Kolmogorov Entropy Research Articles

Kolmogorovʼ entropy to investigate variation of forest soil properties in the Czech Republic

A measure of variation in values of physico-chemical properties of different soil groups is influenced by deterministic uncertainty. In this study, we assessed variation level of soil horizon properties in the most wide-spread forest soil groups on territory of the Czech Republic in the period 1953–2010. The assessment of variation was based on the selection of potentially correlating grain-size and chemical properties and a regression analysis (p < 0.05) between normal values (1981–2010) and Kolmogorov (K-) entropy of the forest soil quantities. Cambisols (73%), Stagnosols (10%) and Leptosols (4%) were the most frequently sampled soil groups in the database of 15,287 pits from the state pilot forest surveys during the normal period. The K-entropies of clay, Ntot, CaO and MgO indicated the mean contents in the soil groups more significantly than the entropies of loam, pH and Al2O3. The low variable clay content differentiated the series of Cambisols-Podzols (CM-PZ) from other soil groups. On the contrary, the higher stable clay content differentiated the Luvisols-Retisols-Stagnosols (LV-RT-ST) series. The low stable clay content and the higher Ntot and CaO contents were characteristic of Fluvisols and Gleysols. The CM-PZ series varied from LV-RT-ST by more variable contents of finer grain particles, Ntot and CaO. Relationships between soil properties means and K-entropies suggest different development series.

A Deformation Prediction Model of High Arch Dams in the Initial Operation Period Based on PSR-SVM-IGWO

The deformation prediction of the dam in the initial stage of operation is very important for the safety of high dams. A hybrid model integrating chaos theory, support vector machine (SVM), and an improved Grey Wolf Optimization (IGWO) algorithm is developed for deformation prediction of dam in the initial operation period. Firstly, the chaotic characteristics of the dam deformation time series will be identified, mainly using the Lyapunov exponent method, the correlation dimension method, and the Kolmogorov entropy method. Secondly, the SVM-IGWO model based on phase space reconstruction (PSR) is established for deformation forecasting of the dam in the initial operation period. Taking SVM as the core, the deformation time series is reconstructed in phase space to determine the input variables of SVM and the GWO algorithm is improved to realize the optimization of SVM parameters. Finally, take the actual monitoring displacement of Xiluodu super-high arch dam as an example. The engineering application example shows that, compared with the existing models, the prediction accuracy of the PSR-SVM-IGWO model established in this paper is improved.

The Methodology of Distinguish Between Random and Chaotic Machine Tool Oscillations

Machine tool oscillations are irregular or aperiodic. Most often, these oscillations are chaotic but, in some cases, they can be quasi-periodic or random. The methodology for characterizing oscillations in the first of two steps uses the nonparametric hypothesis tests which the observed oscillations confirmed as irregular. The methodology for the final characterization of oscillations is based on chaos quantifiers. A time series defined as the measured values of oscillations in the time domain is the basis for calculating the quantifiers of chaos. There are four quantifiers of chaos: the Lyapunov exponent, Kolmogorov entropy, fractal dimension and correlation dimension. The correlation dimension and Kolmogorov entropy are important for distinguishing between random and chaotic oscillations. Other quantifiers of chaos are not used for this purpose. The methodology requires a multidisciplinary approach based on combining Nonlinear Dynamics and Probability Theory and Statistics. The methodology can be applied to many oscillating phenomena. Therefore, the paper mainly used the term oscillations, not vibrations, chatter, etc.

Transition to chaos in lid–driven square cavity flow* *Project supported by theNational Natural Science Foundation of China (Grant No. 12162001), the Natural Science Foundation of Ningxia (Grant No. 2019AAC03129), and the Construction Project of First-ClassDisciplines in Ningxia Higher Education (Grant No. NXYLXK2017B09).

To date, there are very few studies on the transition beyond second Hopf bifurcation in a lid–driven square cavity, due to the difficulties in theoretical analysis and numerical simulations. In this paper, we study the characteristics of the third Hopf bifurcation in a driven square cavity by applying a consistent fourth-order compact finite difference scheme rectently developed by us. We numerically identify the critical Reynolds number of the third Hopf bifurcation located in the interval of (13944.7021,13946.5333) by the method of bisection. Through Fourier analysis, it is discovered that the flow becomes chaotic with a characteristic of period-doubling bifurcation when the Reynolds number is beyond the third bifurcation critical interval. Nonlinear time series analysis further ascertains the flow chaotic behaviors via the phase diagram, Kolmogorov entropy and maximal Lyapunov exponent. The phase diagram changes interestingly from a closed curve with self-intersection to an unclosed curve and the attractor eventually becomes strange when the flow becomes chaotic.

