## Conference submissions

#### Exploring the exact differential equations with GeoGebra software

Jorge Olivares Funes1 , Elvis Valero2

1University of Antofagasta, Department of Mathematics, Chile

###### Abstract

In this paper, we will show the solutions of certain exact differential equations that are obtained through the interactive GeoGebra software. GeoGebra software has been of great motivational support in the processes of teaching and modeling in mathematics in various universities and colleges, especially in the engineering careers of the University of Antofagasta in the courses of differential equations and calculation of several variables.

#### Novel explanation of the Active Galactic Nuclei.The Power Source of Quasars as a result of vacuum polarization by the gravitational singularities on BHs horizon.

Jaykov Foukzon1

1 Israel Institute of Technology , Center for Mathematical Sciences, Technion - Israel Institute of Technology. , Israel

###### Abstract

Novel explanation of the Active Galactic Nuclei.The Power Source of Quasars as a result of vacuum polarization by the gravitational singularities on BHs horizon. J.Foukzon¹, E.Menkova² A.Potapov³ ¹Department of mathematics, Israel Institute of Technology, Haifa, Israel ²All-Russian Research Institute for Optical and Physical Measurements, Moscow,Russia ³Kotel'nikov Institute of Radioengineering and Electronics of the Russian Academy of Sciences, Moscow, Russia In this paper we argue that the current paradigm for AGN and quasars essentially incomplete and rivision is needed. Remind that the current paradigm for AGN and quasars is that their radio emission is explained by synchrotron radiation from relativistic electrons that are Doppler boosted through bulk motion. In this model, the intrinsic brightness temperatures cannot exceed 10¹¹ to 10¹² K. Typical Doppler boosting is expected to be able to raise this temperature by a factor of 10.The observed brightness temperature of the most compact structures in BL Lac, constrained by baselines longer than 5.3Gλ, must indeed exceed 2×10¹³K and can reach as high as ~ 3×10¹⁴K. As well known, these visibilities correspond to the structural scales of 30-40 μas oriented along position angles of 25°-30°.These values are indeed close to the width of the inner jet and the normal to its direction.The observed, T_{b,obs}, and intrinsic, $$T_{b,int}$$, brightness temperatures are related by Eq.1: $$T_{b,obs}=δ(1+z)⁻¹T_{b,int}$$ with δ=7.2.The estimeted by Eq.1 a lower limit of the intrinsic brightness temperature in the core component of our Radio Astron observations of $$T_{b,int}>2.910¹² K$$. It is commonly considered that inverse Compton losses limit the intrinsic brightness temperature for incoherent synchrotron sources, such as AGN, to about 10¹²K [1].In case of a strong flare, the "Compton catastrophe" is calculated to take about one day to drive the brightness temperature below 10¹²K [1]. The estimated lower limit for the intrinsic brightness temperature of the core in the Radio Astron image of $$T_{b,int}>2.910¹²K$$ is therefore more than an order of magnitude larger than the equipartition brightness temperature limit established in [1] and at least several times larger than the limit established by inverse Compton cooling. Remark 1.Note that if the estimate of the maximum brightness temperature given in [1], is closer to actual values, it would imply $$T_{b;int}=5×10¹³K$$. This is difficult to reconcile with current incoherent synchrotron emission models from relativistic electrons, requiring alternative models such as emission from relativistic protons. Remark 2. However the proton, as we know, is 1836 times heavier than an electron and absolutely huge energy is required to accelerated it to sublight speed. We argue that these alternative models such as emission from relativistic protons can be suported by semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved distributional spacetimes presented by rotating gravitational singularities [2]. [1] J. L. Gómez, A. P. Lobanov, G. Bruni, Y. Y. Kovalev,Probing the innermost regions of AGN jets and their magnetic fields with Radio Astron.I.Imaging BL Lacertae at 21 microarcsecond resolution. Astrophysical journal 817 (2016) 96, DOI:10.3847/0004-637X/817/2/96 arXiv:1512.04690 [astro-ph.HE] [2] J.Foukzon,E.Menkova,A.Potapov, Colombeau Solutions to Einstein Field Equations in General Relativity: Gravitational Singularities, Distributional SAdS BH Spacetime-Induced Vacuum Dominance.110 pp. Published November 14, 2019 ISBN-13 (15) 978-93-89562-22-4 https://doi.org/10.9734/bpi/mono/978-93-89562-22-4

#### A study on hidden dimensions, winding number & selected topics of Algebraic Topology in String Theory

SANTANU CHATTERJEE1 , Sanjoy Mukherjee2

1RGM International Pvt. Ltd., Civil Department, India
2Vikram Solar Limited, Innovation, India

###### Abstract

String theory provides an encouraging way to unify all force fields in our universe into a single framework. Different vibrational patterns of a single string resemble different particles. Bosonic strings require 26 spatial dimensions in order to produce particles in a similar fashion as 10 spatial dimensions are required to produce Fermions states. These extra dimensions (beyond 3 spatial & 1-time dimension) are compactified into very small scale and thus in today’s scale of probing energy it is not possible to detect them experimentally. This idea of extra dimensions, hidden from our perception is tempting & worth in-depth theoretical work. Certain topological features and its application on String theory will also be discussed in this paper. We will try to fathom these small plank scale compactified dimensions & will try to throw some light on the various topological aspects of quantum geometry offered by this remarkable theory.

#### MATHEMATICAL MODELLING OF TUMOUR GROWTH UNDER SPHERICAL SYMMETRY

SANTANU CHATTERJEE1 , Sanjoy Mukherjee2

1RGM International Pvt. Ltd., Civil Department, India
2Vikram Solar Limited, Innovation, India

###### Abstract

In today’s era of modern medicines, mathematical modeling plays an important role in analyzing difficult aspects of cell divisions and proliferation to speculate the result of the actual experiment. In these concepts of modeling, differential equation as a technique for determining relation between a function and its derivatives played a very important role. If we know a function and few of its derivatives at a particular point then that information along with differential equation can plot the function over its entire domain [1]. By choosing suitable parameters & boundary conditions different types of differential equations can be formed to represent actual biological process and up-to the complexity levels as we may desire. Thus, mathematical modeling of metastatic cancer & growth of cancerous cell has become an active area of research. In normal cells, hundreds of genes intricately control the process of cell division. Cells become cancerous after mutations accumulate in the various genes that control cell proliferation. In other words, they no longer respond to most of the signals that control cellular growth and death. Mutations in genes can cause cancer by accelerating cell division rates or inhibiting normal controls on the system, such as cell cycle arrest or programmed cell death [2]. In this paper will investigate spherical tumor considering three types of cell formation or layer of cell inside it, namely, proliferating cell layer, quiescent cell layer & necrotic cell layer and we will study their properties & dynamics of expansion or contraction with respect to time. Instead of oxygen (as in; Greenspan, H. (1972)), we will model our case with flow of nutrients that are required for a living cell to be alive and proliferate. Through a simple linear 1st order differential equation, different stages of tumor growth and invasion of tumor cells to surrounding tissue in the form of micro metastatic cells will also be investigated. Our approach will be to modify the 1st order differential equation for standard population dynamics (growth & death type) with considerable changes in the parameters to catch up all three earlier mentioned phases or layers of a growing tumor.

#### The modified bessel functions I_3 / 4 (x) and I_-3/4 (x) in certain fractional differential equations

Jorge Olivares Funes1 , Pablo Martin2 , Elvis Valero3

1University of Antofagasta, Department of Mathematics, Chile
2University of Antofagasta, physics department, Chile

###### Abstract

The fractional derivative of Caputo, has huge and important applications in various areas of science and engineering. In this case, through the definition of the Caputo derivative and the Laplace transform and its inverse, we propose to show the solutions that can be obtained for each specific value of alpha of the following fractional differential equations $$\frac{d^\alpha y}{{dx}^\alpha}= I_3/4(x)$$ , $$\frac{d^\alpha y}{{dx}^\alpha}= I_-3/4(x)$$ , with $$m-1<α\le m$$ , $$m\ \in N$$. Where the non-homogeneous parts I_3 / 4 (x) and I_-3/4 (x) are the modified Bessel functions of the first species.

#### GeoGebra and Wolfram Αlpha in homogeneous reducible differential equations

MARÍA ROJAS1 , Jorge Olivares Funes2 , Daniza Rojas3

2University of Antofagasta, Department of Mathematics, Chile
3University of Antofagasta, Department of Mathematics, Chile

###### Abstract

GeoGebra and Wolfram Αlpha are educational software used in many areas of science, engineering and mathematics. In this work we will use these programs to see the homogeneous reducible differential equations along with their solutions. The present material that will be presented corresponds to the one designed for the course of differential equations for engineering and pedagogy careers in mathematics at the University of Antofagasta-Chile.

#### Anisotropy of glancing angle deposited films: results of atomistic simulation

Fedor Grigoriev1

1M.V. Lomonosov MSU, Russian Federation, RCC, Russian Federation

###### Abstract

Glancing angle deposition (GLAD) is one of the technique for the fabrication of the anisotropic thin films with high porosity and low refractive index. In this technique the incoming flux of the deposited atoms is directed almost parallel to the substrate surface. Due to density fluctuation in the deposited atoms flow and shadow effect, it results in formation of the different separate nanostructures - tree-like column, slanted column, chevron-like structures and so on – on the substrate. GLAD films are widely used in the optical coating due to low reflectance and anisotropy properties. The structural and optical properties of GLAD-films essentially depend on their fabrication conditions. Experimental investigation of these dependencies is still challenge for the existing experimental techniques. On the other hand due to the progress in high performance computing, now it is possible to study thin films deposition process using the atomistic simulation. In the present work the anisotropy of the GLAD SiO2 films is investigated using the classical atomistic simulation and anisotropic Bruggeman effective medium theory. The depolarizing factors defining the difference of the refractive index components are calculated based on the geometry parameters of the pores between the slanted columns forming the large-scale GLAD structure. Averaged shape parameters of these ellipsoids are defined using the density gradient tensor. It is revealed that the values of difference of main components of refractive index tensor are equal 0,03 and 0,05 for deposition angles 60 grad and 80 grad, if the free volume fraction is calculated using the dependence of film density on the deposition angle. The obtained values coincide with experimental results for silicon dioxide films. Simulation is carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University.

