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We present the results of a mathematical model for the aquatic communities which include zooplankton, planktivorous fish and predator fish. The aquatic populations are considered to be body mass- and agestructured, while the trophic relations between the populations to be correspondingly status-specific. The model reproduces diverse dynamic regimes as such steady states and oscillations in the population size. Oscillations in the fish population size are shown to be both regular and irregular. We show that the period of the regular oscillations can be up to decades. The irregular oscillations are shown to be both chaotic and non-chaotic. Analyzing the dynamics in the model parameter space has enabled us to conclude that predictability of fish population dynamics can face difficulties both due to dynamical chaos and to the competition between various dynamical regimes caused by variations in the model parameters, specifically in the zooplankton growth rate.

The mathematical model of a three-layered Co/Cu/Co nanopillar for MRAM cell with one fixed and one free layer was investigated in the approximation of uniformly distributed magnetization. The anisotropy axis is perpendicular to the layers (so-called perpendicular anisotropy). Initially the magnetization of the free layer is oriented along the anisotropy axis in the position accepted to be “zero”. Simultaneous magnetic field and spinpolarized current engaging can reorient the magnetization to another position which in this context can be accepted as “one”. The mathematical description of the effect is based on the classical vector Landau–Lifshits equation with the dissipative term in the Gilbert form. In our model we took into account the interactions of the magnetization with an external magnetic field and such effective magnetic fields as an anisotropy and demagnetization ones. The influence of the spin-polarized injection current is taken into account in the form of Sloczewski–Berger term. The model was reduced to the set of three ordinary differential equations with the first integral. It was shown that at any current and field the dynamical system has two main equilibrium states on the axis coincident with anisotropy axis. It was ascertained that in contrast with the longitudinal-anisotropy model, in the model with perpendicular anisotropy there are no other equilibrium states. The stability analysis of the main equilibrium states was performed. The bifurcation diagrams characterizing the magnetization dynamics at different values of the control parameters were built. The classification of the phase portraits on the unit sphere was performed. The features of the dynamics at different values of the parameters were studied and the conditions of the magnetization reorientation were determined. The trajectories of magnetization switching were calculated numerically using the Runge–Kutta method. The parameter values at which limit cycles exist were determined. The threshold values for the switching current were found analytically. The threshold values for the structures with longitudinal and perpendicular anisotropy were compared. It was established that in the structure with the perpendicular anisotropy at zero field the switching current is an order lower than in the structure with the longitudinal one.

Equations of motion in Newtonian and Hamiltonian forms are used for classical molecular dynamics simulation of particle system time evolution. When Newton equations of motion are used for finding of particle coordinates and velocities in $N$-particle system it takes to solve $3N$ ordinary differential equations of second order at every time step. Traditionally numerical schemes of Verlet method are used for solving Newtonian equations of motion of molecular dynamics. A step of integration is necessary to decrease for Verlet numerical schemes steadiness conservation on sufficiently large time intervals. It leads to a significant increase of the volume of calculations. Numerical schemes of Verlet method with Hamiltonian conservation control (the energy of the system) at every time moment are used in the most software packages of molecular dynamics for numerical integration of equations of motion. It can be used two complement each other approaches to decrease of computational time in molecular dynamics calculations. The first of these approaches is based on enhancement and software optimization of existing software packages of molecular dynamics by using of vectorization, parallelization and special processor construction. The second one is based on the elaboration of efficient methods for numerical integration for equations of motion. A procedure for constructing of explicit, implicit and symmetric symplectic numerical schemes with given approximation accuracy in relation to integration step for solving of molecular dynamic equations of motion in Hamiltonian form is proposed in this work. The approach for construction of proposed in this work procedure is based on the following points: Hamiltonian formulation of equations of motion; usage of Taylor expansion of exact solution; usage of generating functions, for geometrical properties of exact solution conservation, in derivation of numerical schemes. Numerical experiments show that obtained in this work symmetric symplectic third-order accuracy scheme conserves basic properties of the exact solution in the approximate solution. It is more stable for approximation step and conserves Hamiltonian of the system with more accuracy at a large integration interval then second order Verlet numerical schemes.

Verification results of supersonic turbulent jets computational characteristics are presented. Numerical simulation of axisymmetric nozzle operating is realized using FlowVision CFD. Open test cases for CFD are used. The test cases include Seiner tests with exit Mach number of 2.0 both fully-expanded and under-expanded $(P/P_0 = 1.47)$. Fully-expanded nozzle investigated with wide range of flow temperature (300…3000 K). The considered studies include simulation downstream from the nozzle exit diameter. Next numerical investigation is presented at an exit Mach number of 2.02 and a free-stream Mach number of 2.2. Geometric model of convergent- divergent nozzle rebuilt from original Putnam experiment. This study is set with nozzle pressure ratio of 8.12 and total temperature of 317 K.

The paper provides a comparison of obtained FlowVision results with experimental data and another current CFD studies. A comparison of the calculated characteristics and experimental data indicates a good agreement. The best coincidence with Seiner's experimental velocity distribution (about 7 % at far field for the first case) obtained using two-equation $k–\varepsilon$ standard turbulence model with Wilcox compressibility correction. Predicted Mach number distribution at $Y/D = 1$ for Putnam nozzle presents accuracy of 3 %.

