## Conference submissions

Variable dissipation dynamical systems: integrability and analysis
Maxim V. Shamolin, Lomonosov Moscow State University, Institute of Mechanics, Russian Federation
Abstract: In this activity, we systematize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies-pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint, or the center of mass of the body moves rectilinearly and uniformly; this means that there exists a nonconservative couple of forces in the system. We also review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either the energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in dynamics of a multi-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions, which can be expressed through a finite combination of elementary functions. As applications, we study dynamical equations of motion arising in the study of the plane and spatial dynamics of a rigid body interacting with a medium and also a possible generalization of the obtained methods for the study of general systems arising in the qualitative theory of ordinary differential equations, in the theory of dynamical systems, and also in oscillation theory.
Acknowledgements: Institute of Mechanics, Lomonosov Moscow State University
Magnetic Soliton for Single-Ion Anisotropy in SU(3) Group
Yousef Yousefi, Payame Noor University, Physics, Iran, Islamic Republic Of
Abstract: We discuss system with Single-Ion anisotropy Hamiltonian with nearest neighbor exchange within a mean field approximation process that observed in compositions like CSNiF3. We drive Lagrangian and equations describing this model with Path integral technic by using coherent states in real parameters. for small linear excitation from the ground state, dispersion equations of spin wave of dipole and quadrupole branches obtained. If the Single-Ion anisotropy coefficient is zero, we have only dipole dispersion branch and there is no quadruple dispersion. In other word, quadruple dispersion branch obtained only when there is anisotropy term in Hamiltonian. In final, soliton solution for quadrupole branches for these linear equations calculated. This soliton is the solution of nonlinear Klein-Gordon equation and have the form of Hylomorphic soliton. These solitons are like Q-ball solitons. Also this soliton is of the kind of non-topologic ones because their boundary values in ground and infinity are the same from the topological point of view.
Mathematical modeling of polymer flooding using unstructured Voronoi grid
Bulgakova Guzel, Ufa State Aviation Technical University, Mathematics, Russian Federation
Abstract: Nowadays the part of unconventional oil in the total oil reserves in world is more than 60% and continues to grow. Effective recovery of such oil necessitates development of enhanced oil recovery techniques such as polymer flooding. Polymer flooding simulation software is expensive and most of the products uses only rectangular grid for calculations. The study investigated the model of polymer flooding with effects of adsorption and water salinity. The model takes into account six components that includes elements of the classic black oil model. These components are polymer, salt, water, dead oil, dry gas and dissolved gas. The equations of the model and the problem statement are formulated. Solution of the problem is obtained by finite volume method on unstructured Voronoi grid using fully implicit scheme. The discretized nonlinear equations are solved by the Newton’s method. To compare several different grid configurations numerical simulation of polymer flooding is performed. The oil rates obtained by a hexagonal locally refined Voronoi grid are shown to be more accurate than the oil rates obtained by a rectangular grid with the same number of cells. The latter effect is caused by high solution accuracy near the wells due to the local grid refinement. Minimization of the grid orientation effect caused by the hexagonal pattern is also demonstrated. However, in the inter-well regions with large Voronoi cells flood front tends to flatten and the water breakthrough moment is smoothed.
Acknowledgements: This study was supported by the Russian Foundation for Basic Research (Project 17-41-020226 r_a).
The Kirkendall shift during ternary reactive diffusion process - entropy production principle
Bartek Wierzba, Rzeszow University of Technology, Materials Science, Poland
Abstract: In this paper the phenomenological process related to the evolution of the ternary multiphase systems is discussed. The entropy production principle is proposed to choice the reaction path during such diffusion process by means of numerical simulations. The bi-velocity method of the three-component multi-phase system is shown. The simulations present the local entropy production determines the diffusion path in ternary system. The simulations results will be compared with the experiments of diffusion in Fe-Ni-Ti system.
Acknowledgements: This work has been supported by the National Science Centre (NCN) in Poland, decision number 2014/15/B/ST8/00120.