###### Acknowledgements:

The work is supported by the Russian Science Foundation (grant number 19-11-00053).

#### First-order linear partial differential equations using the GeoGebra and GeoGebra 3D graphical calculator

Jorge Olivares Funes1 , Pablo Martin2 , Elvis Valero3

1University of Antofagasta, Department of Mathematics, Chile
2University of Antofagasta, physics department, Chile

###### Abstract

Consider $$A∂z/∂y+B∂z/∂x=0$$, Where A, B 𝜖 R. We will solve these types of linear partial differential equations using the GeoGebra Graphical Calculator and their solutions will be seen by GeoGebra 3D. The use of GeoGebra in the study of PDE is a subject that is still being studied and developed with great potential.

#### Multi-criteria optimization of wind power plant parameters

Andrei Melekhin1

1National Research Moscow State University of Civil Engineering , heat, gas supply and ventilation, Russian Federation

###### Abstract

The use of renewable energy sources to generate electricity is a hot topic. The author has developed a mathematical model of the aerodynamic process of a wind power plant with the solution of a multi-criteria optimization problem. The optimal range of controlled parameters affecting the aerodynamic process with a minimum amount of blown surface and maximum electrical energy production is determined. Regularities of aerodynamic process are established. The convergence of the results of the study in the calculation on the basis of theoretical dependencies and the solution of the mathematical model is determined. To find the optimal controlled parameters of the aerodynamic installation, a complex research method developed by the author is applied, based on multi-criteria optimization of parameters with the introduction of empirically obtained data.The preliminary procedure of IOSO NM 3.8 consists in the formation of an initial plan of the experiment, which can be implemented both in a passive way (using information about various parameters, optimization criteria and constraints obtained earlier) and in an active way, when too much is generated in the initial search area in accordance with a given distribution law. For each vector of variable parameters, the values of optimization criteria and constraints are determined by direct reference to the mathematical model of the object under study. The number of points that make up the initial plan of the experiment depends on the dimension of the problem and the chosen approximation functions. The solution results in a Pareto (optimal) set of solutions.

#### Effects of quantum interference on tunneling magnetoresistance through a single aromatic molecule

1University of Sistan and Baluchestan, physics, Iran (Islamic Republic of)

###### Abstract

The spin-dependent transport properties through a single aromatic molecule sandwiched between two ferromagnetic (FM) electrodes is investigated theoretically. The transmission probability, current–voltage characteristic and tunnel magnetoresistance (TMR) are analyzed by the tight-binding Hamiltonian model and the nonequilibrium Green’s function technique. It is shown that all these characteristics are sensitive to the quantum interference effects originated from the molecule-to-electrode coupling. The spin-dependent transport properties are characterized by several significant factors. One of the important factors in the spin-dependent transport properties of single molecules is the effect of quantum interference that has recently attracted much attention in recent years [1-2]. The effect of quantum interference associated with the molecule-to-electrode interface geometry. In other words, such effects occur when the electronic wavefunctions propagating along the various pathways through ferromagnetic junction. [1] Guimarães M H D, Zomer P J, Vera-Marun I J, van Wees B J 2014 Nano Lett. 14 2952 [2] Stadler R 2009 Phys. Rev. B 80 125401

#### Finite Elements and Finite Differences in some differential equations of second linear order with GeoGebra

Jorge Olivares Funes1 , Elvis Valero2

1University of Antofagasta, Department of Mathematics, Chile

###### Abstract

Let's consider the differential equations of the shape $$-\ \frac{d}{\ dx}(p (x)\frac{dy}{dx\ })\ + q(x)y = f(x)$$, $$y(0)=y(a)=0, a>0.$$ Using GeoGebra software and the numerical methods of finite elements and finite differences, We will display the various numerical approximations they get for each value of "a" along with their absolute and relative error in various applets and examples.

#### Mathematical modelling of flows around the slider body with cavity

Duong Ngoc Hai1 , Nguyen Quang Thai2

1Vietnam Academy of Science and Technology (VAST), Graduate University of Science and Technology, Viet Nam
2Vietnam Academy of Science and Technology (VAST), Institute of Mechanics, Vietnam

###### Abstract

While an object at ambient temperature moving within a fluid, if the relative speed between the object and fluid is large enough, due to evaporation the natural vapour cavities can be appeared on the object’s surface. The mixture flows of such fluids and cavities are called as the natural cavitating flows or cavitating flows. The mathematical modelling for this kind of flows are usually complex because of the transient of laminar to turbulent region in flow near body wall, and moreover the existence of phase transition and vapour cavity with changed shapes. In this case, the mathematical models will contain pairs of models, such as flow pattern model for turbulent transient flows and cavitation model for cavitating flows to achieve the correct calculation results. In this paper, the typical mathematical models of cavitating flow around a slider body in water based on the combining of two possible turbulent flow models (LES – Large Eddy Simulation and RAS – Reynolds Averaged Simulation) and, to a pair, three cavitation models (Kunz, Schnerr-Sauer and Merkle models) are presented. Based on those the numerical simulations for cavitating flow around the two different shape bodies (hemisphere head body and sphere shape bodies) at same flow condition (cavitation number σ = 0.2) are performed by using each above mathematical model. The comparisons of numerical results with published experimental measurement results are performed to evaluate the effect of body shapes (existence of the cylinder body) and the accuracy of numerical results. The paper results might be helpful for investigation of cavitation phenomena

###### Acknowledgements:

This work was supported partly by Grant of the NCVCC42.02/20-20 from the Vietnam Academy of Science and Technology (VAST).

#### Getting and regularizing a hexagonal irregular grid

Sergei V Ryzhkov1 , Victor Kuzenov2 , Sanya Dobrynina3 , V. Shumaev4 , Andrey Starostin5

1Bauman Moscow State Technical University, Thermal Physics Department, Russian Federation
2Dukhov VNIAA, , Russian Federation
3BMSTU, , Russian Federation
4BMSTU, Thermal Physics Department, Russian Federation
5BMSTU, , Russian Federation

###### Abstract

A method is proposed for the transition from a tetrahedral to a hexagonal irregular computational grid. A variant of the elliptic “regularizer” of the grid is developed, which is based on the “mechanical analogy” and is based on the solution of linear equations of the theory of elasticity. The paper presents the initial results of the reconstruction and “regularization” of the computational grid, as well as the distribution of the “angular” criterion for assessing its quality. The hexagonal “regularized” computational grid is shown, as well as the distribution of the “angular” criterion for assessing its quality for the geometric model of a hypersonic aircraft. From the calculation results it follows that the “regularized” grid fills almost the entire volume of the computational domain, and the criterion for assessing the quality of the hexagonal “regularized” computational grid is more than 0.7. Moreover, to achieve this result, ~20 iterations were required only.

#### On instability of three-dimensional dynamic equilibrium states of self-gravitating Vlasov-Poisson gas

Yuriy Gubarev1 , Shuang Sun2

1Lavrentyev Institute for Hydrodynamics, Laboratory for Fluid and Gas Vortex Motions, Russian Federation
2Novosibirsk National Research State University, Department for Differential Equations, Russian Federation

