in increasing the volatility of Turkish stocks.

Local electrical characteristic of memristor structure in a high-resistance state obtained using electrostatic force microscopy: Fractal and multifractal dynamics of surface

A heterostructure BiFeO3/TiO2(Nt)Ti (BFOT) was obtained by the atomic layer deposition (ALD) method. After thermal treatment, the redistribution of Fe/Ti atoms forms an Aurivillius intermediate layered phase, and local charge capture centers are formed in the sample. Due to cationic non-stoichiometry, the BFO film exhibits p-type conductivity, while the nanotubes exhibit n-type conductivity due to oxygen vacancies. It was observed that lateral displacement of the sample can lead to ferroelectric switching, which can, in turn, affect the transition of the memristive structure from high-resistance (HRS) to low-resistance states (LRS). The hysteresis suppression tends to transition to an ohmic character and depends on the amplitude, frequency, and duration of the periodic signal. It has been found that compensation of static charge during resistive switching can affect the transport properties of the material. Fractal dimension analysis showed an acceleration of structure restructuring with increasing voltage, possibly contributing to the transition from an insulator to a metal in certain areas of the film volume. The joint analysis of piezoresponse force microscopy (PFM), electrostatic force microscopy (EFM), and fractal/multifractal dynamics showed a correlation between surface static charge and piezopotential. The new methodology described in this work can help understand the resistive switching processes in ferroelectric/semiconductor memristive structures.

Impact of prolonged chronic social isolation stress on behavior and multifractal complexity of metabolic rate in Octodon degus

Social interaction can improve animal performance through the prevention of stress-related events, the provision of security, and the enhancement of reproductive output and survival. We investigated the effects of prolonged chronic social isolation stress on behavioral, cognitive, and physiological performance in the social, long-lived rodent Octodon degus. Degu pups were separated into two social stress treatments: control (CTRL) and chronically isolated (CI) individuals from post-natal and post-weaning until adulthood. We quantified anxiety-like behavior and cognitive performance with a battery of behavioral tests. Additionally, we measured their basal metabolic rate (BMR) and analyzed the multifractal properties of the oxygen consumption time series using Multifractal Detrended Fluctuation Analysis, a well-known method for assessing the fractal characteristics of biological signals. Our results showed that CI induced a significant increase in anxiety-like behaviors and led to a reduction in social and working memory in male degus. In addition, CI-treated degus reduced the multifractal complexity of BMR compared to CTRL, which implies a decrease in the ability to respond to environmental stressors and, as a result, an unhealthy state. In contrast, we did not observe significant effects of social stress on BMR. Multivariate analyses showed a clear separation of behavior and physiological variables into two clusters, corresponding to CI and CTRL degus. This study provides novel insights into the effects of prolonged chronic social isolation stress on behavior, cognitive performance, and metabolic complexity in this rodent animal model. To the best of our knowledge, it is the first study to integrate cognitive-behavioral performance and multifractal dynamics of a physiological signal in response to prolonged social isolation. These findings highlight the importance of social interactions for the well-being and overall performance of social animals.

Gold and Sustainable Stocks in the US and EU: Nonlinear Analysis Based on Multifractal Detrended Cross-Correlation Analysis and Granger Causality

Geopolitical risks and conflicts wield substantial influence on the global economy and financial markets, fostering uncertainty and volatility. This study investigates the intricate relationship between gold and representatives of green and sustainable stocks in the US and EU during the Russia-Ukraine conflict, employing multifractal detrended cross-correlation analysis (MF-DCCA) and nonlinear Granger causality. MF-DCCA reveals significant multifractal properties and nonlinear cross-correlations across all time series pairs. Notably, conflict weakened the multifractal cross-correlations between US stocks and gold, except for the TESLA/gold pair. A similar significant change in the EU market’s multifractal properties was observed during the conflict. In conjunction with MF-DCCA, Granger causality tests indicate bidirectional nonlinear relationships between gold and green/sustainable stock markets in the USA and EU. Gold’s past movements significantly influence changes in all the considered green and sustainable stocks, enabling predictions of their behavior. These findings shed light on multifractal dynamics during geopolitical crises and emphasize the bidirectional relationships between gold and green and sustainable stock markets. This comprehensive analysis informs investors and policy makers, enhancing their understanding of financial market behavior amid geopolitical instability.