Effect of surface condition on oxidation kinetics of Ni-base superalloy
Wojciech J. Nowak, Rzeszow University of Technology, Department of Material Science, Poland
Abstract: The materials used at high temperature, like in gas turbines or jet engines, need to fulfill a number of requirements, e.g. high creep strength and oxidation resistance at a wide range of operating temperatures, environments and loading condition as well as a suitable ductility at low temperature. Such a properties are obtained in Ni-base superalloys, due to their microstructure consisting of thermodynamically stable γ-Ni matrix with combination of strengthening γ’-Ni3Al phase. However, when one expose the alloys at high temperature, an oxidation process occurs and the material starts to form an oxide scale. The Ni-Cr-Al based alloys can be classified into the three groups of materials in term of formed oxide scales: NiO-forming, chromia forming and an alumina forming alloys. Formation of protective oxides like Al2O3 or Cr2O3 substantially increase the lifetime of the component exposed at high temperature. To provide a resistance against oxidation a protective coatings such as MCrAlY (where M is mainly Ni or Co) or β-NiAl which are an alumina forming materials are applied. However, coatings production is time consuming, results in additional component costs, and can negatively affect alloy mechanical properties, such as fatigue strength. Therefore, another, cheaper method to force material to form a protective oxide scale is proposed in the present study. Namely, a different surface preparation of a Ni-base superalloys, like grinding, polishing, sand blasting etc. on oxide scale formation during exposure at high temperature in Ar-O2 atmosphere will be presented. The model describing the effect of surface treatment on oxidation kinetics and oxide scale formation will be introduced as well.
Acknowledgements: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 665778
“Hypergeomtric solutions for some nonhomogeneous linear differential equations of fractional order” by Jorge Olivares and Pablo Martin, Universidad de Antofagasta, Chile.
Jorge Olivares Funes, University of Antofagasta, Departament of mathematics, Chile
Abstract: In this paper a study is performed to the solution of the linear nonhomogeneous fractional order differential equation (df^α (x))/(dx^α )=I_0 (x) , where I_0 (x) is the modified Bessel function of order zero, the initial condition is f(0)=0 and 0< α<1. Caputo definition for the fractional derivatives is considered [1,2] . Fractional derivatives have become important in physical and chemical phenomena as viscoeslasticity and viscoplasticity, anomalous diffusion and electric circuits. In particular in this work the values of α=1/2, 1/4 and 3/4. are explicitly considered . In these cases Laplace transfom is applied, and later the inverse Laplace transform leads to the solutions of the differential equation, which become hypergeometric functions. In the case of α=1/2 , this is f(x)=2√(x/π) F( 3/4,5/4;1/2,x^2/4). [1]. Podlubny I., “Fractional differential equations “, ( Academic Press, 1990). [2]. Murio D.A . Computers and Mathematics with Applications 51, 1539-1550 (2006).
A new algorithm to sovle nonlocal nonlinear Schrodinger equation
QI GUO, South China Normal University, School of Information and Photoelectronic Science and Engineering, China
Abstract: The propagation of the optical beam in the nonlocal nonlinear media is modeled by the nonlocal nonlinear Schrodinger equation, and the Hermit-Gaussian- like soliton solution (multi-peak soliton solution, MPSS) exists for the strongly nonlocal case. However, it has some limitations to obtain the numerical solutions of the MPSS by the algorithms used, such as the Newton iteration algorithm and the imaginary-time method. We find a new easy and convenient algorithm based on perturbation method for the Schrodinger equation in quantum mechanics to find the numerical MPSS fastly. We can use the algorithm to obtain the numerical MPSS for any response function and in any degree of nonlocality as long as the soliton solution exists. Moreover, the precision of the solutions can be improved by adding higher order perturbation.
Physical Description of Image Conception
Yehuda Roth, Oranim College, Science, Israel
Abstract: We describe a non-linear quantum approach describing a way of conceiving images. Our primary assumption is that although images mostly appear as mixed photon states, each image can be represented by a single coherent photon state. Thus, integrating non-linear dynamics with quantum mechanics, we show how an image that was originally described in a mixed-state form, can be transformed into the pure state description. Being in the pure state phase is regarded as the image's conceived stage.
Dissipative gravitational bouncer on a vibrating surface
Julio S Espinoza-Ortiz, Universidade Federal de Goiás, Physics, Brazil
Roberto E Lagos, Instituto de Geociências e Ciências Exats, UNESP, Physics, Brazil
Abstract: We study the dynamical behavior of a particle flying under the influence of a gravitational field, with dissipation constant {$\lambda$} (Stokes-like), and colliding successive times against a rigid surface vibrating harmonically with restitution coefficient {$\alpha$}. We define re-scaled dimensionless dynamical variables, such as the relative particle velocity {$\Omega$} with respect to the surface's velocity; and the real parameter {$\tau$} accounting for the temporal evolution of the system. At the particle-surface contact point and for the {$k'th$} collision, we construct the mapping described by {$\left(\tau_{k}\,,\Omega_{k}\right)$} in order to analyze the system's nonlinear dynamical behavior. From the dynamical mapping, the fixed point trajectory is computed and its stability is analyzed. We find the dynamical behavior of the fixed point trajectory to be stable or unstable, depending on the values of the re-scaled vibrating surface amplitude {$\Gamma$}, the restitution coefficient {$\alpha$} and the damping constant {$\lambda$}. Other important dynamical aspects such as the phase space volume and the one cycle vibrating surface (decomposed into absorbing and transmitting regions) are also discussed. Furthermore, the model rescues well known results in the limit {$\lambda\,=\,0$}\,.