Constructing Keyed Strong S-Box Using an Enhanced Quadratic Map

As the only nonlinear component for symmetric cryptography, S-Box plays an important role. An S-Box may be vulnerable because of the existence of fixed point, reverse fixed point or short iteration cycles. To construct a keyed strong S-Box, first, a 2D enhanced quadratic map (EQM) was constructed, and its dynamic behaviors were analyzed through phase diagram, Lyapunov exponent, Kolmogorov entropy, bifurcation diagram and randomness testing. The results demonstrated that the state points of EQM have uniform distribution, ergodicity and better randomness. Then a keyed strong S-Box construction algorithm was designed based on EQM, and the fixed point, reverse fixed point, and short cycles were eliminated. Experimental results verified the algorithm’s feasibility and effectiveness.

Identification of drivers’ driving habits and shift schedule correction for vehicles with automatic transmission

Resource conservation has become a hot topic, and driving habits also have a significant impact on a car’s fuel consumption. In view of the different needs of drivers with different driving habits for automatic transmission and shift characteristics, a vehicle automatic transmission, and shift correction control strategy based on driving habit recognition is proposed. Based on the analysis of drivers’ driving behavior, the phase space reconstruction method is first used to reconstruct the time series of driving control signals, and the driving habit identification and gear shift correction control are conducted based on the correlation dimension and Kolmogorov entropy evaluation index. Simulation and real car test show that the identification method based on phase space reconstruction method and driving habits evaluation index can accurately identify drivers’ driving habits. The gear shift correction control strategy based on driving habit recognition fully meets the different requirements of different drivers for the gearshift performance of vehicles and improves the intelligence degree of automatic transmission of vehicles.

Attacks and Defenses for Single-Stage Residue Number System PRNGs

This paper explores the security of a single-stage residue number system (RNS) pseudorandom number generator (PRNG), which has previously been shown to provide extremely high-quality outputs when evaluated through available RNG statistical test suites or in using Shannon and single-stage Kolmogorov entropy metrics. In contrast, rather than blindly performing statistical analyses on the outputs of the single-stage RNS PRNG, this paper provides both white box and black box analyses that facilitate reverse engineering of the underlying RNS number generation algorithm to obtain the residues, or equivalently key, of the RNS algorithm. We develop and demonstrate a conditional entropy analysis that permits extraction of the key given a priori knowledge of state transitions as well as reverse engineering of the RNS PRNG algorithm and parameters (but not the key) in problems where the multiplicative RNS characteristic is too large to obtain a priori state transitions. We then discuss multiple defenses and perturbations for the RNS system that fool the original attack algorithm, including deliberate noise injection and code hopping. We present a modification to the algorithm that accounts for deliberate noise, but rapidly increases the search space and complexity. Lastly, we discuss memory requirements and time required for the attacker and defender to maintain these defenses.

Hydraulic pump fault diagnosis based on chaotic characteristics of speed signals under non-stationary conditions

Pumps are vital components in hydraulic equipment, and their malfunction is directly related to the normal operation of the whole system. The hydraulic system is accompanied by the conversion of multi-domain energy during operation. In particular, it exhibits non-stationarity and nonlinearity under variable operating conditions, which brings difficulties to fault diagnosis. In this paper, the instantaneous angular speed (IAS) signal obtained by the equal-angle measurement is studied to diagnose pump faults under non-stationary conditions. For this purpose, a synchroextracting normal S transform method is proposed to extract the time-frequency feature components properly. Then, the extracted signal is reconstructed into a high-dimensional phase space based on the improved C-C method. On this basis, the quantitative analysis indicators of pump state are obtained by the G-P algorithm and small-data method, including correlation dimension, Kolmogorov entropy, and largest Lyapunov exponent. Pump faults can be classified in a chaotic space by using these sensitive features. The results show that the proposed method is capable of diagnosing different states of pumps and robust to the variation of load values.

Nonlinear Properties Analysis of Metallic Fatigue Damage Process Based on the Combined Nonlinear Parameter

To effectively extract the nonlinear properties of fatigue crack extension and evaluate the fatigue damage of metallic specimens, the chaos and fractal theory is applied to analyze the nonlinear output signal in a nonlinear ultrasonic experiment. A combined nonlinear parameter using the Lyapunov exponent, Kolmogorov entropy, and correlation dimension is proposed to evaluate the nonlinearity of fatigue damage. The nonlinear ultrasonic experiment is carried out on specimens with different stages of fatigue life, wherein the ultrasonic nonlinear output signal is collected and its combined nonlinear parameters are calculated. With the increase of fatigue damage in metallic specimens, the combined nonlinear parameter increases monotonically and tends to become saturated when a macroscopic crack appears, which indicates that complexity and randomness gradually increase during the accumulation of fatigue damage. According to microcrack propagation, the combined nonlinear parameter is very sensitive to the evolution of fatigue damage in metallic specimens, especially earlier fatigue damage. The relationship between the combined nonlinear parameter and the fatigue damage of specimens is established, which can provide an effective analytical method for evaluating fatigue damage and predicting the service life of specimens.

