Kolmogorov Entropy Research Articles

Constructing keyed strong S-box with optimized nonlinearity using nondegenerate 2D hyper chaotic map

Abstract In the realm of information security, the nonlinearity of an S-Box is a pivotal attribute for ensuring its effectiveness. S-Boxes randomly generated with chaotic maps, however, often exhibit insufficient nonlinearity. To construct a robust S-Box with enhanced nonlinearity, we have developed a nondegenerate two-dimensional hyper chaotic map (2D-NDHCM). We rigorously evaluated its dynamic properties through bifurcation and phase diagrams, Lyapunov exponents, Kolmogorov entropy, correlation dimension, and randomness tests to assess the map's resilience. Subsequently, we employed the 2D-NDHCM to devise a novel algorithm for constructing and optimizing a keyed strong S-Box, leveraging simulated annealing technique to achieve superior nonlinearity. Finally, We conducted performance analyses of the optimized S-Box using the classic six criteria, and the results confirmed the algorithm's practicality and effectiveness.

ЭНТРОПИЙНЫЙ ПОДХОД К ПРОГНОЗИРОВАНИЮ ПРИБЫЛИ ОТ РЕАЛИЗАЦИИ ИННОВАЦИОННЫХ ПРОДУКТОВ И УСЛУГ В СЕВЕРО-ЗАПАДНОМ ФЕДЕРАЛЬНОМ ОКРУГЕ РОССИЙСКОЙ ФЕДЕРАЦИИ

The influence of entropy processes in the economy on the profit from the sale of an innovative product (service) is researched, regardless of whether the product creates a new market in the economy of the Northwestern Federal District, or occupies a niche in the established market. The state of the product is described by a vector-matrix differential equation, which can be expressed by a system of differential equations, one of which describes the change in profit. The equation con- tains random components, which are proposed to be determined using Kolmogorov entropy. The implementation of the pro- posed approach will make it possible to develop procedures for an information system of analytics, marketing and promotion of innovation.

Constructing a Nondegenerate m-Dimensional Integer-Domain Chaotic Map Model over GF(2n) with Application in PRNG

To address the problem of dynamic degradation over a finite-precision platform of chaotic maps and the reversibility of linear chaotic maps, we propose an improved model over GF([Formula: see text]) that is called the nondegenerate m-Dimensional [Formula: see text] Integer-Domain Chaotic Maps (mD-IDCMs). This model incorporates modular exponentiation operation, and is capable of constructing nondegenerate IDCMs of any dimension. Moreover, we prove the irreversibility of mD-IDCM and analyze its chaotic behaviors in terms of positive Lyapunov Exponents (LEs). The results of theoretical analysis show that the proposed mD-IDCM model can obtain the desired positive LEs by appropriately configuring its coefficient matrix. Then, we present two instances, and analyze their LEs, Kolmogorov entropy, Sample entropy, Correlation dimension, and the dynamic analysis indicates that the chaotic map constructed by mD-IDCM has ergodicity within a sufficiently large chaotic range. Finally, we design a Pseudo-Random Number Generator (PRNG) with a key to verify the practicability of the mD-IDCM.

Capacity configuration of cascaded hydro-wind-photovoltaic complementary systems considering comprehensive benefits based on a synergetic-orderly framework

The rational configuration of hydro-wind-photovoltaic complementary system is fundamental to fully leveraging the regulation capacity of hydropower and the complementarity between different natural resources. However, complementary systems involving cascaded reservoirs often have comprehensive utilization requirements, which current techno-economic frameworks struggle to address. To solve the above issues, this study proposes a novel framework based on the theory of synergetics to determine the optimal capacities for wind and photovoltaic power considering the comprehensive benefits. Firstly, an inner operation model aimed at maximizing overall order degree is established to optimize the system’s operational performance. Secondly, Kolmogorov entropy is utilized in the outer layer to characterize the synergy of different wind and photovoltaic capacity schemes, thereby selecting the one with the best synergistic effect. Additionally, current techno-economic evaluation indicators are introduced to validate the framework’s effectiveness. A case study of the clean energy base on the upper Yellow River was conducted, and the results indicate that: (1) The proposed framework effectively addresses the comprehensive benefits of multiple subsystems and the optimal capacity scheme performs well in economic and technical aspects. (2) Compared to variations in wind and solar resources, inflow conditions and agricultural water demand have a greater impact on the capacity configuration and operational performance. (3) The optimal capacity ratio of hydro to renewable energy in the upper Yellow River is around 1:0.59. Therefore, this study provides important theoretical support for expanding capacity configuration methods in multi-energy complementary systems.