Acknowledgements: The authors would like to thank the support of the Goi\'as Research Foundation - FAPEG.
Spiraling elliptic beams in nonlocal nonlinear media with linear anisotropy
Guo Liang, Shangqiu Normal University, School of Electrical & Electronic Engineering, China
Zhanmei Ren, South China Normal University, Laboratory of Nanophotonic Functional Materials and Devices, China
QI GUO, South China Normal University, School of Information and Photoelectronic Science and Engineering, China
Abstract: Analytically discussed is the nonlinear propagations of spiraling elliptic beams in nonlocal nonlinear media with linear anisotropy by using an approach of two dimensional asynchronous fractional Fourier transform. The spiraling elliptic beams exhibit a kind of molecule-like libration due to the combined effects of the linear anisotropy and the orbital angular momentum. Depending on the anisotropy parameter of the media, the molecule-like libration mode has two different kinds of dynamical behaviors. When the anisotropy parameter of the media is a rational number, the spiraling elliptic beams evolve as the breathers and can recover their initial shapes after one propagation period. However, when the anisotropy parameter of the media is an irrational number, all the related parameters of spiraling elliptic beams such as the optical intensity, the beam width, and the angular velocity cannot evolve in a periodic manner, and no spiraling elliptic breathers exist in this case.
A note on controllability property for a special class of matrices (Jorge Olivares and Ronald Manríquez)
Jorge Olivares Funes, University of Antofagasta, Departament of mathematics, Chile
Abstract: As is well know, many authors have discussed the controllability property for bilinear control systems when the state variable lies on the plane, see for instance [1], [6], [7]. However, when the dynamic is described on high dimension there are many unsolved problems, and in that sense the search conditions necessary and/or sufficient to characterize the property of controllability have attracted great attention during the last decades, [2], [3], [9], [12], [15]. In this paper, we have establish a characterization of controllability for the class of bilinear systems whose dynamics is determined by a special class of matrices that belong to the semisimple Lie algebra sl(3,R). References 1. Ayala V, and San Martin, L.A.B. Controllability of two{dimensional bilinear system: restricted controls, discrete{time, Proyecciones, 18, 207-223 (1994). 2. C. Bruni, G. Di Pillo, and G. Koch, Bilinear systems: An appealing class of "nearly linear" systems in theory and applications. IEEE Trans. Autom. Control, AC f19, 334-348 (1974). 3. I. Joo and N.M. Tuan, On controllability of some bilinear systems. Comptes Rendus de I'Academie des Sci,. 315, 1393-1398 (1992). 4. J.P. Gauthier and G. Bornard, Controlabilite des Systemes bilineaires, Siam J. Control and Opt., 20(3), 377{384 (1982). 5. L.A.B San Martin, _Algebras de Lie, Unicamp, Campinas, (1999). 6. Rodriguez J. C., V. Ayala, Optimal trajectories for angular systems on the projective line, Optimal Control Applications & Methods, 33, 199{213 (2012). 7. Rodriguez J. C., V. Ayala, L.A.B. San Martin, Optimality on Homogeneous Spaces, and the Angle System Associated with a Bilinear Control System, SIAM Journal on Control and Optimization, 48(4), 2636{2650 (2009). 8. Rodriguez J. C., Control systems, UNICAMP-UCN, (2010). 9. V. Jurdjevic, I. Kupka, Controllability of right invariant systems on semi-simple Lie groups and their homogeneous spaces, Ann. Inst. Fourier, Grenoble, 31(4), 151-179 (1981). 10. V. Jurdjevic, I. Kupka, Accessibility on semi-simple Lie groups and their homogeneous spaces, a paraitre dans "Annales de l'Institut Fourrier " Nov. 1977. 11. V. Jurdjevic, Geometric control theory. Cambridge University Press, (1997.) 12. W. M. Boothby and E. N. Wilson, Determination of the transitivity of bilinear systems, SIAM J. Control Optim., 17, 212-221 (1979). 13. F. Robert, Matrices non_egatives et normes vectorielles. Cours de 3e cycle, INPG ENSIMAG, (1973). 14. Y. L. Sachkov, Control Theory on Lie groups, SISSA, 15/2006/M. 15. Yu. L. Sachkov, Invariant domains of three-dimensional bilinear systems. (Russian) Vest. Mosk. Univ. Ser. Matem., Mekh. 4, 23-26.(1991) English translation: Mosc. Univ. Math.Bulletin.