Smoothing and finite-dimensionality of uniform attractors in Banach spaces

The aim of this paper is to find an upper bound for the fractal dimension of uniform attractors in Banach spaces. The main technique we employ is essentially based on a compact embedding of some auxiliary Banach space into the phase space and a corresponding smoothing effect between these spaces. Our bounds on the fractal dimension of uniform attractors are given in terms of the dimension of the symbol space and the Kolmogorov entropy number of the embedding. In addition, a dynamical analysis on the symbol space is also given, showing that the finite-dimensionality of the hull of a time-dependent function is fully determined by the tails of the function, which allows us to consider more general non-autonomous terms than quasi-periodic functions.As applications, we show that the uniform attractor of the 2D Navier-Stokes equation is finite-dimensional in H and in V, and that of a reaction-diffusion equation is finite-dimensional in L2 and in Lp, with p>2.

Statistical Characterization of Flow Structure of Air–water Two-phase Flow in Airlift Pump–Bubble Generator System

The aim of this study is to identify the flow structure in an airlift pump–bubble generator system by using experimentally obtained differential pressure signals. Differential pressure measurements were applied at both the bottom and top test sections. The normalized time variation of the differential pressure data was analyzed using the probability density function (PDF), power spectral density function (PSDF), Kolmogorov entropy, and discrete wavelet transform (DWT). The results indicate that the water movement mechanism in the riser pipe could be divided into three regions, namely fixed liquid, locally moving liquid, and fully moving liquid, depending on the supplied superficial air velocity. Moreover, the chaotic in riser pipe increases with the increase of the submergence ratio at high supplied superficial air velocity. The chaotic level at the bottom of the test section was higher than that at the top of the test section at low supplied superficial air velocity. Finally, the observed flow patterns closely conformed to the previous definition and flow pattern maps. The bubbly flow was classified into clustered, homogeneous, and cap bubble regimes. The slug flow regime was classified into bubbly stable slug, bubbly unstable slug, and slug-churn, wherein the frequencies of both the liquid and air slugs increased with an increase in the air superficial velocity and submergence ratio.

Probable Relationship between COVID-19, Pollutants and Meteorology: A Case Study at Santiago, Chile

We present here a study about the possible spread of covid-19 pandemic between human’s beings through aerosols contained in urban air polluted by respirable particulate matter and tropospheric ozone, as well as the incidence of local meteorology in an area with orographic basin characteristics and in a certain period of time. Hourly time-series data of three meteorological variables—temperature, relative humidity, wind speed—and three pollutants—PM10, PM2.5 and O3—were considered together with hourly data from the highest number accumulated sick's in seven communes—chosen at random—in Santiago, Chile, studying a probable link between them. From the epidemic perspective, the infected patients number was linked to the hourly time-series of meteorological and pollutant variables, generating new time-series. Nonlinear analysis and the chaos theory formalism was applied to these new time-series, obtaining the largest Lyapunov exponent, correlation dimension, Kolmogorov entropy, Hurst exponent and the Lempel-Ziv complexity. Our preliminary results show meteorological and air pollution variables can be part of the elements fraction that give sustainability to the accumulated growth of infected patients and favor the pandemic spread, making the accumulated sick’s curve chaotic and complex. In addition, environmental pollution could worsen disease conditions like coronavirus (COVID-19) infection. For all time-series, the Lempel-Ziv complexity turned out to be between 0 and 1 which is indicative of connectivity and chaos. The largest Lyapunov exponent as well as the Kolmogorov entropy were positive which also exhibits chaos. The Hurst exponent was found to be greater than 0.5 and less than 1 for all time-series, indicating positive long-term autocorrelation. Finally, the correlation dimension was less than 5, revealing that new time-series constructed are not random.

