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Browsing Статті КІТ by Author "Belozyorov, Vasiliy Ye."
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Item Chaos in Essentially Singular 3D Dynamical Systems with Two Quadratic Nonlinearities(Springer Science + Business Media, 2016) Belozyorov, Vasiliy Ye.EN: A new class of 3D autonomous quadratic systems, the dynamics of which demonstrate a chaotic behavior, is found. This class is a generalization of the well-known class of Lorenz-like systems. The existence conditions of limit cycles in systems of the mentioned class are found. In addition, it is shown that, with the change of the appropriate parameters of systems of the indicated class, chaotic attractors different from the Lorenz attractor can be generated (these attractors are the result of the cascade of limit cycles bifurcations). Examples are given.Item Development of an Approach to Ensure Stability of the Traction Direct Current System(PC "Technology Center", Ukrainian State University of Railway Transport, Kharkov, 2018) Sychenko, Viktor G.; Kuznetsov, Valeriy; Kosariev, Yevhen M.; Hubskyi, Petro V.; Belozyorov, Vasiliy Ye.; Zaytsev, Vadym G.; Pulin, Mykola M.EN: The result of applying the quantitative approach to the calculation of static stability of the traction power system helped us establish that when a train runs along an actual section there emerge zones with lack of stability in terms of voltage. Exact solution to the task of evaluating the stability is extremely difficult because of the need to compute the nonlinear dependences determining the modes of operation of the traction power system and electric rolling stock. In this work, we constructed a system of four autonomous nonlinear differential equations based on experimental data that simulate the behavior of current and voltage in the contact network. We also calculated stability regions for voltage regulators in the traction network, which stabilize voltage at pantographs of electric rolling stock. The obtained stability regions of voltage regulators made it possible to estimate resource of stability and to find the most robust regulators out of those constructed. The study revealed that the non-linear regulator has better robust properties than the linear one. In this case, stability of the linear regulator is very narrow ‒ Δk=0.000004, which is an order of magnitude lower than for the non-linear regulator. When applying the non-linear regulator, voltage in the contact network stabilizes 3 times faster regardless of the place of its location. Application of the devised approach would make it possible to calculate the stability regions for various schematics of the traction network in the implementation of high-speed motion and to narrow the range of voltage fluctuations. The developed dynamic model of power consumption processes, as well as the voltage regulator, could be used when constructing an intelligent, adaptive traction power system for high-speed motion.Item Discrete Processes and Chaos in Systems of Ordinary Differential Equations(Oles Honchar Dnipro National University, Dnipro, Ukraine, 2022) Belozyorov, Vasiliy Ye.; Volkova, Svetlana A.ENG: A method for constructing a one-dimensional discrete mapping describing a certain periodic process in a general system of ordinary autonomous differential equations is proposed. The resulting discrete mapping is then used to prove the existence of chaos in the original system of differential equations.Item General Method of Construction of Implicit Discrete Maps Generating Chaos in 3D Quadratic Systems of Differential Equations(World Scientific Publishing Company, 2014) Belozyorov, Vasiliy Ye.EN: A method allowing to study the dynamics of 3D systems of quadratic differential equations by the reduction of these systems to the special 2D systems is presented. The mentioned 2D systems are used for the construction of new types of discrete maps generating the chaotic dynamics in some 3D autonomous systems of quadratic differential equations. Strong simplification of all results gives an introduction of the Lambert function. Due to this function some implicit discrete maps become explicit. Examples are given.Item Generating Chaos in 3D Systems Of Quadratic Differential Equations With 1D Exponential Maps(World Scientific Publishing Company, 2013) Belozyorov, Vasiliy Ye.; Chernyshenko, Sergey V.EN: New existence conditions of homoclinic orbits for some systems of ordinary quadratic differential equations with singular linear part are found. A realization of these conditions guarantees the existence of chaotic attractors at 3D autonomous quadratic systems. Examples of chaotic attractors are given.Item Implicit One-Dimensional Discrete Maps and Their Connection with Existence Problem of Chaotic Dynamics in 3-D Systems of Differential Equations(Elsevier, 2012) Belozyorov, Vasiliy Ye.EN: New types of chaotic attractors for some 3-D autonomous systems of ordinary quadratic differential equations are founded. Examples are given.Item A New Approach to the Problem of Diagnostics of Cerebral Cortex Diseases Using Chaotic Dynamics Methods(Richtmann Ldt, London, 2017) Belozyorov, Vasiliy Ye.; Pohorielov, Oleksii V.; Serdiuk, Valerii N.; Zaytsev, Vadym G.EN: An modeling attempt of behavior process of brain electric impulses for some patient by solutions of 3Dsystem of autonomous quadratic differential equations is undertaken. This system of differential equations was got from a multivariatetimes series with the help of polynomial averages and least squares method. Further, with the help of the got system a question about existence of chaotic attractor in this system is studied. In this case, the presence of chaotic attractor makes it possible to interpretas the absence of disease and vice versa.Item A New Mathematical Model of Dynamic Processes in Direct Current Traction Power Supply System(Дніпровський національний університет ім. О. Гончара, 2019) Belozyorov, Vasiliy Ye.; Kosariev, Yevhen M.; Pulin, Mykola M.; Sychenko, Viktor G.; Zaytsev, Vadym