Multifractality and scale-free network topology in a noise-perturbed laminar jet

We present experimental evidence of multifractality and scale-free network topology in a noise-perturbed laminar jet operated in a globally stable regime, prior to the critical point of a supercritical Hopf bifurcation and prior to the saddle-node point of a subcritical Hopf bifurcation. For both types of bifurcation, we find that (i) the degree of multifractality peaks at intermediate noise intensities, (ii) the conditions for peak multifractality produce a complex network whose node degree distribution obeys an inverse power-law scaling with an exponent of $2 < \gamma < 3$ , indicating scale-free topology and (iii) the Hurst exponent and the global clustering coefficient can serve as early warning indicators of global instability under specific operating and forcing conditions. By characterising the noise-induced dynamics of a canonical shear flow, we demonstrate that the multifractal and scale-free network dynamics commonly observed in turbulent flows can also be observed in laminar flows under certain stochastic forcing conditions.

MULTIFRACTAL ANALYSIS OF EMG FOR CLASSIFICATION AND PROGRESSIVE ASSESSMENT OF BICEPS BRACHII MUSCLE STRENGTH

The proposed research demonstrates an attempt to introduce a systematic procedure for studying the multifractal dynamics of biceps brachii muscle actions during light exercises. The intrinsic patterns in the Surface Electromyogram (sEMG) signals were extracted by fruitfully exploiting the Multifractal features of the signal. The Multifractal features are derived from the multifractal singularity spectrum of the EMG signals. This multifractal feature vector could be utilized for signal characterization, which was successfully extended for the classification of EMG signals. Experimental verification has been done to validate the feature extraction and classification algorithm proposed in this article. A pilot study was conducted on signals from the Physionet database, which was then extended to a medium database developed with biceps brachii EMG signals of 32 healthy male subjects. From this study, we could validate that the Multifractal features fit as a differentiating multi-feature set and also for the progressive assessment of EMG signals of different classes. The observations of the proposed method revealed that the strength of multifractality and area under the spectrum increase as a result of fast movements of the forearm or increases in muscle fatigue. The classification is performed using well-recognized supervised classification algorithms such as k-Nearest Neighbor and Support Vector Machine (SVM) Classifiers. The performance analysis of the classifiers are studied on various measures such as Accuracy, Precision, Recall, F1 score. The statistical significance analysis of the feature vector was carried out by one-way ANOVA test.

Interplay of multifractal dynamics between shadow policy rates and stock markets

Stock markets are generally perceived as a barometer of the economy and respond to international monetary policies even before economic activities. Many central banks have turned to unconventional policy measures in response to various financial crises such as the global financial crisis of 2007–2009 or the recent crisis caused by COVID-19. To examine the cross-correlation of overall international monetary policies with stock markets, we employ the daily shadow short rate (SSR), which has the advantage of allowing comparison across unconventional and conventional regimes. The analysis is made through a multifractal context using multifractal detrended cross correlation analysis (MF-DXA), considering daily data from 1st January 2000 to 31st March 2022 and country specific SSR and the stock markets of eight developed economies. The main empirical findings are the following: (i) all the country specific pairs of SSR with stock markets have significant multifractal characteristics (ii) the pairs of NZ-SSR/NZX50, US-SSR/DJIA, and CN-SSR/S&P TSX have the highest multifractal patterns while EU-SSR/Euro-area Index has the lowest multifractal patterns (iii) Australian and New Zealand stock markets exhibit anti-persistent cross-correlation with SSR while the remainder have persistent cross-correlation in their multifractality. Lastly, the findings of this study have several important implications for central banks and stock market participants.

Multifractal analysis of standardized precipitation index in Northeast Brazil

Northeast Brazil (NEB) is one of the world's most densely populous dryland regions most vulnerable to the impacts of climate change. It is expected that the risk of natural disasters related to excess or lack of water (flash floods and severe droughts) will continue increasing in the region until the end of the century. In this work, we study multifractal properties of the Standardized precipitation index (SPI) which is commonly used to evaluate the degree of severity of drought and humid conditions at different time scales and can be compared among regions with distinct climatic conditions. The method Multifractal detrended fluctuation analysis (MFDFA) is used to analyze SPI for 1, 3, 6, and 12-month accumulation periods at 133 locations across the state of Pernambuco, with climate varying from tropical humid in the coastal area to semiarid in the deep inland region. The results show that all SPI series (independently of time scale and climatic conditions) display multifractal dynamics, which can be described by properties of the multifractal spectrum. It was found that SPI series exhibit long-term persistence and multifractality that becomes stronger as accumulation time increases. For all temporal scales, the values of Hurst exponent increase in the west-east direction indicating stronger persistence of dry/wet conditions in the coastal region Zona da Mata, with a tropical humid climate, than in the dry semiarid inland Sertão region. For SPI-1, SPI-3, and SPI-6 which describe short and medium conditions the width of the multifractal spectrum (degree of multifractality) doesn't vary across the studied area, while for SPI-12 which describes long-term conditions the width of the multifractal spectrum decreases in a west-east direction, indicating stronger multifractality and higher complexity in long duration precipitation anomalies in the semiarid Sertão region.