СИНЕРГЕТИЧНІ МЕТОДИ В ПРОГНОЗУВАННІ ДИНАМІКИ ПРОЦЕСІВ ФОНДОВИХ РИНКІВ

This article is devoted to forecasting the dynamics of processes occurring in the stock market, the methods of chaotic dynamics. Chaotic dynamics is one of the areas of synergetics. Although there are many effective methods for modeling securities price forecasts, emerging stock markets are characterized by long-term stability and short-term instability. In this situation, to study the processes occurring in the markets, chaos theory can be used, which includes methods of theory of dynamical systems and fractal analysis. An example of practical implementation of this approach using neural networks for the share price of the company "Pharmacy 36 and 6" at the close of trading for 2014–2019. The company "Pharmacy 36 and 6" is a retail network of pharmacies "Lekpharm", which is actively working and developing in the pharmaceutical market of Ukraine since 2001. The initial data were obtained on the website of the holding "FINAM". Neural networks have the ability to learn. At the same time, in the process of learning there is a phenomenon similar to self-organization. This property is characteristic of chaotic systems. The most important characteristics of chaos-dynamic analysis are calculated: Hausdorff dimension and Kolmogorov entropy. Their values indicate that the studied statistical series contains a chaotic component. The algorithm of technical application of neural networks is offered in the article. The model is built using neural network technology. Some of the input parameters of the network are determined based on the analysis performed. Comparison of the results obtained on the model deviate slightly from the actual data. It could be considered. that the model is adequate and can be used in practice. Forecasting the dynamics of securities can give a certain economic effect, a specific financial benefit to the owners of securities. Synergistic methods do not always lead to a solution to a problem or task. But they lead to a deeper and more accurate understanding of reality, which allows one to give an explanation for incomprehensible phenomena, such as pre-construction stability and short-term instability. Therefore, the synergistic economic paradigm has an interest on the part of researchers and policy makers.

Kolmogorov Entropy of Random Attractors for Stochastic Lattice Systems With White Noise

Based on the asymptotic behavior theory for solutions of dissipative stochastic lattice systems, the Kolmogorov entropy of random attractors in stochastic Klein-Gordon-Schrdinger lattice dynamic systems with white noise was studied with the element decomposition method and the topological properties of polyhedral sphere covering in the finite dimensional space, and an upper bound for the Kolmogorov entropy of random attractors was obtained.

Reliable identification of the first transition velocity in various bubble columns based on accurate sophisticated methods

In bubble columns (BCs), the reliable identification of the first transition velocity Utrans-1, marking the end of the homogeneous regime, is an essential prerequisite for the successful BC design since this parameter marks the onset of bubble coalescence. In this work, a new database with 13 experimental Utrans-1 values at various operating conditions is reported. They are identified precisely by different sophisticated methods, which are considered most suitable and powerful for the analysis of the specific experimental data used. The BCs have different column diameters (and bed aspect ratios) and they are equipped with different perforated plate (PP) gas distributors. For Utrans-1 identification, eight different time-dependent signals recorded in air-deionized water, air-tap water, nitrogen-tap water, nitrogen-ethanol (96%), air-therminol and air-benzonitrile systems have been analyzed. The hidden information in the time series has been quantified by reliable identification parameters such as degree of randomness of the signal, Kolmogorov, information and reconstruction entropies and degree of disorder of the signal. Their profiles as a function of superficial gas velocity have been used for flow regime identification.In the case of column diameters between 0.1 m and 0.46 m and bed aspect ratios beyond 5, a simple empirical rule-of-thumb has been established: when the open area (OA) of the PP gas distributors is ≤1.0%, the Utrans-1 value should occur at around 0.03 m/s. On the other hand, when the OA is greater than 1.0%, then the Utrans-1 value is identifiable at 0.04 m/s. It has been found that this new rule yields better predictions than the most popular empirical correlation of Reilly et al. (1994).

Estimation the Entropies Parameters of Electroencephalogram Operator Influenced by Noise Electromagnetic Radiation

Parameters of the electroencephalogram (EEG) operator under the influence of noise electromagnetic radiation have been analyzed. The electroencephalogram was analyzed in the following EEG leads: Fp1-A1, Fp2-A2, F3-A1, F4-A2, C3-A1, C4-A2, P3-A1, P4-A2, O1- A1, O2-A2, F7- A1, F8-A2, T3-A1, T4-A2, T5-A1, T6-A2. The background state (healthy person) and the state in the presence of noise electromagnetic radiation were analyzed. Quantitative parameters of influence were the Kolmogorov entropy, approximate entropy and sample entropy; spectral power density of theta-rhythms. The results indicate a possible negative effect of noise electromagnetic radiation on the central nervous system.