###### Abstract

The Vlasov-Poisson model of boundless collisionless gas of neutral particles in a self-consistent gravitational field continues to be one of the basic models of modern astrophysics. This is due to simplicity, clarity, and obvious effectiveness of the model in describing large-scale processes in the Universe. Despite the fact that this model has been intensively studied for a long time, from the point of view of the mathematical stability theory, it was possible to establish, by and large, only sufficient conditions for the theoretical stability (at semi-infinite time intervals) of a number of dynamic equilibrium states with respect to both small and finite perturbations, but from incomplete unclosed subclasses. In this work, we consider the spatial motions of the boundless collisionless self-gravitating Vlasov-Poisson gas of neutral particles in a three-dimensional Cartesian coordinate system: $$\frac{\partial f}{\partial t} + v_i\frac{\partial f}{\partial x_i} - \frac{\partial \varphi }{\partial x_i}\frac{\partial f}{\partial v_i} = 0, \frac{\partial ^2\varphi }{\partial x^2_i} = 4\pi \int \limits _{\mathbb{R}^3}f({\bf x}, {\bf v}, t)d{\bf v} \ (1)$$ $$i = 1, 2, 3; f = f({\bf x}, {\bf v}, t) \geq 0; f({\bf x}, {\bf v}, 0) = f_0({\bf x}, {\bf v})$$ Here $f$ denotes the distribution function of neutral particles (for reasons of convenience, their masses are assumed to be the same and equal to unity); $t$ is time; ${\bf x} = (x_1, x_2, x_3)$ and ${\bf v} = (v_1, v_2, v_3)$ denote coordinates and velocities of neutral particles; $\varphi ({\bf x}, t)$ is the potential of a self-consistent gravitational field; $f_0({\bf x}, {\bf v})$ denotes the initial data for the function $f$. We suppose that the distribution function $f$ asymptotically approaches zero as $|{\bf v}| \rightarrow \infty$, and this function along with the potential $\varphi$ are periodic in argument ${\bf x}$ or asymptotically approach zero as $|{\bf x}| \rightarrow \infty$ too. Summation is performed on repeating lower index $i$ throughout the work. It is assumed that the mixed problem (1) has the following exact stationary solutions: $$f = f^0({\bf x}, {\bf v}) \geq 0, \varphi = \varphi ^0({\bf x}) \ (2)$$ $$v_i\frac{\partial f^0}{\partial x_i} = \frac{\partial \varphi ^0}{\partial x_i}\frac{\partial f^0}{\partial v_i}, \frac{\partial ^2\varphi ^0}{\partial x^2_i} = 4\pi \int \limits _{\mathbb{R}^3}f^0({\bf x}, {\bf v})d{\bf v}$$ The aim of this work is to prove the absolute linear instability for the spatial states of dynamic equilibrium (2) of the boundless collisionless self-gravitating Vlasov-Poisson gas with respect to small three-dimensional perturbations $f^\prime ({\bf x}, {\bf v}, t)$ and $\varphi ^\prime ({\bf x}, t)$: $$\frac{\partial f^\prime }{\partial t} + v_i\frac{\partial f^\prime }{\partial x_i} - \frac{\partial \varphi ^\prime }{\partial x_i}\frac{\partial f^0}{\partial v_i} - \frac{\partial \varphi ^0}{\partial x_i}\frac{\partial f^\prime }{\partial v_i} = 0 \ (3)$$ $$\frac{\partial ^2\varphi ^\prime }{\partial x^2_i} = 4\pi \int \limits _{\mathbb{R}^3}f^\prime ({\bf x}, {\bf v}, t)d{\bf v}; f^\prime ({\bf x}, {\bf v}, 0) = f_0^\prime ({\bf x}, {\bf v})$$ where $f_0^\prime ({\bf x}, {\bf v})$ denotes the initial data for the function $f^\prime$. In the work, a transition from kinetic equations (1) which describe the motion of the gas under study to an infinite system of relations similar to the equations of isentropic flow of a compressible fluid medium in the “vortex shallow water” and Boussinesq approximations was carried out. In the course of the instability proof, the well-known sufficient Newcomb-Gardner-Rosenbluth condition for stability of dynamic equilibrium states (2) with respect to one incomplete unclosed subclass of small spatial perturbations was conversed. Also, a linear ordinary differential second-order inequality with constant coefficients was obtained. An a priori exponential lower estimate for the growth of small three-dimensional perturbations (3) follows from this inequality when the sufficient conditions for linear practical instability of the considered dynamic equilibrium states found in this work are satisfied. Since the obtained estimate was deduced without any additional restrictions on the equilibrium states under study, then the absolute linear instability of the spatial states (2) of the dynamic equilibrium of the Vlasov-Poisson gas with respect to small three-dimensional perturbations (3) was thereby proved. The results of the work are fully consistent with the classical Earnshaw instability theorem. This theorem states that any equilibrium configuration of point electric charges is unstable if, besides its own Coulomb forces of attraction and repulsion, no other forces act on them. Now the area of applicability for the Earnshaw theorem is expanded from electrostatics to kinetics, namely, to the boundless collisionless self-gravitating Vlasov-Poisson gas of neutral particles. Constructiveness is inherent in the sufficient conditions for linear practical instability established here, which allows them to be used as a testing and control mechanism for conducting physical experiments and performing numerical calculations.

#### Quantum Implementation for Comparing Sets of Data

yehuda Roth1

###### Abstract

Based on entangled states, quantum computers have the advantage of simultaneously implementing a large number of processes. The coherence of entanglement enables a single operator (logical gate) to be activated simultaneously on all of the states in the superposition Consequently, to implement a quantum computer, a quantum algorithm has to be implemented. In this paper, we propose a different quantum approach that can simultaneously analyze a large amount of data. Although our process allows many processes to work simultaneously, it is not within the conventional frame of quantum computers.

#### Research of the plasma characteristics in the magnetooperated hollow arc cathode

Mikhail Dokukin1

1Bauman Moscow State Technical University, Department of Physics, Russian Federation

###### Abstract

Need of a research of processes for the heavy current discharge with the refractory hollow cathode when pumping through it inert working gas is dictated by the requirements of technologies of welding and melting of the chemically active metals in a vacuum. In the offered work the experimental and theoretical research of the magnetooperated hollow cathode for the purpose of determination of the intra cathodic plasma parameters and the influence of the last on the power characteristics of the external discharge is conducted. Good compliance of the settlement quantities of this plasma with the results of experiments on the model device is obtained. Recommendations for improvement of the operational characteristics of the used arc vacuum technological devices are made.

#### The open unsymmetrical stadium billiard

Julio S Espinoza-Ortiz1 , Roberto E. Lagos-Monaco2

1Federal University of Goias-Catalão, Physics, Brazil

###### Abstract

In open billiards a particle can escape from the cavity through a leak. This type of systems have received special attention because of their applications to a wide variety of physical phenomena ranging from hydrodynamics to quantum chaos and astronomy. Chaotic leaked billiards are characterized by a so called transient behavior, i.e. by the presence of chaotic motion with a finite life time impossible to be studied just through the analysis of its asymptotic behavior. Under this scenario, we consider the quarter stadium billiard and study the influence of its leaking marginal unstable periodic orbits on the survival trajectories. A rigorous statistical analysis of the survival probability is presented. To pursue this objectives, we set up the classical trajectories' solution in such a way that the system only depends on its partial separability and then from it we pass to construct the Birkhoff map. The possibility of more than one leak into a billiard is also considered.

#### Simulator development of a rotary magneto-caloric refrigerator by stepwise regenerator modeling approach

L. Diógenes T. Câmara1

1IPRJ_UERJ, Dep. Mechanical Eng. and Energy, Brazil

###### Abstract

Authors Tedesco, J.C.G. and Câmara, L.D.T. Affiliations Polytecnique Institute of UERJ, Nova Friburgo-RJ, Brazil Abstract Magnetic refrigeration is a new promising technology based on the magneto-caloric effect of solid materials like gadolinium that offers smaller global environmental impact if compared to conventional refrigeration vapor compression processes which utilizes in general the ozone depletion chlorofluorocarbons refrigerants. The rotary refrigerators presents a new challenge in terms of complexity if compared to reciprocating ones which is compensated by refrigeration capacity, steady process operation, performance etc. The modeling and simulation of magneto-caloric refrigerator processes can provide important data in the development and optimization of the experimental units which are in general the only research step carried out by the researchers. A novel full process simulator of a magneto-caloric refrigerator was implemented to simulate the process performance over different conditions of rotating frequency, pump flow rate, room temperature etc. A stepwise modeling approach of the clockwise regenerator movement was implemented which simplifies the phenomena of heat transfers in the regenerators leading to ordinary differential equations which are solved more easily if compared to the partial differential equations in general applied to such complex process. The magneto-caloric bed material utilized in the rotating clockwise wheel was gadolinium with an anticlockwise closed flow loop of water which percolates the six gadolinium porous beds and also the hot and cold heat exchanger. The simulator was able to represent the transient aspects as well as the steady state conditions of the magneto-caloric refrigerator processes in terms of both low time performance and numerical stability. The inversion in heat transfer profiles along the process can be used as a limit in the calculation of the maximum heat transfer absorption in the refrigerator cold exchanger according to the operating conditions assumed.

#### Algorithm for solutions of nonlinear equations of strongly monotone type and applications to convex minimization and variational inequality problems

Mathew Aibinu1

1Durban University of Technology, Institute for Systems science, South Africa

###### Abstract

Real life problems are governed by equations which are nonlinear in nature. Nonlinear equations occur in modeling problems, such as minimizing costs in industries and minimizing risks in businesses. A technique which does not involve the assumption of existence of a real constant whose calculation is unclear is used to obtain a strong convergence result for nonlinear equations of (p, η)-strongly monotone type, where η > 0, p > 1. As a consequence of the main result, the solutions of convex minimization and variational inequality problems are obtained. This solution has applications in other fields such as engineering, Physics, Biology, Chemistry, Economics and game theory.

#### An invisible DWT watermarking algorithm using noise removal with application to dermoscopic images

Simona Moldovanu1

1Dunarea de Jos University, Computer Science and Information Technology, Romania

###### Abstract

A new approach for the digital watermarking process is proposed to be part of the pre-processing stage of a computer-aided diagnosis system. We propose to embed a denoised image acting as watermark image in the original host image with the final goal of improving the quality of demoscopic images for further image processing operation related to CAD. The proposed algorithm uses Discrete Wavelet Transform (DWT) corroborated with some basic properties of Human Visual System such as Contrast Sensitive Function (CSF) and Noise Visibility Function (NVF) with the goal of correlating the texture properties and noise. This approach hides the watermark (i.e. denoised version of the host image) in high-pass subbands that are focused on image features. The main concern is to evaluate the distortion produced to the host image by watermarking and an objective quality measure function, i.e. Weighted Peak Signal-to-Noise Ratio (WPSNR), is used to evaluate the existing differences between the original and watermarked images. The proposed approach is tested using the available skin lesion images from the digital image archive of the Department of Dermatology of the University Medical Center Groningen. The experiment results show the improved performance of the proposed scheme against a 3 3 median filtering attack in comparison with the a 5 5 median filtering attack.