Anti-persistent adherence dynamic of the COVID-19 vaccines

This research explores the multifractal dynamics of time series of the daily number of vaccinees for COVID-19, considering six European countries (Belgium, Denmark, France, Germany, Greece and Italy) using the Multifractal Detrended Fluctuations Analysis (MF-DFA). We calculate the multifractal spectrum f(α) and apply a fourth-degree polynomial regression fit to estimate the complexity parameters that describe the degree of multifractality of the underlying process. We found that the multifractal dynamics of all these countries are characterized by strongly anti-persistent behavior (α 0 < 0.5) a lower degree of multifractality, and small fluctuations are dominant in the multifractal spectrum. From an immunization perspective, it means that a panorama that encompasses the population’s behaviour is marked by the dynamics of anti-persistent adherence to COVID-19 vaccines. Our findings confirm that the period of immunization of the population that adhered to the vaccination campaigns is short and that the application of new doses of vaccines must obey this phenomenology to keep people safe. In addition, we used the multifractal efficiency coefficient to rank countries that are most proactive in developing campaigns that promote greater adherence and loyalty to COVID-19 vaccines. Our findings indicate that Germany, Belgium and France were more efficient than Greece, Denmark and Italy.

Evaluation of non-linear heart rate variability using multi-scale multi-fractal detrended fluctuation analysis in mice: Roles of the autonomic nervous system and sinoatrial node.

Nonlinear analyses of heart rate variability (HRV) can be used to quantify the unpredictability, fractal properties and complexity of heart rate. Fractality and its analysis provides valuable information about cardiovascular health. Multi-Scale Multi-Fractal Detrended Fluctuation Analysis (MSMFDFA) is a complexity-based algorithm that can be used to quantify the multi-fractal dynamics of the HRV time series through investigating characteristic exponents at different time scales. This method is applicable to short time series and it is robust to noise and nonstationarity. We have used MSMFDFA, which enables assessment of HRV in the frequency ranges encompassing the very-low frequency and ultra-low frequency bands, to jointly assess multi-scale and multi-fractal dynamics of HRV signals obtained from telemetric ECG recordings in wildtype mice at baseline and after autonomic nervous system (ANS) blockade, from electrograms recorded from isolated atrial preparations and from spontaneous action potential recordings in isolated sinoatrial node myocytes. Data demonstrate that the fractal profile of the intrinsic heart rate is significantly different from the baseline heart rate in vivo, and it is also altered after ANS blockade at specific scales and fractal order domains. For beating rate in isolated atrial preparations and intrinsic heart rate in vivo, the average fractal structure of the HRV increased and multi-fractality strength decreased. These data demonstrate that fractal properties of the HRV depend on both ANS activity and intrinsic sinoatrial node function and that assessing multi-fractality at different time scales is an effective approach for HRV assessment.

Mathematical Modeling of Stress Using Fractal Geometry; The Power Laws and Fractal Complexity of Stress

In this study, we analyze the physiological data during real-world driving tasks to determine whether driver’s relative stress is mono-fractal or multi-fractal. We use the PhysioNet database including long term ECG recordings from 15 healthy volunteers, taken while they were driving on a prescribed route including city streets and highways in and around Boston, Massachusetts. The vibration analysis such as power spectral densities (PSD) analysis has been performed to estimate the exponent from realizations of these pro- cesses and to find out if the signal of interest exhibits a power-law PSD. Multifractal dynamics of heartbeat interval signals have been assessed by multifractal spectrum analysis to explore the possibility that ECG recordings belong to class of multi-fractal process for which a large number of scaling exponents are re- quired to characterize their scaling structures. We apply Higuchi algorithm to find the fractal complexity of each cardiac rhythm for different time intervals. According to our analysis, we investigate that driver’s ECG signals under relative stress follows fractal behavior unlike control healthy signals which are multi-fractal. Our findings provide a comprehensive framework for detect stress and differentiate people who experience stress with normal people without stress which is crucial in finding the best diagnostic and controlling strat- egy in fight against many health problems due to stress, such as high blood pressure, heart disease, obesity and diabetes. Moreover, being able to recognize stress can help us to manage it.