G.EN: Abstract. A new autonomous 4D nonlinear model with two nonlinearities describing the dynamics of change of voltage and current in the contact railway electric network is offered. This model is a connection of two 2D oscillatory circuits for current and voltage in the contact electric network. In the found system for the defined values of parameters an existence of limit cycles is proved. By introduction of new variables this system can be reduced to 5D system only with one quadratic nonlinearity. The constructed model may be used for the control by voltage stability in a direct current power supply system.Item A Novel Search Method of Chaotic Autonomous Quadratic Dynamical Systems without Equilibrium Points(Springer Netherlands, 2016) Belozyorov, Vasiliy Ye.EN: A wide class of autonomous real quadratic dynamic system without real (but with two complex conjugate or even imaginary) equilibrium points is considered. For any system of this class, a new idea of the uniquely definite degenerate autonomous real quadratic dynamic system having exactly one real double equilibrium point (there are no complex equilibrium points) is introduced. It is shown that if the degenerate system demonstrates the chaotic behavior, then for the original (not degenerate) system, a similar chaotic behavior also takes place. The idea of the degenerate system for researches of the real quadratic systems, for which number of complex conjugate equilibrium points more than two, is also used. The same idea can be adapted to research of any autonomous real quadratic system having at least one pair complex conjugate equilibrium points. An attempt to apply some derived results to a search problem of hidden chaotic attractors was undertaken. Examples are given.Item Odd and Even Functions in the Design Problem of New Chaotic Attractors(World Scientific Publishing Company, Singapore, 2022) Belozyorov, Vasiliy Ye.; Volkova, Svetlana A.ENG: Let A ⊂ Rn be a chaotic attractor generated by a quadratic system of ordinary differential equations x˙ = f(x). A method for constructing new chaotic attractors based on the attractor A is proposed. The idea of the method is to replace the state vector x = (x1,...,xn)T located on the right side of the original system with new vector u(x); where u(x) = K ·(h1(x1),...,hn(xn))T , K ∈ Rn×n, and hi(xi) are odd power functions; i = 1,...,n. (In other words, a state feedback x → u(x) is introduced into the right side of the system under study: x˙ = f(x) → x˙ = f(u(x)).) As a result, the newly obtained system generates new chaotic attractors, which are topologically not equivalent (generally speaking) to the attractor A. In addition, for an antisymmetric neural ODE system with a homoclinic orbit connected at a saddle point, the conditions for the occurrence of chaotic dynamics are found.Item Quadratic Model of Inter-Population Interaction: Investigation of Stability Areas(Elsevier, 2013) Belozyorov, Vasiliy Ye.; Chernyshenko, Sergey V.EN: A development quadratic model of the heterogeneous biological population consisting of a few sub-populations is investigated. The classic stabilization problem of solutions for the system of ordinary quadratic differential equations, which describing this model, is solved. Examples are given.Item Recurrence Analysis of Time Series Generated by 3D Autonomous Quadratic Dynamical System Depending on Parameters(Дніпропетровський національний університет імені Олеся Гончара, 2016) Belozyorov, Vasiliy Ye.; Zaytsev, V. G.EN: For the wide class of 3D autonomous quadratic dynamical systems depending on parameters the sufficient conditions of boundedness of solutions of any system from this class are found. A connection between change of one of the parameters and a recurrence plot structure, which was built on the time series for any system of this class, is determined. Due to this connection it is possible to find bifurcation values of the parameter of any system from the considered class only on its time series without knowledge of differential equations of this system. Examples are given.Item Role of Logistic and Ricker’s Maps in Appearance of Chaos in Autonomous Quadratic Dynamical Systems(Springer Netherlands, 2016) Belozyorov, Vasiliy Ye.; Volkova, Svetlana A.EN: New existence conditions of a chaotic behavior for wide class of (n+1)-dimensional autonomous quadratic dynamical systems are suggested. It is shown that in all such systems the chaotic dynamics is generated by 1D discrete map by some combination of the logistic map f (x) = λx(1−x);λ> 0 and Ricker’s map g(x) = x exp(μ−x);μ>0.Item Singular Differential Equations and their Applications for Modeling Strongly Oscillating Processes(Oles Honchar Dnipro National Universuty, Dnipro, Ukraine, 2023) Belozyorov, Vasiliy Ye.; Volkova, Svetlana A.; Zaytsev, Vadym G.ENG: The normal system of ordinary differential equations, whose right-hand sides are the ratios of linear and nonlinear positive functions, is considered. A feature of these ratios is that some of their denominators can take on arbitrarily small nonzero values. (Thus, the modules of the corresponding derivatives can take arbitrarily large value.) In the sequel, the constructed system of differential equations is used to model strongly oscillating processes (for example, processes determined by the rhythms of electroencephalograms measured at certain points in the cerebral cortex). The obtained results can be used to diagnose human brain diseases.Item Study of the Dynamics of Product Sales Process with the Help of Zolotas Model(Oles Honchar Dnipro National University, Dnipro, 2024) Belozyorov, Vasiliy Ye.; Volkova, Svetlana A.ENG: . A new mathematical model describing the dynamics of sales of goods on the market has been proposed. This model takes into account the following characteristics: the average welfare of buyers, the maximum welfare of buyers, the number of buyers who know about the incoming product and the level of market saturation with this product. Examples demonstrating the features of the constructed model are given.