#### Analytic Approximation to the modified Bessel function I1(x) with high accuracy

Pablo Martin1 , Jorge Olivares Funes2 , Adrian Sotomayor3

1University of Antofagasta, Physics department, Chile
2University of Antofagasta, Department of Mathematics, Chile
3University Antofagasta, Mathematic, Chile

###### Abstract

In a previous work a method to approximate modified Bessel functions of integer order were presented and applied to the modified Bessel function I1(x) [1]. An improvement to this technique is carry out in the present work. The procedure is, as in previous work , to use simultaneously power series and asymptotic expansions . Starting from both expansions an analytic function is determined as a combination of rational functions and elementary ones [2,3]. This last one in the present work was selected as the hyperbolic function sinh(x), this change produce a lot differences Fist ( where x is the independent variable) , and in previous work was cosh(x)[1]. The parameters of the actual rational function are different to those presented in our previous work . Besides, the important point, is that the efficiency now is better, and the results , using the same number of parameters that in previous one , have now much lower error . In the simplest case of the three parameters , the largest relative error is now 0.007 instead of 0.011 . The evolution of both relative errors can be shown graphically as a function of the independent variable . There is a confluence of the relative errors of both approximations for large values of the independent variable , and this relative error is about 0.004 . Reference [1]. P. Martin, J. Olivares and A. Sotomayor. “Precise analytic approximation for the modified Bessel function I1(x)”. Rev. Mex. Fisica 63, 130-133 (2017). [2]. P. Martin, JJ. Rodriguez-Nunez, JL. Marquez, “Two-dimensional hydrogenlike atoms in the presence of a magnetic field: Quasifractional approximations”. Phys. Rev. B45 , 8359-8362 (1992). [3]. P. Martin, E. Castro, J.L. Paz, “Multi-point quasi-rational approximants for the energy eigenvalues of two-power potentials “. Rev. Mex. Física 58, 301-307 (2012).

#### To the optimized approach to get the fundamental property of the gravity assists maneuvers from the Jacobi integral

Alexey Grushevskii1

1Keldysh Institue of Applied Mathematics - KIAM, Russian Academy of Sciences, Russian Federation

###### Abstract

The design of interplanetary trajectories using a series of gravity assists maneuvers begins with the ballistic mission design. It is reasonable to construct the corresponding initial approximation using the patched conics method within the model of the circular restricted three-body problem. Such a construction requires the calculation of the “transfer parameter” Vinf - the asymptotic velocity of the spacecraft relative the target planet, when switching from heliocentric arcs to planetocentric segments and vice versa. In the circular restricted three-body problem model, can be calculated using the Jacobi integral J (or using it's analogue - the Tisserand parameter Ti ) and the basic property of the Jacobi integral for the gravity assists maneuvers within the framework of the circular restricted three body problem: J=3 - Vinf*Vinf . According to this property, the J value does not change during the multiple gravity assists maneuvers that preserve the Jacobi integral constant are performed. This fact is known in astrodynamics but it is classically derived in a rather cumbersome method. In this study, a shorter method for it’s obtaining is proposed. The modifications of the representation of the Jacobi integral in the circular restricted three-body problem for the various configurations of three bodies and the table of transformations of the Jacobi integral and the Tisserand parameter are presented for all cases.

#### Several Classes of Plain Dynamic Systems Qualitative Investigation

Irina Andreeva1

1Peter the Great St.Petersburg Polytechnic University, Higher Mathematics, Russian Federation

###### Abstract

Dynamic systems in applications are useful as mathematical models of those processes and phenomena, where statistical events, or fluctuations, may be disregarded. Dynamic systems may be divided into the two main categories - the systems with continuous time (the flows), and systems with discrete time (the cascades). During the investigations of, first of all, flows, normal autonomous systems of ordinary differential equations are used. The present work is devoted to the original rigorous research of some important family of dynamic systems having reciprocal polynomial right parts, which are the forms of their phase variables. The whole wide family under consideration is being split into numeric subfamilies belonging to different hierarchical levels, and is subjected to the first and the second Poincare transformations, or mappings. As a result, the full qualitative pattern of trajectories is constructed - using the Poincare sphere - in the Poincare disk. A series of new special investigation methods was developed, useful for further investigations of similar dynamic systems’ classes. References. 1. Andreeva,Irina, Andreev, Alexey. Investigation of a Family of Cubic Dynamic Systems. //Vibroengineering Procedia. Vol.15. Dec.2017. Pp.88-93. 2. Andreeva I.A., Andreev A.F. Phase Portraits of One Family of Cubic Systems in a Poincare Circle. I.//Vestnic RAEN. 2017. Vol.17. №4. Pp.8–18. 3. Andreeva I.A., Andreev A.F. Phase Portraits of a Family of Cubic Systems in a Poincare Circle.I I.//Vestnic RAEN. 2018. Vol.18. №4. Pp.11–15. 4. Andreev A.F., Andreeva I.A. Phase Portraits of Some Family of Cubic Dynamic Systems in a Poincare Circle. III.//Vestnic RAEN. 2019. Vol.19. №2. Pp.20–24. 5. Andreev A.F., Andreeva I.A. On a Behavior of Trajectories of a Certain Family of Cubic Dynamic Systems in a Poincare Circle. // IOP Journal of Physics, Conference Series, 2018, 1141. 6. Andreeva I.A., Efimova T.O. Phase Portraits of a Special Class of Dynamic Systems in a Poincare Circle.//IOP Journal of Physics, Conference Series, 2019, 1236.

###### Acknowledgements:

Prof. Dr. Alexey F. Andreev, St.Petersburg State University, Russia

#### Simple math model for calculation about possibility to disclose Stealth

Oleksandr Denisov1

1Harbin Institute of Technology, Microwave Engineering, China

###### Abstract

. Stealth coating is the antidote against of the radar. But it can be disclosed by the microwave radiometer because their job based on the measuring the radio-brightness contrast between Stealth object and the background of the environments (sky is cold, Earth is warm). This short report presents some math model for the simplest calculation of the possible disclosing distance till the Stealth object.

###### Acknowledgements:

Authors wishing to acknowledge assistance from colleagues in Harbin Institute of Technology.

#### Notes on the Introduction of Parallel Definitions of Energetic and Information Producing Systems

Valeriy Zakharov1

1Lomonosov Moscow State University, Mathematical Analysis, Russian Federation

###### Abstract

Informatics reached extraordinary heights in its development. However, a satisfactory solid theoretical foundation for this science has not been established until now. The reason is the absence of satisfactory general definitions of the notions of information and information system. The distinguished peculiarity is inherent not to Informatics only. In Physics despite of its longer existence the situation with the solid theoretical foundation gets on in the same way: there are no satisfactory general definitions of the notions of energy and energy system. In 1964 the outstanding American physicist Richard Feynman in his famous lectures [Feynman R.F., Leighton R.S., Sands M. The Feynman Lectures on Physics. V. 1: Mainly Mechanics, Radiation, and Heat. − United States of America: Addison Wesley Publishing Company, 1964. − 270 p.] in §1 of Chapter 4 has written: «It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, …». The following conclusion may be done from this apparently not random coincidence: the cause of the described situation is founded not in Informatics itself and not in Physics itself, but in the absence of satisfactory general solid theoretical conception of the world and its being, in which the notions of information and energy might appear in some natural deductive way. Some joint (synthetic) closed in itself idea about the world and its being is expounded in the report, which gives the opportunity to invent the parallel definitions of the energy and the information not exceeding the bounds of the united world. This allows us to introduce some sufficiently general notion of the produciпg (conservatively-dynamic surrounded stream) system, described by the proper system of evolutional equations. In the capacity of important partial cases of such systems the proper notions of the energetic producing system and the informatic producing system are introduced. The explicitly analyzed examples of the heating stove (as the energetic producing system) and the personal computer (as the informatic producing system) expose the applicability of proposed idea to a generalized and formalized description of some wide class of systems really existing.

#### Inspection Planning Optimization Method of Power Distribution Equipment

Seyed Morteza Moghimi1

1IAU, Electrical Engineering department, Iran, Islamic Republic Of

###### Abstract

Power distribution companies should provide an optimal maintenance plan that balances the reliability and efficiency of their widespread and distributed facilities. It is common for power companies to distribute their distribution areas to multiple inspections and maintain their high reliability through periodic inspections. Such a time-based maintenance plan is generally inefficient. This paper provides a method for optimizing the power distribution inspection method. A time interval inspection is proposed that estimates the time interval of each facility using previous inspection data. In addition, an algorithm is proposed to reduce the cost of inspection based on the estimated time interval. This algorithm effectively defines the optimal combination of two inspection plans, a regional inspection and a single inspection. The evaluation of the results with experimental data showed that this algorithm reduces the cost of inspection while maintaining the reliability of the network. The proposed method has led to a more effective inspection program.

#### A Novel Graph Theory based Algorithm for Power Management of the 5G Ultra Dense Networks by Distribution Networks Reconfiguration

Seyed Morteza Moghimi1

1IAU, Electrical Engineering department, Iran, Islamic Republic Of

###### Abstract

A novel algorithm using the graph theory to optimize the efficiency of the power distribution network is proposed in this paper that alongside with the power consumption controlling of the 5G radio networks offers a promising energy solution towards the increasing power requirement of the 5G ultra dense networks. As the 5G is required to serve billions of associated gadgets and the Internet of Things (IoT), with the correct compromise between speed, latency, and vitality of energy consumption at a reasonable expense. 5G networks will unequivocally rely upon utilizing a very high density of embedded Small Cells (SCs). In addition, to that they depend on the Macro Cells (MCs). This sort of Ultra-Dense Networks (UDN) comprising of an enormous number of MCs and SCs require a huge amount of energy to power them up. Improving the efficiency of the power distribution network and controlling the energy consumption in the 5G radio network can move us towards the expansion of the 5G solution is all aspects of technology. After reconfiguration using the graph algorithm method, the voltage profile of the system has shown improvement, with an overall decreased system loss by 15.23 percent. Also, in order to compare the results of this algorithm with other papers, there is no limit to the voltage of the bus and the flow of the lines. By performing the reconfiguration, the optimal arrangement loss is reduced to 0.001609 per unit. The reconfiguration solution is constructed by the algorithm representing a computation data of the power grid topology built upon a radial network.