Maximally Chaotic Dynamical Systems of Anosov–Kolmogorov

The maximally chaotic K-systems are dynamical systems which have nonzero Kolmogorov entropy. On the other hand, the hyperbolic dynamical systems that fulfil the Anosov C-condition have exponential instability of phase trajectories, mixing of all orders, countable Lebesgue spectrum and positive Kolmogorov entropy. The C-condition defines a rich class of maximally chaotic systems which span an open set in the space of all dynamical systems. The interest in Anosov-Kolmogorov C-K systems is associated with the attempts to understand the relaxation phenomena, the foundation of the statistical mechanics, the appearance of turbulence in fluid dynamics, the non-linear dynamics of the Yang-Mills field as well as the dynamical properties of gravitating N-body systems and the Black hole thermodynamics. In this respect of special interest are C-K systems that are defined on Reimannian manifolds of negative sectional curvature and on a high-dimensional tori. Here we shall review the classical- and quantum-mechanical properties of maximally chaotic dynamical systems, the application of the C-K theory to the investigation of the Yang-Mills dynamics and gravitational systems as well as their application in the Monte Carlo method.

Observers and relative entropy of G-sets

In the current study, a new type of relative entropy is presented through observable objects in quantum mechanics as the probability measures. The relative probability measures as an extension of probability measures are considered via one-dimensional observers. The notion of relative entropy of the semigroup on a relative measure space is introduced using the mathematical modeling of an observable object. Moreover, it is proved that this nonnegative quantity is invariant under $$(\Theta _1 ,\Theta _2 )$$-isomorphism. Finally, we applied this concept to specific semigroups to extract Kolmogorov entropy of a dynamical system as a special case.

Correlation between jerky flow and jerky dynamics in a nanoscratch on a metallic glass film

Chaos has been well understood in dynamic system, however, how the chaotic behavior occur in jerky flow in material, is still not clear, and is lack of specific chaotic attractor. Here the jerky evolution of lateral force and the stair-like fluctuation of lateral displacement are observed for Ni62Nb38(at.%) metallic glass film during nanoscratch process. This jerky flow is investigated by using the largest Lyapunov exponent, Kolmogorov entropy and fractal dimension, and chaotic behavior of lateral force-time and normal displacement-lateral displacement sequences is verified. In addition to time series analysis, it is found that jerk equation can be used to describe the jerky flow of the metallic-glass film during nanoscratch. More importantly, unambiguous chaotic attractor is presented by jerky dynamics using “jerk”-singularities, namely the total change rate of lateral force relative to scratch time. These reveal an inner connection between jerky flow and jerky dynamics in nanoscratch of a metallic-glass film.

On Temporal Scale Separation in Coupled Data Assimilation with the Ensemble Kalman Filter

Data assimilation for systems possessing many scales of motions is a substantial methodological and technological challenge. Systems with these features are found in many areas of computational physics and are becoming common thanks to increased computational power allowing to resolve finer scales and to couple together several sub-components. Coupled data assimilation (CDA) distinctively appears as a main concern in numerical weather and climate prediction with major efforts put forward by meteo services worldwide. The core issue is the scale separation acting as a barrier that hampers the propagation of the information across model components (e.g. ocean and atmosphere). We provide a brief survey of CDA, and then focus on CDA using the ensemble Kalman filter (EnKF), a widely used Monte Carlo Gaussian method. Our goal is to elucidate the mechanisms behind information propagation across model components. We consider first a coupled system of equations with temporal scale difference, and deduce that: (i) cross components effects are strong from the slow to the fast scale, but, (ii) intra-component effects are much stronger in the fast scale. While observing the slow scale is desirable and benefits the fast, the latter must be observed with high frequency otherwise the error will grow up to affect the slow scale. Numerical experiments are performed using the atmosphere-ocean model, MAOOAM. Six configurations are considered, differing for the strength of the atmosphere-ocean coupling and/or the number of model modes. The performance of the EnKF depends on the model configuration, i.e. on its dynamical features. A comprehensive dynamical characterisation of the model configurations is provided by examining the Lyapunov spectrum, Kolmogorov entropy and Kaplan–Yorke attractor dimension. We also compute the covariant Lyapunov vectors and use them to explain how model instabilities act on different model’s modes according to the coupling strength. The experiments confirm the importance of observing the fast scale, but show also that, despite its slow temporal scale, frequent observations in the ocean are beneficial. The relation between the ensemble size, N, and the unstable subspace dimension, n_0, has been studied. Results largely ratify what known for uncoupled system: the condition Nge n_0 is necessary for the EnKF to work satisfactorily. Nevertheless the quasi-degeneracy of the Lyapunov spectrum of MAOOAM, with many near-zero exponents, is potentially the cause of the smooth gradual reduction of the analysis error observed for some model configurations, even when N>n_0. Future prospects for the EnKF in the context of coupled ocean-atmosphere systems are finally discussed.