Nonlinear Characteristics of Radio Variability in Quasar 3C 446

Quasars are characterized by violent and large amplitude variability in all the observation wavelengths, and the analysis of optical variability is useful for developing theoretical models. Long-term optical variability data of quasar 3C 446 were collected from the University of Michigan Radio Astronomy Observatory database in the 4.8, 8.0, and 14.5 GHz radio bands from 1976 to 2012. Due to the complexity of the variability data, it's hard to study by the linear time series analysis methods. For learning more about non-linear characteristics of the temporal evolution of quasar variability, the Ensemble Empirical Mode Decomposition and nonlinear analysis are used to analyse chaotic dynamics, fractal properties, and periodicity. This paper focuses on whether there is a significant difference between the periodic and non-linear properties of the quasar variability before and after the removal of the periodic or chaotic components. It turns out that the variability of quasar 3C 446 in the radio bands consists of periodic, trend, and chaotic components, and the periodic and trend components are dominant. The periods of the variability after removing the chaotic and trend components are exactly the same as the periods of the original variability data, but the chaotic and fractal characteristics of the two are significantly different. The saturated correlation dimension indicates that the reconstruction of the dynamical system requires more independent parameters than the original optical variables after the removal of the periodic and trend components. The Kolmogorov entropy indicates that the loss of information is greater for the former than the latter, and the system is more chaotic and more complex. The Hurst value indicates that the self-similarity and long-range correlation are slightly stronger for the latter than the former.

Generalized kinetic theory of coarse-grained systems. I. Partial equilibrium and Markov approximations

The general kinetic theory of coarse-grained systems is presented in the abstract formalism of communication theory developed by Shannon and Weaver, Khinchin and Kolmogorov. The martingale theory shows that, under reasonable, general hypotheses, coarse-grained systems can be approximated by generalized Markov systems. For mixing systems, the Kolmogorov entropy production can be defined for nonstationary processes as Kolmogorov defined it for stationary processes.

Analysis of application of entropy methods of vibration diagnostic signal processing to assess technical condition of pipelines

The purpose of this article is to review the existing reliability problems of pipeline systems of power complexes. The article considers the existing statistical and logistics systems, which allow to process diagnostic information when assessing the technical condition of pipelines. Modern diagnostic methods are mainly based on the use of vibration, sound, and ultrasonic sensors. The presence of a defect in a pipeline is determined by analysis of the amplitude of a diagnostic signal. Higher efficiency in detecting defects was shown by probability-statistical methods of signal analysis, which are based on chaos theory. One such method is entropy analysis. Analysis of modern signal processing methods has shown that methods based on chaos theory are the most effective. The possibility of using entropy indices as sensitive diagnostic signs is considered. Comparative analysis of signal processing was carried out using entropy methods (Shannon entropy, Kolmogorov entropy) and using known statistical and logistic methods (Fourier Transform, Wavelet Transform, Hilbert-Huang Transform). The analysis results showed that entropy indicators respond to a change in signal structure caused by the presence of a defect in the pipeline or Entropy analysis is a promising method of processing diagnostic signals when assessing the technical condition of pipelines.

A non-degenerate n-dimensional integer domain chaotic map model with application to PRNG

To address the limitations of existing chaotic maps, we proposed a non-degenerate n-dimensional (n ≥ 2) integer domain chaotic map (nD-IDCM) model that can construct any non-degenerate n-dimensional integer domain chaotic maps. Moreover, we analyzed its chaotic behavior through Lyapunov exponent, and found that the nD-IDCM generates chaotic sequences in the integer domain, which effectively resolves the issue of finite precision effect when implementing existing chaotic maps on computers or digital devices. To verify the effectiveness of nD-IDCM, we presented two instances to demonstrate how the positive Lyapunov exponents can be regulated by manipulating the parameter matrix. Subsequently, we have scrutinized their dynamical behavior using Kolmogorov entropy, sample entropy, correlation dimension and randomness testing via TestU01. Finally, to assess the feasibility of nD-IDCM, we devised a keyed pseudo random number generator (PRNG) based on a 3D-IDCM that can ensure superior randomness and unpredictability. Experimental results indicated that integer domain chaotic maps constructed using nD-IDCM have desirable Lyapunov exponents and exhibit ergodicity within a sufficient larger chaotic range.

Rail Flaw Detection via Kolmogorov Entropy of Chaotic Oscillator Based on Ultrasonic Guided Waves.

