Class Of Differential Systems Research Articles

Global centres in a class of quintic polynomial differential systems

A centre of a differential system in the plane $ {\mathbb {R}}^2$ is an equilibrium point $p$ having a neighbourhood $U$ such that $U\setminus \{p\}$ is filled with periodic orbits. A centre $p$ is global when $ {\mathbb {R}}^2\setminus \{p\}$ is filled with periodic orbits. In general, it is a difficult problem to distinguish the centres from the foci for a given class of differential systems, and also it is difficult to distinguish the global centres inside the centres. The goal of this paper is to classify the centres and the global centres of the following class of quintic polynomial differential systems \begin{align*} \dot{x}= y,\quad \dot{y}={-}x+a_{05}\,y^5+a_{14}\,x\,y^4+a_{23}\,x^2\,y^3+a_{32}\,x^3\,y^2+a_{41}\,x^4\,y+a_{50}\,x^5, \end{align*} in the plane $ {\mathbb {R}}^2$ .

Four Limit Cycles of Three-Dimensional Discontinuous Piecewise Differential Systems Having a Sphere as Switching Manifold

Because of their applications, the study of piecewise-linear differential systems has become increasingly important in recent years. This type of system already exists to model many different natural phenomena in physics, biology, economics, etc. As is well known, the study of the qualitative theory of piecewise differential systems focuses mainly on limit cycles. Most papers studying the problem of existence and the maximum number of limit cycles of piecewise differential systems have precisely considered planar systems. However, few papers have examined this problem in [Formula: see text]. In this paper, our main goal is to examine a class of discontinuous piecewise differential systems in [Formula: see text], where we consider the unit sphere as the separation surface that divides the entire space into two regions, each one has a linear vector field analogous to planar center. In general, it is hard to determine an exact upper bound for the number of limit cycles that a class of differential systems can exhibit. We prove that this class of differential systems can have at most four limit cycles. We show that there are examples of such differential systems with exactly 1, 2, 3 and 4 limit cycles.

Remarkable relation between melting entropy and kinetic viscosity in metallic glasses

There has been so far no consensus regarding the roles of melting entropy on glass formation. For this purpose, a study has been conducted on three compositions of the pseudo-binary alloys, Zr2Ni-Ti2Ni, viz., the eutectic compositions (Zr2Ni)0.206(Ti2Ni)0.794, (Zr2Ni)0.654(Ti2Ni)0.346, and the Laves compound (Zr2Ni)0.44(Ti2Ni)0.56. The study results of melting entropies, melting viscosities, and glass formabilities of the alloys unequivocally shows the fact that low melting entropy and high melting viscosity facilitate glass formation. From the data of different classes of systems, a unique relationship between melting entropy and melting viscosity has arisen in glass forming systems, which highlights the positive roles of low melting entropy on glass formation from a special perspective. This study of thermodynamics, kinetics, and glass formation highly suggests the melting entropy should be taken as a crucial reference for designing metallic glasses.

Геопорталы как инструмент доступа к облачным хранилищам данных о метагеосистемах

The article is devoted to solving the scientific problem of spatial data storage management using geoportal systems on the basis of solving the scientific problem of analyzing the patterns of spatiotemporal organization of complex dynamic hierarchically organized metageosystems. The paper proves that the problem of integrating large arrays of spatial information in spatial data infrastructures (SDIs) can be solved by designing multi-model storages operating on the basis of database management systems of different classes. To improve the reliability of spatial data integration systems, it is advisable to focus on a microservice architecture that involves organizing the interaction of separate services, each of which performs its own specific task. The following microservices usage contexts can be distinguished in the IPD: geoportals, spatial data analysis, processing and management systems, as well as distributed cloud storages.

On the asymptotic and practical stability of Persidskii-type systems with switching

In the paper, one class of differential systems with nonlinearities satisfying sector constraints is considered. We study the case where some of the sector constraints are given by linear inequalities, and some by nonlinear ones. It is assumed that the coefficients in the system can switch from one set of values to another. Sufficient conditions for the asymptotic and practical stability of the zero solution of the system are investigated using the direct Lyapunov method and the theory of differential inequalities. Restrictions on the switching law that provide a given region of attraction and ultimate bound for solutions of the system are obtained. An approach based on the construction of different differential inequalities for the Lyapunov function in different parts of the phase space is proposed, which makes it possible to improve the results obtained. The results are applied to the analysis of one automatic control system.

Application of Box and Voronoi CAPs for Metastable Electronic States in Molecular Clusters.