#### Geesthacht Coupled Coastal Model System (GCOAST)

Joanna Staneva1

1HZG, Hydrodynamics and Data Assimilation, Germany

###### Abstract

The GCOAST (Geesthacht Coupled cOAstal model SysTem) is built upon a flexible and comprehensive coupled model system, integrating most important key components of the regional and coastal systems, that enable to include information from observations. It encompasses: (i) atmosphere-ocean-waves interactions, (ii) the dynamics and fluxes in the land-sea transition, (iii) the coupling of the marine hydrosphere and biosphere. Triggered by the need for novel modelling capacity, GCOAST system is designed to handle cross compartment fluxes of water and energy between the atmosphere and ocean thought the dynamic wave interface, dynamics and biogeochemistry in the land-ocean transition and marine ecosystems and benthic-pelagic coupling, transport and transformation of environmental pollutants. Coupling of different models is a commonly used approach when addressing the complex interactions between different components of the earth system. We focus on the nonlinear feedback between strong tidal currents and wind -waves, which can no longer be ignored, in particular in the coastal zone where its role seems to be dominant. In NEMO stand-alone model, the momentum flux from the atmosphere, which is related to the wind speed, is passed directly to the ocean and this is controlled by the drag coefficient. However, in the real ocean, the waves also play the role of a reservoir for momentum and energy because different amounts of the momentum flux from the atmosphere is taken up by the waves. In the coupled model system the momentum transferred into the ocean model is estimated as the fraction of the total flux that goes directly to the currents plus the momentum lost from wave dissipation. Additionally, we demonstrate that the wave-induced Stokes Coriolis force leads to a deflection of the current. During extreme events, the Stokes velocity is comparable in magnitude to the current velocity. The resulting wave-induced drift is crucial for the transport of particles in the upper ocean. The performance of the coupled modelling system also illustrated for the cases of several extreme events and assessment of coupled model versus newly available data (e.g. Sentinel) is performed. The comparisons with in-situ and satellite data showed that the implementation of wave model component into the coupled systems reduces the errors, especially under severe storm conditions.

###### Acknowledgements:

This work is supported by the Initiative and Networking Fund of the Helmholtz Association through the project “Advanced Earth System Modelling Capacity (ESM)” and Cluster of Excellence “Climate, Climatic Change, and Society” (CLICCS) .

#### Dynamic model of ultrasonic micro-scale impact processing by two coaxial longitudinal waveguides

Ildar vagapov1

1Kazan Federal University , Naberezhnye Chelny Institute, Russian Federation

###### Abstract

A dynamic model of the ultrasonic vibro-impact processing using an oscillation system with two longitudinal coaxial waveguides mounted with a gap between their working ends is considered. One waveguide is connected with an ultrasonic transducer and the other serves as a passive resonant anvil. Waveguides are described as of visco-elastic rods with close natural frequencies. The material being deformed in the gap between the vibrating ends of waveguides is approximated by a rigid-plastic rheology. Vibro-impact processing is presented as the forced oscillations of two coupled resonant subsystems, where the workpiece is assumed both as the processing load and the nonlinear coupling link. The amplitude and phase responses of the longitudinal oscillations of the two-rod system are calculated, the ranges of oscillatory stability are determined. The difference in the natural frequencies of the waveguides is taken into account. Two cases of the relation between the natural frequencies of interchangeable waveguides are considered: the natural frequency of the forced waveguide is higher or lower than that of the passive waveguide. It is shown that in the first case resonance of the anti-phase oscillations of waveguides take place. In the second case the oscillation system resonates on the in-phase eigen-mode. The anti-phase oscillations are preferable for vibro-impact processing because the working ends of waveguides move toward each other at the impact moment. Application of ultrasonic micro-forging to cutting edge sharpening is exemplified. The threshold value of the impact force, which is necessary to overcome the resistance to plastic flow, is derived analytically. The maximum achievable degree of plastic deformation is estimated. It is shown that the anti-resonant mode of operation of the oscillatory system is most effective. Recommendations for design parameters and resonance tuning of ultrasonic equipment are given.

#### Multi-physics simulation of shedding of in-flight ice

Andrea Rausa1 , Alberto Guardone2

1Politecnico di Milano, Department of Aerospace Science and Technology, Italy
2Politecnico di Milano, Department of Aerospace Science & Technology, Italy

###### Abstract

In-flight ice accretion may possibly jeopardise the safety of fixed- and rotary-wing aircraft. Icing can possibly occur if supercooled water droplets in clouds impinge on the aircraft surfaces and freeze upon impact. It may result in instrument failures and degradation of the aerodynamic performances. A major problem related to ice accretion is the possibility of ice shedding from the main body and impacting other parts of the aircraft or being ingested by the engines. In fixed-wing aircraft, shedding is caused by the action of the aerodynamic forces or the activation of an Ice Protection System. In the present work, a multi-physics framework is presented to simulate ice accretion and shedding from wings and engine nacelles. The aerodynamics is computed using the open-source tool-kit SU2. Cloud droplet trajectories are computed using the arbitrary-precision Lagrangian in-house solver PoliDrop. Then, the in-house ice accretion tool-kit PoliMIce is used to determine the ice layer. A FEM structural analysis is performed on the accreted ice shape by means of the open-source code MoFEM. Internal stresses within the ice geometry due to aerodynamic forces are computed. The possibility of the occurrence of cracks in the ice layer is assessed and its propagation is determined numerically. Two-dimensional ice accretion simulations are performed to check the validity of the present approach and compare fairly well with available results.

#### Modeling the influence of the Earth rotation axis position on the global climate variations

Valeriy Parkhomenko1

1The Federal Research Center "Computer Science And Control" of The Russian Academy Of Sciences, Bauman Moscow State Technical University, Dorodnicyn Computing Centre, Russian Federation

###### Abstract

This study presents the results of numerical experiments to determine the Earth’s climate when its rotation axis is displaced without changing the axis tilt to the ecliptic plane. There is some evidence of the possibility of this shift in the past. The calculations were carried out using a hydrodynamic three-dimensional global climate model, including blocks of atmosphere, thermohaline large-scale ocean circulation and sea ice. Numerical experiments demonstrate a significant temperature changes throughout the world. A large area of Antarctica warmed up to temperatures above 15 ° C. This is reason of intense melting of glaciers for a long time. Significant warming of the Arctic Ocean will lead to sea ice melting in the Arctic. Strong changes in temperature and ice cover lead to significant changes in horizontal ocean circulation. A procedure is proposed for calculating wind speed in atmosphere energy - moisture balance model. It is based on the geostrophic approach, taking into account the thermal component of the wind, and introducing the mechanism of friction on the underlying surface. A technique has been developed for the formation of the necessary maps and the relationships between them when turning the Earth rotation axis or using new cartographic data.

#### On dark stars, galactic rotation curves and fast radio bursts

Igor Nikitin1

1Fraunhofer Institute for Algorithms and Scientific Computing, SCAI, Germany

###### Abstract

This paper is a continuation of our recent work on Radial Dark Matter stars (RDM-stars), black holes, coupled with radial flows of dark matter. As a galaxy model, it produces flat rotation curves, approximately valid for many galaxies far from the center. In this paper, more detailed modeling is carried out, including the vicinity of the galactic center. Assuming that the distribution of stellar black holes repeats the distribution of luminous matter, we get a perfect match between the model rotation curves and the observed ones. Further, using numerical integration, we examine the gravitational field of an individual RDM-star. The computation shows the event horizon being erased and rapidly increasing mass density arising instead (mass inflation). In this regime, we apply the previously constructed Planck star model, where at high densities a repulsive force occurs (quantum bounce). In our stationary model, the evolution of a Planck star has stopped under the pressure of dark matter flows. This system is considered as a possible source of Fast Radio Bursts (FRBs). In a scenario involving an asteroid falling onto an RDM-star, the model reproduces the correct frequency range of FRBs. Their total energy, coherence and short duration are explained as well.

#### Mathematical modeling and visualization of topologically non-trivial solutions in general relativity

Igor Nikitin1

1Fraunhofer Institute for Algorithms and Scientific Computing, SCAI, Germany

###### Abstract

In general relativity, there is a class of solutions that currently do not have observed analogues, but on which the theory is shaped, giving an understanding what is fundamentally possible within its framework. Such solutions include wormholes, tunnels that connect distant regions in spacetime. Although not a single wormhole has yet been discovered, there is a large number of works devoted to their study, thanks to which wormholes as a class of solutions become firmly established in modern science. In this paper, we consider two topologically nontrivial types of solutions related to wormholes. First: wormholes that can open and close. In this relation, we will discuss topological censorship theorems, which under certain conditions prohibit changing topology. We will also discuss known ways to circumvent these theorems. Using analytical and numerical methods, as well as visualization, we will construct an example of an opening and closing wormhole with the dimensions of the central black hole in the Milky Way galaxy. Our construction continues the work by Kardashev, Novikov and Shatskiy, in which a static wormhole with the same parameters was considered. The second type is a modification of Visser's dihedral wormhole solution for a dynamic case.

#### Predicting entanglement and coherent times in FMO complex using the HEOM method

Francisco Delgado1 , Alan Anaya-Morales2 , Bruno Gonzalez-Soria3

1Tecnologico de Monterrey, Physics and Mathematics, Mexico
2Tecnologico de Monterrey, Physics and Mathematics, Mexico
3Tecnologico de Monterrey, Physics and Mathematics, Mexico

###### Abstract

Advances in ultrafast-spectorscopy techniques have revealed long time quantum coherences between electronic states in Fenna-Matthews-Olson (FMO) bacteriochlorophylls, molecules responsible of the energy transfer in the photosynthetic process of green sulfur bacteria. Several methods have been explored to model this quantum phenomenon, mainly using quantum open systems theory. Most of these methods studied do not take into account the memory effects of the surrounding, commonly approximated as a phonon bath on thermal equilibrium. This article applies the Hierarchical Equations of Motion method (HEOM), a non Markovian approach, for the modelling of the system evolution to perform predictions about the coherence times scales together with the global or semi-local entanglement measures involved during the quantum excitation process analysed in terms of some relevant parameters in such system. This analysis suggests a possible roadmap to track or to fit genetic modifications improving the photosynthetic performance.