Ultrasonic guided wave (UGW) inspection is an emerging non-destructive testing(NDT) technique for rail flaw detection, where weak UGW signals under strong noise backgrounds are difficult to detect. In this study, a UGW signal identification model based on a chaotic oscillator is established. The approach integrates the UGW response into the critical state of the Duffing system to serve as a disturbance control variable. By evaluating the system's motion state before and after introducing the UGW response, identification of UGW signals can be realized. Thus, the parameters defining the critical state of Duffing oscillators are determined by Ke. Moreover, an electromagnetic transducer was specifically devised to enable unidirectional excitation for UGWs targeted at both the rail base and rail head. Experimental studies showed that the proposed methodology effectively detected and located a 0.46 mm notch at the rail base and a 1.78 mm notch at the rail head. Furthermore, Ke was directly proportional to the notch size, which could be used as a quantitative index to characterize the rail flaw.

Urban Meteorology, Pollutants, Geomorphology, Fractality, and Anomalous Diffusion

The measurements, recorded as time series (TS), of urban meteorology, including temperature (T), relative humidity (RH), wind speed (WS), and pollutants (PM10, PM2.5, and CO), in three different geographical morphologies (basin, mountain range, and coast) are analyzed through chaos theory. The parameters calculated at TS, including the Lyapunov exponent (λ > 0), the correlation dimension (DC < 5), Kolmogorov entropy (SK > 0), the Hurst exponent (0.5 < H < 1), Lempel–Ziv complexity (LZ > 0), the loss of information (<ΔI> < 0), and the fractal dimension (D), show that they are chaotic. For the different locations of data recording, CK is constructed, which is a proportion between the sum of the Kolmogorov entropies of urban meteorology and the sum of the Kolmogorov entropies of the pollutants. It is shown that, for the three morphologies studied, the numerical value of the CK quotient is compatible with the values of the exponent α of time t in the expression of anomalous diffusion applied to the diffusive behavior of atmospheric pollutants in basins, mountain ranges, and coasts. Through the Fréchet heavy tail study, it is possible to define, in each morphology, whether urban meteorology or pollutants exert the greatest influence on the diffusion processes.

The Effects of Lockdown, Urban Meteorology, Pollutants, and Anomalous Diffusion on the SARS-CoV-2 Pandemic in Santiago de Chile

A study was carried out in Santiago de Chile, located in a geographic basin, on the sustainability and diffusion of the recent SARS-CoV-2 pandemic. Hourly measurements were used (carried out for 3.25 years in seven communes of the city) to quantify the accumulated sick (AS) population, urban meteorology variables (MVs) (temperature (T), relative humidity (RH), and magnitude of wind speed (WS)), and air pollution (P) (PM10, PM2.5, 03). Time series (TS) were constructed for each commune, which related AS to MVs, called AS/VM, and to P, noted AS/P. Chaos theory was applied to each TS, requiring the following variables: the Lyapunov exponent (λ > 0), the correlation dimension (DC < 5), Kolmogorov entropy (SK > 0), the Hurst exponent (H, such that 0 < H < 1), Lempel–Ziv complexity (LZ > 0), and information loss (<ΔI> < 0). Every TS complied with chaos theory. For each commune, CK was calculated as a quotient between the sum of AS/T, AS/WS, and AS/RH entropies and the sum of AS/PM10, AS/PM2.5, and AS/O3 entropies. The results show that the entropy for the AS/P ratio is lower than that of the AS/VM ratio in three of the seven communes, since between 2020 and early 2022, the population was confined, reducing pollution. The TS of the AS/P ratio is more persistent and complex. The predictability times of the ratios are comparable in four of the seven communes. The TS of the AS/MV ratios shows greater information loss and chaos. According to the calculated CK values, it is possible to relate it to anomalous diffusion (sub/super-diffusion) and the context that favored the expansion of the pandemic: urban densification, pollution, urban meteorology, population density, etc. Using Fréchet heavy-tailed probability, the compatibility of the results with CK is verified.

Using Kolmogorov Entropy to Verify the Description Completeness of Traffic Dynamics of Highly Autonomous Driving

In this paper, we outline the analysis of a fully provable traffic system based on the Kolmogorov entropy. The completeness of the traffic node dynamics is realized in the form of a nonlinear dynamical model of the participating transport objects. The goal of this study is to determine the completeness of transport nodes based on the Kolmogorov entropy of the traffic trajectories of a node with an unspecified number of actors, like cars and pedestrians. The completeness of a highly autonomous driving detection system describing a traffic node could be realized if the entropy-based error-doubling time of the trajectories of the Euler–Lagrange equation interpreted at the transport junction is less than 1.3.