Numerical approximation of the modified zero-order Bessel differential equation by means of Lagrange Interpolation and Maple software at various intervals (By Jorge Olivares and Elvis Valero)
Jorge Olivares Funes, University of Antofagasta, Departament of mathematics, Chile
Abstract: In the present paper it is shown how the modified differential equation of Bessel of order zero x^ 2 (d ^ 2 y) / (dx ^ 2) + x dy / dx-x ^ 2 y = 0, y (0) = 1, dy / dx (0) = 0 Is approximated by means of the Lagrange interpolation and with the help of maple software commands in the intervals [0,1], [0,1 / 2] and [0,1 / 4]. Reference [1] G. N. Watson, "A treatise on the theory of Bessel functions ", 2nd. ed., Cambrigde,England: Cambrigde University Press(1966). [2] J. D. Jackson, C_ lasical Electrodynamics",2nd. ed., John Wiley and Sons, Inc.,Ch.3 (1975). [3] P. Martin, E. Castro, J. L. Paz and A.De Freitas, "Multipoint quasi-rational approximants in Quantum Chemistry", Chapter 3 of "New Developments in Quantum Chemistry"(Transworld Research Network,Kerala, India, 2009) pp. 55-78. [4] P. Martin, J. Olivares, L. Cortes Vega and A. Sotomayor,"Multi-point quasirational approximants for the midi_ed Bessel function I1(x)", Journal of Physics: Conference Series 738: 012066 (2016). .
Photon wave function and intrinsic electromagnetic properties
CONSTANTIN MEIS, CEA - Saclay, National Institute for Nuclear Science and Technology, France
Abstract: Constantin Meis(1) and Pierre Richard Dahoo(2) (1) National Institute for Nuclear Science and Technology, CEA - Saclay, Université Paris Saclay, 91191 Gif-sur-Yvette, France. (2) LATMOS /IPSL, UVSQ Université Paris-Saclay, F-78280, Guyancourt, France. The permanent violation of Bell’s inequality by the experimental evidence with single photon states has demonstrated that hidden variables within a local representation are excluded. However, new variables within a non-local representation for the photon through a real wave function are not explicitly excluded. We consider here the vector potential, enhanced at a single k-mode photon state beyond the standard description in QED, with the quantized amplitude proportional to the angular frequency times a quantization constant. Within this representation the vector potential function for a k-mode and lamda-polarization photon satisfies the classical electromagnetic wave propagation equation as well as Schrodinger’s equation for the relativistic massless Hamiltonian and finally a quantum equation for the vector potential amplitude operator. Consequently, taking into account the left (L) and right (R) circularly polarized states for a single photon we define a six components function as a general wave function for a k-mode photon in a non-local representation that can be suitably normalized. It is shown that the established photon wave function satisfies Schrodinger’s equation with the relativistic massless Hamiltonian coupled to Pauli spin 1 matrices. We deduce that the square of the modulus of this wave function gives the energy density at a given coordinate while the probability for detecting a k-mode photon around a point on the propagation axis depends on the fourth power of the angular frequency Furthermore, the amplitudes of the electric and magnetic fields of single k-mode photon are also expressed through the square of the angular frequency. This representation confers precise properties to a single photon state opening perspectives for further experimental investigations in order to understand the real nature of a single photon state.
Understanding dynamics of polar vortices on Mars and Saturn with "improved" rotating shallow water model
Vladimir Zeitlin, Laboratory of Dynamical Meteorology, Earth and Planetary Sciences, France
Abstract: We show how the salient features of the atmospheres of Mars and Saturn, namely the surprising symmetry and stability of Martian polar vortex, and the longevity of the hexagonal vortex at the North pole of Saturn, which is being observed for decades, can be explained with the help of a simple shallow-water type model resulting from vertical averaging of primitive equations of planetary atmospheres, with addition of phase transitions of water, and mosit convection, for Saturn, and addition of phase transitions of CO2 with dust nuclei, for Mars. Thus "improved" multi-phase rotating shallow water model allows for detailed stability analysis of vortex structures, and for efficient finite-volume numerical implementations and high-resolution long-time simulations at low cost. The former are used to identify the unstable modes of polar vortices, and the latter - to follow nonlinear evolution of the instabilities and life-cycles of the resulting structures.