###### Acknowledgements:

Bruno Gonzalez-Soria, Francisco Delgado, Alan Anaya-Morales

#### Performance of two redundant quantum channels for single qubits under indefinite causal order

Francisco Delgado1 , Carlos Cardoso-Isidoro2 , Marco Enriquez-Flores3

1Tecnologico de Monterrey, Physics and Mathematics, Mexico
2Tecnologico de Monterrey, Physics and Mathematics, Mexico
3Tecnologico de Monterrey, Physics and Mathematics, Mexico

###### Abstract

Indefinite causal order has introduced novel procedures to improve the quality of quantum communication. This procedure introduces the superposition of paths on a set of consecutive quantum channels. It has demonstrated enhancement in communication on well characterized quantum channels as depolarizing, dephasing-noise and teleportation ones. This work modelling a generic quantum channel for single qubits in terms of Kraus operators for the channels in the form of Pauli operators. An output quantum state for a single qubit states going through a certain imperfect quantum communication channel can be obtained analytically. Then, the quality of such outputs are analysed using the quantum fidelity measure.

#### Visual reasoning and the perception of forms

Sara Vesely1

1CNR, ITB, Italy

###### Abstract

Whenever a new device enables our senses to access an uncharted sensible world, our experience needs to be widened to be able to embrace it. Many animals don't recognize themselves in mirrors. Therefore, mirrors found in ancient Egyptian tombs bear witness to a device that involved a widening of human experience. While by then reflectors were commonly believed to hold the spirit of their beholder, they also provided the motivating force for use of geometry as a logical framework, rather than the form of the outer world. Euclid of Alexandria conceived of geometric constructions and their rules as the connecting link between visual world and hypothetical-deductive reasoning. Yet, he didn't have a clue about receivers. In our opinion the problem of linking received information in an image format to a mathematical space cannot be solved once and for all, but rather needs to be posed and understood afresh once in a while. All the more so in an information and telecommunication era, when the techniques of acquisition and rendering of visual information have been extended well beyond the domain of optical instruments, and the language of mathematics has advanced to a different level of proficiency.

#### Optimal control and stabilization of nonlinear control-affine systems

Armen Bagdasaryan1

1American University of the Middle East, Department of Mathematics, Kuwait

###### Abstract

In this talk we will discuss the problems of optimal control and stabilization for nonlinear control-affine systems of the form $$\dot{x} = A(x)+B(x)u$$ where $$x=x(t), \; x\in\mathbb{R}^n, \; u\in \mathbb{R}^1, \; |u|\leq 1$$ and the vector functions $A(x), B(x)$ are assumed to be smooth in the domain $D\subset \mathbb{R}^n$, $0\in D$, $A(0)=0$. We assume that the system is small-time locally controllable (STLC) at $x = 0$, that is, if $x = 0$ is locally continuously reachable in small time with small control. We give an in-depth analysis of optimal control techniques for the above systems and then consider the problem of synthesis of continuous control $u=u(x)$, $u(0)=0$, that stabilizes the system at the equilibrium point $(x,u)=(0,0)$. The solution to the problem is based on the transformation of the system to the canonical form, and on the usage of nonlinear stabilization.

#### STOCHASTIC SIMULATION TO PROJECT THE KAZAKHSTAN GENERAL PRACTITIONER WORKFORCE SUPPLY TO 2030

Berik Koichubekov1

1Karaganda medical university, informatics and biostatistics, Kazakhstan

###### Abstract

Berik Koichubekov, Azamat Kharin, Marina Sorokina, Ilya Korshukov, Bauyrzhan Omarkulov Department of Informatics and Biostatistics, Karaganda Medical University, 100000 Gogol st. 40, Karaganda, Kazakhstan. Health human resource planning includes assessment of current situation, forecasting future demand and developing appropriate strategies to balance supply and demand for manpower. In different countries different forecasting methods are used and each approach has its own strengths and weaknesses. One of the general weaknesses is incomplete data on the past and present. Therefore, any assumption about the future has a chance of being wrong. Our aim was to explore the effect of the uncertainty of some parameters on total General Practitioner (GP) supply in Kazakhstan to 2030. System dynamics simulation was used to develop model for General Practitioner workforce forecasting. Sensitivity analysis (Monte Carlo method) was performed to account for changes in the number of GP as each and set of model parameters is varied. This produced output values that are value ranges rather than point estimates. Next, the median, minimum and maximum values at 95% confidence level for each run were used to show the credible interval. Three key input parameters were explored: retirement rate, attrition rate, recruitment rate. For each parameter relative sensitivity S(t) was calculated. As our results showed, the created model is the least sensitive to the parameter of the retirement rate. The relative sensitivity S (t) in the forecast period ranges from 1% to 10%. Changing this parameter within the confidence interval will not allow to bridge the gap between supply and demand. Currently, about 7% of retirement age doctors work in the primary healthcare (PHC). However, in Kazakhstan it is hardly possible to increase their share, because in the PHC there is an active process of replacing doctors of therapists and pediatricians of the Soviet period with general practitioners with other competencies. Both the inflows and outflows affect the health workforce and it is very important that the dynamics of the labor market is understood if countries can formulate effective human resources policies and strategies. Attrition - in the broad sense, is defined as the exit from the workforce, which may be associated with emigration, voluntary exit (for example, to other sectors of employment), illness, death - is an important element of the outflow from the labor market and what governments can directly influence implementing strategies to motivate and retain health workers. Our model is very sensitive to this parameter - S (t) ranged from 27% in 2020 to 113% in 2030. Given the current level of attrition, scenarios of both GP deficit (high limit scenario) and excess supply (most likely, low limit scenario) are possible. Staff turnover is one of the main problems of primary health care. The attrition rate parameter has great potential for controlling the flow of labor. The level of recruitment (other than new GPs) is another factor that has a significant impact on forecasting GPs. According to our data, the share of such individuals is 10.3% (CI: 9.7; 10.9). The sensitivity S (t) to this parameter is comparable to the sensitivity to the rate of depletion (table). We believe that measures taken to reduce dropouts will simultaneously increase recruitment. We also evaluated the effect of all three parameters simultaneously. If the most likely scenario is realized, then the proposal of a General Practitioner will almost completely cover the needs with a small deficit of 68 to 305 doctors. If the high limit or low limit scenarios in the labor market are implemented, serious problems may arise that the Ministry of Health will face. The advantage of system dynamics is its flexibility. As new data accumulate, it is necessary to adjust the basic parameters of the model, which will increase its reliability

#### Bifurcation diagram of stationary solutions of the 2D Kuramoto-Sivashinsky equation in periodic domains

Nikolay Evstigneev1 , Oleg Ryabkov2

1Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, 81 Dynamics of Macrosystems, Russian Federation
2Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, 81 Dynamics of Macrosystems, Russian Federation

###### Abstract

Stationary 2D Kuramoto-Sivashinsky equation for scalar function $u:\mathbb{T}(L)^2 \to \mathbb{R}$ is considered: $$\lambda \left( 2 uu_x + 2 uu_y + \bigtriangleup u \right) + 4 \bigtriangleup^2 u=0,$$ $$u \in \mathbb{T}(L)^2: \mathbf{x} \in [0, L\pi] \times [0, \pi],$$ where $L$ is the integer stretch factor, $\lambda$ is the bifurcation parameter, $()_{j}$ is the derivative in the $j$-direction and $\bigtriangleup$ is the Laplace operator. The constants used in the equations are widely used in other papers and the zero mean is assumed. This equation is physically relevant in terms of model equations for turbulence as well as chaotic dynamical systems. These equations can also describe the behaviour of the thin film hydrodynamics. The equation in question was analyzed by many authors, including very detail numerical analysis of time dependent solutions by A. Kalogirou, E. E. Keaveny and D. T. Papageorgiou as well as analytical analysis of stationary solutions by Cao, Titi; Foias, Titi; Nicolaenko; Ambrose, Azzucato; Temam. A closely related paper by Changpin and Zhonghua discusses the bifurcation of the nontrivial solution from the stationary one at particular points. This paper is dealing with analytical as well as numerical analysis of bifurcations of stationary including the search for dislocated solution curves. For periodic domain the equation is transfered to the Fourier domain with the ansatz $u(\mathbf{x}) = \sum_{ \{j,k\} \in \mathbb{Z}^2} \hat{u}_{j,k} e^{\mathrm{i}(j/L+k)}$, $\hat{u}_{0,0} = 0$ and $\hat{u}_{j,-k} = \left( \hat{u}_{j,k} \right)^*$, $\hat{u}_{-j,0} = \left(\hat{u}_{j,0} \right)^*$ due to reality condition that results in the following discrete infinite dimensional operator: $$F(\hat{u}, \lambda):=\lambda \left[ 2 \sum_{\{l,m\} \in \mathbb{Z}^2 } \left( \frac{\mathrm{i}l}{L} \hat{u}_{l,m} \hat{u}_{l-j,m-k} + \mathrm{i} m \hat{u}_{l,m} \hat{u}_{l-j,m-k} \right) - \left( \frac{1}{L^2} j^2 + k^2 \right) \hat{u}_{j,k} \right] +$$ $$+ 4 \left( \frac{1}{L^4} j^4 + \frac{2}{L^2} j^2 k^2 + k^4 \right)\hat{u}_{j,k} = 0, \forall \{j,k\} \in \mathbb{Z}^2,$$ with $\hat{u0} = \hat{u}_{j,k} = 0$ as a trivial solution.\\ One can immediately observe, that the linear operator $F_u(\hat{u0})$ has eigenvalues $\lambda_{j,k} = \lambda(j^2/L^2 + k^2) - 4(j^4/L^4+2j^2/L^2k^2+k^2)$. Hence the possible bifurcation points of the trivial solution are those points, where $\lambda_{j,k} = 0$. This paper reveals the complexity of the bifurcation structure and analyzes the points of possible bifurcations and linear operator kernel dimensions, which is high. Next, the problem is reduced to finite dimensional grid of $2048 \times 2048$ Fourier harmonics that is solved numerically using deflated pseudo-arclength continuation method. The paper describes heuristics that are used to pass high order degenerate points, where $\text{dim}(\text{ker}(F_u))>1$. The analysis is done for $\lambda \in [0; 30]$, where high order bifurcations as well as dislocated curves of solutions are found.