Rolling bearing fault diagnosis based on fine-grained multi-scale Kolmogorov entropy and WOA-MSVM

In allusion to solve the issue of fault diagnosis for bearing and other rotatory machinery, a technique based on fined-grained multi-scale Kolmogorov entropy and whale optimized multi-class support vector machine (abbreviated as FGMKE-WOA-MSVM) is proposed. Firstly, vibration signals are decomposed by fine-grained multi-scale decomposition, and the Kolmogorov entropy of the sub-signals at different analysis scales is calculated as the multi-dimension feature vector, which quantitatively characterize the complexity of the signal at multi-scales. Aiming at the problem of sensitive parameters selection for multi-class support vector machine model (abbreviated as MSVM), the whale optimization algorithm (abbreviated as WOA) is introduced to optimize the penalty factor and kernel function parameter, and constructing optimal WOA-MSVM model. Finally, an instance analysis is carried out with Jiangnan University bearing datasets to verify the effectiveness and superiority of this technique. The results show that compared with different feature vectors and models such as K nearest neighbors (abbreviated as KNN) and Decision Tree (abbreviated as RF), the proposed technique is superior with fast computation speed and high diagnostic efficiency.

Examining the interfacial behavior of non-Newtonian gas-liquid two-phase flow in horizontal square microchannels

This study investigates the flow dynamics of a non-Newtonian two-phase flow within a horizontal square microchannel. The experimental apparatus employed an acrylic channel with a side length of 8 × 10−4 m. Nitrogen served as the test gas, and the test liquids consisted of water and CMC (Carboxymethylcellulose) aqueous solutions at concentrations of 0.2 and 0.4 %wt. The superficial velocity of working gas and liquids spanned ranges of 0.26–20 m/s and 0.05–1.0 m/s, respectively. A differential pressure transducer was used to measure pressure gradients, while a high-speed video camera captured the flow behavior for analysis with a custom image processing technique. Nonlinear regression statistical analysis was applied to establish the criteria for flow pattern transition. Furthermore, signal processing techniques such as PSD (Power Spectral Density), DWT (Discrete Wavelet Transform), ANN (Artificial Neural Network), and Kolmogorov entropy were employed to analyze the flow behavior based on the pressure gradient signals from the differential pressure transducers. Our findings reveal the occurrence of bubbly, slug, slug-annular, and churn flow patterns. The nonlinear regression statistical analysis offered ambiguous results for determining flow pattern transitions. However, an empirical coefficient was obtained for the nondimensional liquid height. Moreover, the empirical nondimensional liquid height aligned well with the experimental values. The PSD and DWT analysis corroborated each other in clearly characterizing the flow pattern based on pressure gradient fluctuation and wavelet energy. The Kolmogorov entropy effectively captured the chaotic flow pattern. Furthermore, the ANN analysis demonstrated a solid agreement between the predicted and actual flow patterns.

Exploiting robust quadratic polynomial hyperchaotic map and pixel fusion strategy for efficient image encryption

Recently, to offer better protection for images, new image encryption (IE) algorithms have been continuously proposed. However, these algorithms still exhibit some problems, such as poor chaotic performance, flawed key design, and low efficiency. To solve these problems, we first constructed a robust hyperchaotic map called 2D-SQPM. Then, we further developed an efficient IE algorithm based on 2D-SQPM and a pixel fusion strategy (IEASP). The robustness and superiority of 2D-SQPM were verified through several chaotic indicators. These indicators include the Lyapunov exponent, sample entropy, Kolmogorov entropy, and permutation entropy, with average values of 11.5129, 2.8861, 2.4262, and 0.99989, respectively. Several targeted measures were devised in IEASP. Specifically, the common keystream avoids the need to constantly change secret keys. Pixel fusion reduces the computational cost of encryption operations. Two rounds of vector-level image filtering, chaotic pixel superposition, and intra-vector quick scrambling speed up encryption while strengthening security. The security and superiority of IEASP were confirmed through extensive experiments. IEASP achieved excellent scores for correlation coefficients (<0.005), information entropy (7.9998), NPCR (99.6096), and UACI (33.4643). Notably, IEASP also achieved an impressive average encryption rate of 56.3358 Mbps. Due to our targeted designs, IEASP outperforms many recently reported leading IE algorithms in several aspects, particularly in encryption efficiency.