Correction to the Wills-Harrison approach: Influence on the Fe-based liquid alloys thermodynamics
Nikolay Dubinin, Ural Federal University, Engineering centre , Russian Federation
Abstract: Some years ago, we applied the Wills-Harrison (WH) [1] approach in conjunction with the variational method of the thermodynamic perturbation theory to calculate thermodynamic properties of Fe-Co and Fe-Ni liquid alloys [2, 3]. Later, we introduce the correction to the WH model due to the non-diagonal coupling between d electrons on different atoms and applied this correction to investigate the WH effective pair interactions in liquid Fe, Co and Ni [4]. Here, the influence of this correction on the thermodynamics of Fe-Co and Fe-Ni liquid alloys near their melting temperatures at different component concentrations is studied. This work is supported by the federal target project “R&D for Priority Areas of the Russian Science-and-Technology Complex Development for 2014-2020”, government contract № 14.578.21.0200 on the subject “Development of ceramic components and parts production by selective laser melting technology, using innovative diagnostic processes of products and methods” (Application Code «2016–14–579–0009–3076»). 1. J.M. Wills, W.A. Harrison, Phys. Rev. B 28, 4363 (1983). 2. N.E. Dubinin, L.D. Son, N.A. Vatolin, J. Phys.: Condens. Matter 20, 114111 (2008). 3. N.E. Dubinin, J. Phys.: Conf. Ser. 144, 012115 (2009). 4. N.E. Dubinin, J. Phys.: Conf. Ser. 338, 012004 (2012).
Acknowledgements: This work is supported by the federal target project “R&D for Priority Areas of the Russian Science-and-Technology Complex Development for 2014-2020”, government contract № 14.578.21.0200 (Application Code «2016–14–579–0009–3076»).
Diffusion of neutrons in the toroidal nuclear electrogenerator
Vladimir Tertychny-Dauri, Saint-Petersburg National Research University of Information Technologies, Mechanics and Optics, Physics and Engineering, Russian Federation
Abstract: Diffusion of neutrons in the toroidal nuclear electrogenerator (nuclegen) is investigated. Important conclusion about the practically complete absence of the diffusion of neutrons and charged fission particles through the external boundary under toroidal motion is substantiated by the solution of corresponding equation diffusion with the aid of Fourier's standard division method.
Explanation of Rotation Curves in Galaxies and Clusters of Them, by Generalization of Schwarzschild Metric and Combination with MOND, eliminating Dark Matter
Spyridon Vossos, NKUA, Chemistry, Greece
Elias Vossos, NKUA, Physics, Greece
Abstract: Schwarzschild Metric is the first and the most important solution of Einstein vacuum field equations. This is associated with Lorentz metric of flat spacetime and produces the relativistic potential (Φ) and the field strength (g) outside a spherically symmetric mass or a non-rotating black hole. It has many applications such as gravitational red shift, the precession of Mercury’s orbit, Shapiro time delay etc [1]. However, it is inefficient to explain the rotation curves in large galaxies and clusters of them, causing the necessity for dark matter [2-6]. On the other hand, Modified Newtonian Dynamics (MOND) has already explained these rotation curves in many cases, using suitable Interpolating function (μ) in Milgrom’s Law [7-11]. In this presentation, we initially produce a Generalized Schwarzschild potential and the corresponding Metric of spacetime, in order to be in accordance with any isotropic metric of flat spacetime (including Galilean Metric of spacetime which is associated with Galilean Transformation of spacetime). Then, we are limited to the case of flat spacetime with Lorentz metric (Minkowski space), because the experimental data have been extracted using the Relativistic Doppler Shift and the gravitational red shift of Classic Relativity (CR). From this Generalized Schwarzschild potential (Φ), we calculate the corresponding field strength (g), which has two terms. The first is significant, near to Schwarzschild radius, while the second is significant at very long distance from the center of gravity. This second term is associated with the Interpolating function (μ). With that, a new relativistic potential is obtained (let us call 2nd Generalized Schwarzschild potential) which describes the gravitational interaction at any distance. Thus, not only the necessity for Dark Matter is eliminated, but also MOND becomes a specialization at very long distance of a pure Relativistic Gravitational Interaction associated with Lorentz metric. This relativistic potential and the corresponding metric of spacetime have been obtained by the light of Euclidean Closed Linear Transformations of Complex Spacetime endowed with the Corresponding Metric [12,13]. Of course, may also be applied by scientists who prefer the hyperbolic geometry of Classic Relativity (CR).