###### Acknowledgements:

This work is supported by RFBR grant no. 18-29-10008 mk.

#### Disconnected stationary solutions for 2D Kolmogorov flow problem in periodic domain

Nikolay Evstigneev1

1Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, 81 Dynamics of Macrosystems, Russian Federation

###### Abstract

The classical A.N. Kolmogorov's flow problem for the stationary 2D Navier-Stokes equations on a stretched torus for velocity vector function $\mathbf{u}:\mathbb{T}(\alpha)^2 \to \mathbb{R}^2$ and pressure scalar function $p:\mathbb{T}(\alpha)^2 \to \mathbb{R}$ is considered as: $$(\mathbf{u}, \nabla) \mathbf{u} + \nabla p - \frac{1}{R}\bigtriangleup \mathbf{u} - (\sin(\beta y);0)^\mathrm{T} = 0,$$ $$\nabla \cdot \mathbf{u} = 0,$$ where $R$ is the Reynolds number (bifurcation parameter), $\alpha$ is the stretch factor for the domain $\mathbb{T}(\alpha)^2:=[0;2\pi/\alpha] \times [0;2\pi]$ and $\bigtriangleup$ is the Laplace operator. The force vector field depends only on the second spatial variable $y$ and coefficient $\beta$ is an integer. We designate that $\mathbf{u} = (u,v)^\mathrm{T}$, the problem has a trivial solution $u=\frac{R}{\beta^2} \sin(\beta y), v = 0$. An equivalent infinite dimensional system of equations can be obtained in the Fourier space using Galerkin method as: $$\sum_{\{l,m\} \in \mathbb{Z}^2 }\left[ \mathrm{i} \alpha (j-l)\left(1 - \frac{\alpha^2 l^2 + \alpha l m}{\alpha^2 l^2 + m^2} \right) \hat{u}_{l,m}\hat{u}_{j-l,k-m} + \mathrm{i} (k-m)\left(1 - \frac{\alpha l m + m^2}{\alpha^2 l^2 + m^2} \right) \hat{v}_{l,m}\hat{u}_{j-l,k-m} \right] +$$ $$+ \frac{1}{R} (\alpha^2 j^2 + k^2) \hat{u}_{j,k} - \frac{\beta}{2}\delta_{\beta}^{|k|} = 0,$$ $$\sum_{\{l,m\} \in \mathbb{Z}^2 }\left[ \mathrm{i} \alpha (j-l)\left(1 - \frac{\alpha^2 l^2 + \alpha l m}{\alpha^2 l^2 + m^2} \right) \hat{u}_{l,m}\hat{v}_{j-l,k-m} + \mathrm{i} (k-m)\left(1 - \frac{\alpha l m + m^2}{\alpha^2 l^2 + m^2} \right) \hat{v}_{l,m}\hat{v}_{j-l,k-m} \right] +$$ $$+ \frac{1}{R} (\alpha^2 j^2 + k^2) \hat{v}_{j,k} = 0, \forall {j,k} \in \mathbb{Z}^2,$$ where the pressure is eliminated from the system by the projection operator $\mathbb{P}:=(id - \nabla \bigtriangleup^{-1} \nabla \cdot)$ applied to the nonlinear term and the system is obtained by the following ansatz: $f(\mathbf{x}) = \sum_{ \{j,k\} \in \mathbb{Z}^2} \hat{f}_{j,k} e^{\mathrm{i}(\alpha j+k)}$, $\hat{f}_{0,0} = 0$ and $\hat{f}_{j,-k} = \left( \hat{f}_{j,k} \right)^*$, $\hat{f}_{-j,0} = \left(\hat{f}_{j,0} \right)^*$ due to reality condition. \par Many papers are dedicated to the problem at hand. First, early results of Meshalkin and Senai demonstrated that the system is asymptotically stable for any $\alpha>1$. Further researches were conducted by many authors that shown complex system behaviour, infinite number of pitchfork bifurcations as $\alpha \to 0$, existence of recurrent flows for high Reynolds numbers, complex nonlinear dynamics involving cascades of limited cycles and invariant tori of period three as well as chaotic behaviour was found. \par This paper is focusing on the numerical investigation of the finite dimensional Fourier-Galerkin system (using $(512/\alpha) \times (512)$ Fourier harmonics), construction of the solution curves in the parameter-phase space and analysis of disconnected solutions. The system of equations is transformed to the problem $F(\hat{\mathbf{u}}, R) = \mathbf{0}$ for the fixed values of $\alpha$ and $\beta = 2$, where $\hat{\mathbf{u}} = (\hat{u}, \hat{v})^{\mathrm{T}}$ and $R \in [1;20]$. The nontrivial solution curve $\hat{\mathbf{u}}(R)$ that bifurcated from the trivial solution $\hat{\mathbf{u}}_0(R)$ at point $R_0$ is called connected solution curve and $\hat{\mathbf{u}}(R_0) = \hat{\mathbf{u}}_0(R_0)$ at the bifurcation curve. All other curves that bifurcated from the connected curve are also called connected solution curves. Note, that the stability of these solutions is irrelevant meaning that connected curves are applicable to both subcritical and supercritical bifurcations. The disconnected solution curve $\hat{\mathbf{u}}_d(R)$ is such a solution curve that $\hat{\mathbf{u}}_d,R \neq \hat{\mathbf{u}},R$ for any admissible value of $R$, where $\hat{\mathbf{u}}$ is a connected curve. The paper presents bifurcation diagrams for $\alpha = \{1,2,3,4\}$ and shows the location of such disconnected curves. These curves can be responsible for the multistability in chaotic regimes.

###### Acknowledgements:

This work was supported by the Russian Foundation for Basic Research, grants: 18-29-10008mk and 20-07-00066.

#### Modelling and Sizing of a Y-shaped laminar flow Micro-fluidic fuel cell

Antonio Sornoza1 , Jonathan Yepez2 , Mayken Espinoza-Andaluz3 , Martin Andersson4

1Escuela Superior Politecnica del Litoral, ESPOL, , Ecuador
2Escuela Superior Politecnica del Litoral, ESPOL, , Ecuador
3Escuela Superior Politecnica del Litoral, ESPOL, Facultad de Ingenieria Mecanica y Ciencias de la Produccion, FIMP. Centro de Energias Renovables y Alternativas, CERA, Ecuador
4Lund University, Department of Energy Sciences, Sweden

###### Abstract

The energy demand to supply micro devices has been increasing during the last years. Considering the power output of the laminar flow microfluidic fuel (LFFC), it appears as a suitable solution to provide the required electrical energy in small devices. Absence of electrolyte and not requirement of platinum as catalytic material are two of the most important advantages of this type of fuel cells. The current study aims to provide a detailed information about the design and characteristics of a LFFC. A complete analysis of the different shape channels has been considered in this study being selected the best option as the Y-shaped channels. The impact of the inclination degree for the inlet channels has been considered to evaluate the average velocity that the flow can acquire into the channel. In addition, the voltage-current behaviour considering the materials, fuel/oxidant and design characteristics has been obtained from a modelling point of view.

###### Acknowledgements:

The authors kindly acknowledge the financial support from FIMCP-CERA-05-2017. In addition, Åforsk project No 17-331 is gratefully acknowledged.

#### Gravitation in the theory of compressible oscillating ether

Nikolai Magnitskii1

1Federal Research Center "Computer Science and Control", Laboratory of Chaotic Dynamics, Russian Federation

###### Abstract

Previously, the basic laws and equations of electrodynamics, atomic physics, and elementary particles theory were derived from the theory and equations of compressible oscillating ether [1-3]. In this work, the ethereal theory of gravitation is constructed, the similarities and differences between gravitational and electrostatic fields are explained. It is shown that in gravitation there are no attractive forces, but there are pressing forces, and that the gravitational constant is not really constant, but weakly depends on the chemical composition of interacting bodies. Gravitational interactions between bodies do not propagate from one body to another at a certain speed, and at any moment, the stationary gravitational fields of any bodies exist with and around the bodies, and therefore neither gravitational waves nor gravitons exist in nature. The values of all parameters of the ether, including the density of its unperturbed state, are found. 1. Magnitskii N.A. Theory of compressible oscillating ether. Results in Physics, 12 (2019), p.1436–1445. 2. Magnitskii N.A. Fundamentals of the theory of compressible oscillating ether. IOP Conf. Series: Journal of Physics: Conf. Series 1141 (2018) 012052. 3. Magnitskii N.A. Structure and properties of atomic nuclei in the theory of compressible oscillating ether. IOP Conf. Series: Journal of Physics: Conf. Series 1391 (2019) 012084.