Multi-Time-Scale Analysis of Chaos and Predictability in vTEC

Theoretical modelling of the local ionospheric medium (LIM) is made difficult by the occurrence of irregular ionospheric behaviours at many space and time scales, making prior hypotheses uncertain. Investigating the LIM from scratch with the tools of dynamical system theory may be an option, using the vertical total electron content (vTEC) as an appropriate tracer of the system variability. An embedding procedure is applied to vTEC time series to obtain the finite dimension (m∈N) of the phase space of an LIM-equivalent dynamical system, as well as its correlation dimension (D2) and Kolmogorov entropy rate (K2). In this paper, the dynamical features (m,D2,K2) are studied for the vTEC on the top of three GNSS stations depending on the time scale (τ) at which the vTEC is observed. First, the vTEC undergoes empirical mode decomposition; then (m,D2,K2) are calculated as functions of τ. This captures the multi-scale structure of the Earth’s ionospheric dynamics, demonstrating a net distinction between the behaviour at τ≤24h and τ≥24h. In particular, sub-diurnal-scale modes are assimilated to much more chaotic systems than over-diurnal-scale modes.

Chaotic dynamics of diatomic systems in an electromagnetic field: Dynamical and topological invariants

An advanced combined quantum-dynamic and chaos-geometric method for analysis, modelling, and forecasting of the chaotic dynamics of diatomic molecules in an intense electromagnetic field is presented. The method is based on the use of the non-stationary theory of the Schrödinger equation in the approximation of the density functional and the methods of the theory of chaos and dynamic systems for the analysis of time series of polarization and other characteristics of diatomic molecules in an intense electromagnetic field. In particular, the latter includes the Gottwald-Melbourne test, the correlation integral method , fractal and multifractal formalism, average mutual information, false nearest neighbours, surrogate data algorithms, analysis on the basis of the Lyapunov's exponents, Kolmogorov entropy, nonlinear forecast models based on algorithms of optimized predicted trajectories, B-spline approximations. As an illustration, the advanced data for the dynamical and topological invariants (correlation dimension, embedding dimension, Kaplan-York dimension, Lyapunov's exponents, Kolmogorov entropy, etc.) for the diatomic ZrO molecule in a linearly polarized electromagnetic field are listed.

Identification of Local Isotropic Turbulence Conditions in Various Bubble Columns Based on Several Reliable Parameters

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.

A Robust Memristor-Enhanced Polynomial Hyper-Chaotic Map and Its Multi-Channel Image Encryption Application.

Nowadays, the utilization of memristors to enhance the dynamical properties of chaotic systems has become a popular research topic. In this paper, we present the design of a novel 2D memristor-enhanced polynomial hyper-chaotic map (2D-MPHM) by utilizing the cross-coupling of two TiO2 memristors. The dynamical properties of the 2D-MPHM were investigated using Lyapunov exponents, bifurcation diagrams, and trajectory diagrams. Additionally, Kolmogorov entropy and sample entropy were also employed to evaluate the complexity of the 2D-MPHM. Numerical analysis has demonstrated the superiority of the 2D-MPHM. Subsequently, the proposed 2D-MPHM was applied to a multi-channel image encryption algorithm (MIEA-MPHM) whose excellent security was demonstrated by key space, key sensitivity, plaintext sensitivity, information entropy, pixel distribution, correlation analysis, and robustness analysis. Finally, the encryption efficiency of the MIEA-MPHM was evaluated via numerous encryption efficiency tests. These tests demonstrate that the MIEA-MPHM not only possesses excellent security but also offers significant efficiency advantages, boasting an average encryption rate of up to 87.2798 Mbps.

Human–machine fusion–based operational complexity measurement approach to assembly lines for smart manufacturing

Complexity is an important quantification of uncertain operation in assembly lines and the key source of invisible uncertainty problems in smart manufacturing. The purpose of this paper is to propose a complexity measurement approach to assess the complexity of assembly lines integrating humans, machines and configurations. First, the complexity models of the three states of the operation related to humans and machines are built based on information entropy and the operation time model. Then, an operational complexity model is built at the station level; it is constructed with a single station, parallel stations and sublines based on Kolmogorov entropy. The model quantitatively describes the cumulative complexity along with the material flow. Furthermore, the complexity model of the overall system is given, and the Lempel–Ziv algorithm is applied to measure the complexity flow along with the stations. The complexity equilibrium index is derived to quantify the balancing degree among the stations. The model incorporates uncertain operation into system modeling to quantify the influence of uncertainties on the state of the assembly line. An engine assembly line is used to validate that the approach can measure the complexity from operation to station to system.