## 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, Department 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.