#### Space-time chaos in the nonlinear Schrödinger equation

Nikolai Magnitskii1

1Federal Research Center "Computer Science and Control", Laboratory of Chaotic Dynamics, Russian Federation

###### Abstract

The paper provides an analytical and numerical analysis of the transition to space-time chaos in the generalized nonlinear Schrödinger equation $$i ∂ψ/∂t+c_1 (∂^2 ψ)/(∂y^2 )+c_2 ψ+c_3 |ψ|^2 ψ=0 (1)$$ with complex, in the general case, parameters. Equation (1) describes, in particular, the wave amplitude of a surface plasmon polariton propagating over the contact surface of a metal with a dielectric [1]. It is proved that equation (1) has an infinite number of different stable wave solutions running along the spatial axis with arbitrary velocities, as well as an infinite number of different modes of space-time chaos in full accordance with the universal Feigenbaum-Sharkovsky-Magnitskii bifurcation theory [2-4]. In this case, the bifurcation parameter is the value of the speed of propagation of traveling waves along the spatial axis, which is clearly not included in the original equation. 1. Burov D.A., Evstigneev N. M., Magnitskii N. A. On the chaotic dynamics in two coupled partial differential equations for evolution of surface plasmon polaritons. Comm. Nonlin. Sci. Numer. Simul., ELSEVIER, 2017,v.46, p. 26-36. 2. Magnitskii N.A. Universality of Transition to Chaos in All Kinds of Nonlinear Differential Equations. Chapter in monograph Nonlinearity, Bifurcation and Chaos - Theory and Applications, Chapter 6, INTECH, 2012, p. 133-174. 3. Magnitskii N.A. Bifurcation Theory of Dynamical Chaos. Chapter in monograph Chaos Theory, Chapter 11, INTECH, 2018, p.197-215. 4. Magnitskii N.A. Traveling Waves and Space-Time Chaos in the Kuramoto–Sivashinsky Equation. Differential Equations, 2018, Vol. 54, No. 9, pp. 1266–1270.

#### Explanation of Light Deflection, Precession of Mercury’s Perihelion, Gravitational Red Shift and Rotation Curves in Galaxies, by using General Relativity or equivalent Generalized Scalar Gravitational Potential, according to Special Relativity and Newtonian Physics

Spyridon Vossos1 , Elias Vossos2 , Christos Massouros3

1National and Kapodistrian University of Athens, Core Department, Greece
2National and Kapodistrian University of Athens, Core Department, Greece
3National and Kapodistrian University of Athens, Core Department, Greece

###### Abstract

The development of Geometric theories of gravitation and the application of the Dynamics of General Relativity (GR) is the mainstream approach of gravitational field. Besides, the Generalized Special Relativity (GSR) contains the fundamental parameter (ξI) of Theories of Physics (TPs). Thus, it expresses at the same time Newtonian Physics (NPs) for ξI→0 and Einstein Relativity Theory (ERT) for ξI=1. Moreover, the weak Equivalence Principle (EP) in the context of GSR, has the interpretation: mG=m (1), where mG and m are the gravitational mass and the inertial rest mass, respectively. In this paper, we bridges GR with GSR. This is achieved, by using a GSR-Lagrangian, which contains the proper time of the corresponding GR-Lagrangian. Thus, we obtain a new generalized central scalar GSR-gravitational potential V=V(k,l,r,r_dot,φ_dot), where k=k(ξI), l=l(ξI), r is the distance from the center of gravity and r_dot, φ_dot are the radial and angular velocity, respectively. We demand that ‘this new GSR-gravitational field in accordance with EP (1), gives the same equation of orbit as Schwarzschild Metric (SM) does’ and we obtain k=1 and l=ξI^2. Thus, it emerges that both the NPs and Einsteinian Special Relativity (SR) have the horizon at one Schwarzschild radius (rS). The procedure described in this paper can be applied to any other spacetime metric of GR, in order to find out the corresponding GSR-gravitational potential. We modify the aforementioned central scalar GSR-gravitational potential as V=V(h,k,l,r_dot,φ_dot), where h=h(r). The combination of the above with MOND interpolating functions, or distributions of Dark Matter (DM) in galaxies, provides the functions h=h(r). Thus, we obtain a new central GSR-Gravitational field strength g=g(h,k,l,r_dot,φ_dot), which not only explains the Precession of Mercury’s perihelion, Deflection of Light, Gravitational Red Shift and Rotation Curves in Galaxies, eliminating Dark Matter, but also gives a possible explanation to the Dark Energy problem, by using parameter ξI that depends on cosmic time. .................. COMMENTS: ..................... It is considered that Special Relativity (SR) cannot explain the Gravitational phenomena and only General Relativity (GR) can do this (by using curved spacetime) [1] (pp. 90, 111, 116), [2] (pp. 34, 109), [3] (p. 249). In this paper, we prove that there exist suitable generalized gravitational potentials, according to SR or Newtonian Physics (NPs) that can produce exactly the same results as GR. This is achieved by using the GR-time dilation and the Lagrangian of Generalized Special Relativity (GSR). The procedure is analytically developed in case of Schwarzschild metric, but can also be applied to any kind of metrics. ................ [1] Einstein, A 1920 Relativity: The Special and General Theory; (Holt, New York, USA). Translated by Robert W. Lawson. https://ibiblio.org/ebooks/Einstein/Einstein_Relativity.pdf [2] Rindler W 2006 Relativity: Special, General and Cosmological (New York: Oxford University Press). [ISBN: 978-0-19-856732-5]. [3] Tsamparlis M 2010 Special relativity: An introduction with 200 problems and solutions (Berlin Heidelberg: Springer-Verlag) [ISBN: 978-3-642-03836-5, e-ISBN: 978-3-642-03837-2]

#### Fatigue tests simulation of materials with a random endurance limit

1Perm National Research Polytechnic University , Department of Applied Mathematics, Russian Federation
2Perm National Research Polytechnic University , Department of Applied Mathematics, Russian Federation

###### Abstract

Abstract. Fatigue tests of materials are characterized by long duration and high cost. In this regard, it is relevant to develop methods for modeling test results in the widest possible range of loads. Fatigue curve mathematical model includes the equation of the relationship between the amplitude of the stress and the durability of the samples, considering the random nature of the values of durability and the limit of unbounded endurance. At the first stage, the values of the model parameters are determined using the maximum likelihood function using real test data in a limited range of stress amplitudes. At the second stage, the problem solution is found considering the random value of the limit of unbounded endurance. Moreover, the mean estimate is obtained from the solution of the first part of the problem. The estimate of the unbounded endurance limit variance is obtained by calculation from the variance balance conditions. The results of modeling fatigue tests for aluminum alloy samples in a wide range of stress amplitude values are presented. Simulation results for determining the values of the fatigue curve left tolerance are considered.

#### A variational method for studying the stability of one nonlinear dynamical system

1Lomonosov Moscow State University, Department of Computational Math & Cybernatics, Russian Federation
2Lomonosov Moscow State University, Department of Computational Math & Cybernatics, Russian Federation

###### Abstract

In this work using the variational method the stability of the equilibrium states of a nonlinear dynamic system was studied: $$\frac{dx_{1}}{dt}=x_{1}(\mu_{1}-(x_{1}^{2}+2x_{2}^{2})),$$ $$\frac{dx_{2}}{dt}=x_{2}(\mu_{2}-(2x_{1}^{2}+x_{2}^{2})),$$ where $\mu_{1} ,\; \mu_{2} >0$ - fixed parameters. This system is, of course, nonlinear and more perspective from the point of view of its analysis than the well-known mathematical model the brusselator'' considered in many works. The variational method is particularly effective when the Lyapunov's method does not give the desired result or creates insurmountable difficulties or causes inaccuracies in its application. The proposed method is simple to implement and allows you to serve as an incentive for further research in this direction. proposed nonlinear dynamic system has five stationary points (equilibrium positions): $(0,0)$, $(a,0)$ and $(0,b)$, where $a=\pm \sqrt{\mu_{1}}$, $b=\pm \sqrt{\mu_{2}}$. For the zero (trivial) equilibrium position $(0,0)$ just prove that the perturbed solution is stable only if $\mu_{1}=\mu_{2}=0$. For the equilibrium position $(a,0)$ it is proved that the necessary stability conditions for the perturbed solution are valid if the inequality $0<\mu_{2} <2\mu_{1}$. For the equilibrium position $(0,b)$ it is established that the necessary stability conditions for the perturbed solution are valid if it is satisfied that $0<\mu_{1}<2\mu_{2}$.

#### Numerical study of the effect of stochastic disturbances on the behavior of solutions of some differential equations

Arsenij Firsov 1 , Vladimir Nefedov2 , Igor Inovenkov3 , Vasilij Tikhomirov4

1Lomonosov Moscow State University, Computational math & Cybernatics, Russian Federation
2Lomonosov Moscow State University, Department of Computational Math & Cybernatics, Russian Federation
3Lomonosov Moscow State University, Department of Computational Math & Cybernatics, Russian Federation
4Lomonosov Moscow State University, Department of Computational Math & Cybernatics, Russian Federation

###### Abstract

Nowadays interest of the deterministic differential system of Lorentz equations is still primarily due to the problem of gas and fluid turbulence. Despite a large number of existing systems for calculating turbulent flows, new modifications of already known models are constantly being investigated. In this paper we consider the effect of stochastic additive perturbations on the Lorentz convective turbulence model. To implement this and subsequent interpretation of the results obtained, a numerical simulation of the Lorentz system perturbed by adding a stochastic differential to its right side is carried out using the programming capabilities of the MATLAB programming environment.

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Dimitrios Vlachos1

1University of Peloponnese, Department of Informatics and Telecommunications, Greece

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