The complex absorbing potential (CAP) approach offers a practical tool for characterization of energies and lifetimes of metastable electronic states, such as temporary anions and core ionized states. Here, we present an implementation of the smooth Voronoi CAP combined with the equation-of-motion coupled cluster with single and double substitutions method for metastable states. The performances of the smooth Voronoi CAP and box CAP are compared for different classes of systems: resonances in isolated molecules and localized and delocalized resonances in molecular clusters. The benchmark calculations show that the Voronoi CAP is generally more robust when applied to molecular clusters, but box CAPs are equally reliable for localized resonances or in the cases when the resonance does not exhibit significant electron density delocalization into the intramolecular region. As such, the choice of the CAP shape and onset should be guided by the character of the metastable states.

On bounded solutions of differential systems

The question of the existence of bounded solutions on an infinite interval of a linear inhomogeneous system of ordinary differential equations in a finite-dimensional space is considered. The study of bounded solutions of systems of ordinary differential equations is one of the most important problems in the qualitative theory of differential equations. In the study of the asymptotic behavior of solutions to differential systems, the works of A. Poincare and A.M. Lyapunov. Various conditions for the existence of bounded solutions of a linear system of ordinary differential equations have been obtained by many authors. Note the works of O. Perron, A. Walter, H. Shpet, D. Caligo, N.I. Gavrilova, M. Hukukara, M. Nagumo, M. Caratheodori, U. Barbouti, N. Ya. Lyashchenko,  B.P. Demidovich, A. Wintner, R. Bellman, Yu.S. Bogdanov, Z. Vazhevsky,          N. Levinson, M. Markus, L. Cesari and others. In this paper, we establish sufficient conditions for the boundedness of all solutions of a linear inhomogeneous system of differential equations on an infinite interval. A coefficient criterion for the boundedness of all solutions on an infinite interval of a linear inhomogeneous system of differential equations in a certain class of differential systems is given. Applied methods of differential equations and function theory. The results obtained are used in applications of differential equations and are of practical value.

Emerging scale invariance in a model of turbulence of vortices and waves.

This note is devoted to broken and emerging scale invariance of turbulence. Pumping breaks the symmetry: the statistics of every mode explicitly depend on the distance from the pumping. And yet the ratios of mode amplitudes, called Kolmogorov multipliers, are known to approach scale-invariant statistics away from the pumping. This emergent scale invariance deserves an explanation and a detailed study. We put forward the hypothesis that the invariance of multipliers is due to an extreme non-locality of their interactions (similar to the appearance of mean-field properties in the thermodynamic limit for systems with long-range interaction). We analyse this phenomenon in a family of models that connects two very different classes of systems: resonantly interacting waves and wave-free incompressible flows. The connection is algebraic and turns into an identity for properly discretized models. We show that this family provides a unique opportunity for an analytic (perturbative) study of emerging scale invariance in a system with strong interactions.This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.

Prediction of the geomechanical state of the rock mass when mining salt deposits with stowing

The technogenic impact of mining on the environment is analyzed and the transition to geotechnology with stowing to reduce the impact of mining operations is proposed. The results of the research work devoted to the justification of parameters of the development of salt deposits with stowing and the definition of the influence of stowing on the dynamics of deformation of the underworked rock massif are presented. The relevance of research aimed at creating a safe and efficient technology for the transition from systems with natural maintenance of stoping space to systems with stowing has been substantiated.  The results of studies on qualitative and quantitative assessment of the state of the rock massif (by the finite element method using FLAC3D software), worked out by combines, are given and the dynamics of the impact of mining operations on the rock mass and the change in the maximum stresses during the hardening of the stowing in the chambers are revealed. The numerical modeling method is used to analyze the conditions of change in the state of the underworked rock mass, to establish the mechanisms of its deformation at various stages of development. It is recommended to use this approach for geotechnical assessment of the rock mass state in conditions of using development systems of different classes.

Stable Hyperbolic Limit Cycles for a Class of Differential Systems

Stable Hyperbolic Limit Cycles for a Class of Differential Systems

Substantiation of process schemes of field development taking into account technogenic conversion

Introduction. Mining disturbs the natural state of the rock mass. Besides, the field reserves are of a different degree and nature of man-induced changes, which directly affects mining prime cost and ore marketable value. The scientific and practical relevance of the research consists in ensuring increased completeness and economic efficiency of field development by applying the development systems of different classes and taking into account the man-induced changes in reserves in mining dynamics. Research aim is to substantiate the technological schemes of mining, that take into account the possibility and feasibility of developing the reserves which have been affected by the man-induced changes, in particular when switching from one class of development systems to another. Research methods. In this research, based on the analysis of the mining and geological conditions of copper sulfide deposits, economic and mathematical modeling of developing an ore deposit by development systems of different classes was carried out, and the man-induced changes in underground reserves in the process of mining were taken into account. Results. Based on the modeling results, it was found that the increase in the economic efficiency of the deposit development is provided by justifying the type, sequence, and limits of applying different geotechnologies related to different classes of development systems within the entire deposit and a separate ore body. Conclusion. The research results can be used when designing fields in terms of selecting and substantiating the applied development systems.