Interplay multifractal dynamics among metal commodities and US-EPU

This research employs the Multifractal Detrended Fluctuations Analysis (MF-DFA) and the Multifractal Detrended Cross-Correlation Analysis (MF-DCCA) to study the components of the observed multifractality considering eight metals commodities time series (Gold, Silver, Platinum, Palladium, Copper, Lead, Nickle and tin) and economic policy uncertainty (US-EPU) index time series. For both multifractal methods, we analyze the generalized Hurst exponent h(q) and the Rényi exponent τ(q), and evaluate their statistical properties, which allow us to examine separately the small-scale contributing (primarily via the negative moments q) and the large scale (via the positive moments q). Also, we calculate the multifractal spectrum f(α) and perform a fourth-degree polynomial regression fit to evaluate the complexity parameters that describe the degree of multifractality, considering the volatility time series and the volatility time series pairs. We find that these volatility time-series (α0>0.5) and these volatility time series pairs (αxy(0)>0.5) are characterized by overall persistent behavior. We discover that Gold seems more related to the other commodities than (US-EPU) index, which is relevant for the different market agents, mainly investors, which could retrieve important information from these results, for example, for diversification purposes.

On the fractal geometry of different heart rhythms

In this study, we explore the possibility that ECG recordings belong to class of multifractal process for which a large number of scaling exponents are required to characterize their scaling structures. We use the BIDMC Congestive Heart Failure database including long term ECG recordings from 11 men, aged 22 to 71, and 4 women, aged 54 to 63 with severe congestive heart failure and the MIT-BIH Arrhythmia database that contains 48 half-hour excerpts of two-channel ambulatory ECG recordings, obtained from 47 subjects studied by the BIH Arrhythmia Laboratory between 1975 and 1979. We compare these two chronic heart diseases with the control people in the MIT-BIH Normal Sinus Rhythm database which includes 18 long-term ECG recordings of 5 men, aged 26 to 45, and 13 women, aged 20 to 50 without significant arrhythmia. The vibration analysis such as power spectral densities (PSD) analysis has been performed for differentiating the time series. Multifractal dynamics of heartbeat interval signals have been assessed by multifractal spectrum analysis to differentiate normal signals with arrhythmia and severe congestive heart failure patients. We apply Higuchi algorithm to find the fractal complexity of each cardiac rhythm and then compare the signals for different time intervals. According to our analysis, we investigate that neither the power spectral densities nor the Higuchi algorithm to find the fractal dimension alone were sufficient to separate different classes of patients and healthy people. However, when multifractal analysis and scaling exponent were used as a classifier, the three classes were well separated. In addition, multifractal analysis revealed that we have a narrow range of exponents for arrhythmia and congestive heart failure subjects and as a result, a clear loss of multifractality for them. It was of great significance to show that we have a narrower range of exponents for arrhythmia subject compared to congestive heart failure subject which is useful to recognize these two classes of patients with heart disease. Our findings provide a comprehensive framework for diagnostic and classifying different patients with cardiac disease such as arrhythmia and congestive heart failure and differentiate them with normal people without heart disease which is crucial in finding the best diagnostic and controlling strategy in fight against chronic heart disease.

Modeling the multifractal dynamics of COVID-19 pandemic

To describe the COVID-19 pandemic, we propose to use a mathematical model of multifractal dynamics, which is alternative to other models and free of their shortcomings. It is based on the fractal properties of pandemics only and allows describing their time behavior using no hypotheses and assumptions about the structure of the disease process. The model is applied to describe the dynamics of the COVID-19 pandemic from day 1 to day 699 from the beginning of the pandemic. The calculated parameters of the model accurately determine the parameters of the trend and the large jump in daily diseases in this time interval. Within the framework of this model and finite-difference parametric nonlinear equations of the reduced SIR (Susceptible-Infected-Removed) model, the fractal dimensions of various segments of daily incidence in the world and variations in the main reproduction number of COVID-19 were calculated based on the data of COVID-19 world statistics.