Limit cycles of 3-dimensional discontinuous piecewise differential systems formed by linear centers

In this paper we deal with 3-dimensional discontinuous piecewise differential systems formed by linear centers and separated by one plane or two parallel planes. We prove that these systems separated by one plane have no limit cycles, besides the systems separated by two parallel planes have at most one limit cycle, and that there are systems having such a limit cycle. So we solve the extension of the 16th Hilbert problem to this class of differential systems.

Improving the field homogeneity of fixed- and variable-diameter discrete Halbach magnet arrays for MRI via optimization of the angular magnetization distribution

The aim of this work was to maximize the homogeneity of fixed- or variable-diameter Halbach array of discrete magnets by optimizing the angular rotation of individual magnets within each ring of the array. Numerical simulations have been performed for magnet arrays with various length:radius ratios (L/R) using a dipole-approximation model. These simulations used an uninformed random-search algorithm, with the initial state corresponding to the classical Halbach dipole configuration. Two different classes of systems were studied, one with magnet rings of constant radius, and the other in which the radius of the rings was allowed to vary to increase the homogeneity. Simulation results showed that for a fixed-diameter array optimization of the angular orientation of individual magnets increased the homogeneity by ~17% for very short magnets, with the improvement dropping to ~5% for L/R values greater than ~3:1, where the homogeneity was measured over a region-of-interest equal to one-half the diameter of the magnet array. An empirical formula was derived which allows easy estimation of the required magnetization angles for any L/R. For a 23-ring variable diameter magnet with L/R of ~4:1 the optimization procedure produces an increase in homogeneity of ~18%.

Limit cycles of discontinuous piecewise differential systems formed by linear centers in [formula omitted] and separated by two circles

We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or three limit cycles.

Iterative Learning Based Modulating Functions Method for Distributed Solar Source Estimation

Modulating functions method is a non asymptotic estimation method, which provides accurate and robust estimations of states, parameters and inputs for different classes of systems, which include unknown linear ordinary differential systems, fractional systems and linear partial differential equations. In the case of time or space varying unknown, the method requires the decomposition of the unknown into predefined basis functions. However, the estimation performance will depend on the nature of the basis functions which in some cases are not easy to determine. This letter proposes a new iterative learning based modulating functions method, which combines the standard modulating functions with a dictionary learning procedure. The dictionary learning step allows the determination of appropriate set of functions to decompose the unknown, while the modulating function step allows the non-asymptotic and robust estimation of the projection coefficients. The performance of the proposed method is illustrated in a distributed solar collector application, modeled by partial differential equations and where the unknown solar irradiance is estimated.

ЭФФЕКТИВНОСТЬ КОМБИНИРОВАНИЯ ТЕХНОЛОГИЙ ВЫЕМКИ РУД В ПРЕДЕЛАХ РУДНОГО ПОЛЯ

Актуальность исследования. Статья посвящена актуальной проблеме совершенствования технологии разработки сложно-структурных месторождений скального типа для увеличения производства содержащихся в рудах редких и ценных металлов, дефицитность которых возрастает с развитием научно-технического прогресса. Цель исследования – определение условий совмещения систем разработки разных классов для увеличения возможностей селективизации добычных работ при использовании различных вариантов извлечения руд в смежных добычных участках. Методика исследования. Основу исследования составляет экономический расчет возможных вариантов совмещения в пределах рудного поля на основе полученных экспериментально сведений о рудовмещающем массиве. Результаты. Дана характеристика условий залегания руд на конкретном скальном месторождении сложной структуры. Произведено инженерно-геологическое районирование рудовмещающего массива с дифференцированной оценкой геолого-структурных нарушений по величине и направленности. Дан прогноз возможного по аналогии с практикой поведения горных выработок под влиянием разломных структур и макротрещин. На уровне экспертной оценки определена целесообразность применения слоевой или камерной систем разработки с закладкой технологических пустот твердеющими смесями. Выводы. Эффективность выемки руд достигается селективизацией выемки запасов в пределах одного рудного поля при комбинировании систем разработки при максимальном учете геолого-структурных факторов локализации руд и механо-физических свойств рудовмещающих пород.

Eclipses of jets and discs of X-ray binaries as a powerful tool for understanding jet physics and binary parameters

ABSTRACT We calculate the expected effects on the spectral energy distributions and light curves in X-ray binary jets from eclipses by the donor stars. Jets will be eclipsed for all inclination angles, with just the height along the jet where the eclipse takes place being set by the orbital parameters. Typically, eclipses will lead to 5–10 per cent reductions in the jet emission over a range of a factor of few in wavelength with a periodic modulation. In ideal systems with high inclination angles, relatively even mass ratios, and modest jet speeds, the eclipses may be deeper. We discuss how eclipses can be used to measure binary system parameters, as well as the height of the bases of the jets. We also discuss how, with data sets that will likely require future facilities, more detailed tests of models of jet physics could be made by establishing deviations from the standard recipes for compact conical flat spectrum jets and by determining the ingress and egress durations of the eclipses and measuring the transverse size of the jets. We provide representative calculations of expectations for different classes of systems, demonstrating that the most promising target for showing this effect in the radio band is the longer period ‘atoll’-class neutron star X-ray binaries, while in the optical and infrared bands, the best candidates are likely to be the most edge-on black hole X-ray binaries. We also discuss the effects of the outer accretion disc eclipsing the inner jet.

Multivariate multiscale complexity-entropy causality plane analysis for complex time series

The multivariate multiscale complexity-entropy causality plane (MMCECP) is introduced for evaluating the dynamical complexity and long-range correlations of multivariate nonlinear systems. Numerical simulations from different classes of systems are applied to confirm the effectiveness of the proposed measure. We observe that the MMCECP not only can characterize the deterministic properties of the systems, but also can distinguish Gaussian and non-Gaussian processes. Moreover, it is immune to varying degrees of noises at large scales. Then we apply it to financial time series analysis, mainly investigating the classification of stock market dynamics. Empirical results illustrate that the MMCECP is robust and valid to detect the physical structures of stock markets.

PARADIGM OF DIFFERENCE: PERSON AND TECHNOLOGY IN THECONDITIONS OF INFORMATION REDUNDANCY

Purpose. The paper solves an important methodological and practical problem, which is to increase the safety of interaction of different classes systems in the conditions of information redundancy on the basic principles of thinking in complexity. The aim of the work is to investigate the possibility of a non-traumatic connection of different classes’ systems “person”, “technology”, “environment” into a single macrosystem. Theoretical basis. The study of the key provisions of the safety problem of the complex system “person–technology–environment” is considered in the context of the basic principles of post-non-classics and “thinking in complexity”, which allowed us to take into account the dimensionality and diversity of the systems included in the macrosystem “person–technology–environment”. It is shown that the macrosystem “person–technology–environment” is complex, characterized by openness, self-organization, human- and psycho-dimensionality, non-linear development and instability. It is hypothesized that the main cause of the accident is a certain incompatibility within the macrosystem“person–technology–environment” of the systems “person”, “technology”, “environment” connected in it in terms of membership in different classes, which causes the emergence of a critical difference for the interaction of complex systems of different classes. Originality. For the first time, the concept of a complex macrosystem of a new type is introduced, connecting systems of different classes as independent “whole” on the basis of the conceptual model of post-non-classical “whole in a whole”. Conclusions. The growth of road traffic accidents is formed by the joint interaction of “different-quality” systems “person/driver”, “technology/car”, “environment”“environment” into a single macrosystem. The new quality of the macrosystem “person/driver-technology/car-environment” is determined by the nature of the bonds and the emergence of consistency/or mismatch between different integrity in a single macrosystem. It was established the need to accept a difference paradigm as a scientific branch on the basis of the methodology of non-traumatic/ecological connection without combining the multiclass subsystems into a single macrosystem with a mega-control. The proposed recommendations contribute to improving road safety.

Equivalence between nonlinear dynamical systems and urn processes

An equivalence is shown between a large class of deterministic dynamical systems and a class of stochastic processes, the balanced urn processes. These dynamical systems are governed by quasi-polynomial differential systems that are widely used in mathematical modeling while urn processes are actively studied in combinatorics and probability theory. The presented equivalence extends a theorem by Flajolet et al (2006 Discrete Mathematics and Theoretical Computer Science, AG (DMTCS Proc.) pp 59–118) already establishing an isomorphism between urn processes and a particular class of differential systems with monomial vector fields. The present result is based on the fact that such monomial differential systems are canonical forms for more general dynamical systems.