General guidelines for simulation of supersonic turbulent jets in the FlowVision software are formulated in the given paper. Grid convergence determined the optimal cell rate. In order to calculate the design regime, it is recommended to build a grid, containing not less than 40 cells from the axis of symmetry to the nozzle wall. In order to calculate an off-design regime, it is necessary to resolve the shock waves. For this purpose, not less than 80 cells is required in the radial direction. Investigation of the influence of turbulence model on the flow characteristics has shown that the version of the SST $k–\omega$ turbulence model implemented in the FlowVision software essentially underpredicts the axial velocity. The standard $k–\varepsilon$ model without compressibility correction also underpredicts the axial velocity. These calculations agree well with calculations in other CFD codes using the standard $k–\varepsilon$ model. The in-home $k–\varepsilon$ turbulence model KEFV with compressibility correction a little bit overpredicts the axial velocity. Since, the best results are obtained using the standard $k–\varepsilon$ model combined with the Wilcox compressibility correction, this model is recommended for the problems discussed.

The developed methodology can be regarded as a basis for numerical investigations of more complex nozzle flows.

One- and two-dimensional hydrodynamic models of turbulent transfer within the atmospheric surface layer under neutral thermal stratification are considered. Both models are based on the solution of system of the timeaveraged equations of Navier – Stokes and continuity using a 1.5-order closure scheme as well as equations for turbulent kinetic energy and the rate of its dissipation. The influence of the upper and lower boundary conditions on vertical profiles of wind speed and turbulence parameters within the atmospheric surface layer was derived using an one-dimensional model usually applied in case of an uniform ground surface. The boundary conditions in the model were prescribed in such way that the vertical wind and turbulence patterns were well agreed with widely used logarithmic vertical profile of wind speed, linear dependence of turbulent exchange coefficient on height above ground surface level and constancy of turbulent kinetic energy within the atmospheric surface layer under neutral atmospheric conditions. On the basis of the classical one-dimensional model it is possible to obtain a number of relationships which link the vertical wind speed gradient, turbulent kinetic energy and the rate of its dissipation. Each of these relationships can be used as a boundary condition in our hydrodynamic model. The boundary conditions for the wind speed and the rate of dissipation of turbulent kinetic energy were selected as parameters to provide the smallest deviations of model calculations from classical distributions of wind and turbulence parameters. The corresponding upper and lower boundary conditions were used to define the initial and boundary value problem in the two-dimensional hydrodynamic model allowing to consider complex topography and horizontal vegetation heterogeneity. The two-dimensional model with selected optimal boundary conditions was used to describe the spatial pattern of turbulent air flow when it interacted with the forest edge. The dynamics of the air flow establishment depending on the distance from the forest edge was analyzed. For all considered initial and boundary value problems the unconditionally stable implicit finite-difference schemes of their numerical solution were developed and implemented.

Reduction of the fire hazard of polymeric materials is one of the important scientific and technical problems. Since complexity of experimental procedures associated with flame spread, establishing reacting flows theoretical basics turned out to be crucial field of modern fundamental science. In order to determine parameters of flame spread over solid combustible materials numerical modelling methods have to be improved. Large amount of physical and chemical processes taking place needed to be resolved not just separately one by one but in connection with each other in gas and solid phases.

Upward flame spread over vertical solid combustible material is followed by unsteady eddy structures of gas flow in the vicinity of flame zone caused by thermal instability and natural convection forces accelerating hot combustion products. At every moment different amount of heat energy is transferred from hot gas-phase flame to solid material because of eddy flow structures. Therefore, satisfactory heat flux and eddy flow modelling are important to estimate flame spread rate.

In the current study we evaluated parameters of numerical method for flame spread over solid combustible material problem taking into account coupled nature of complex interaction between gas phase, solid material and eddy flow resulted from natural convection. We studied aspects of different approximation schemes used in differential equations integration process over space and time, of fields relaxation during iterations procedure carried out inside time step, of different time step values.

Mathematical model formulated allows to simulate flame spread over solid combustible material. Fluid dynamics is modeled by Navier – Stokes system of equations, eddy flow is described by combined turbulent model RANS–LES (DDES), turbulent combustion is resolved by modified turbulent combustion model Eddy Break-Up taking into account kinetic effects, radiation transfer is modeled by spherical harmonics method of first order approximation (P1). The equations presented are solved in OpenFOAM software.

The well-known evolutionary equation of mathematical physics, which in modern mathematical literature is called the Kuramoto – Sivashinsky equation, is considered. In this paper, this equation is studied in the original edition of the authors, where it was proposed, together with the homogeneous Neumann boundary conditions.

The question of the existence and stability of local attractors formed by spatially inhomogeneous solutions of the boundary value problem under study has been studied. This issue has become particularly relevant recently in connection with the simulation of the formation of nanostructures on the surface of semiconductors under the influence of an ion flux or laser radiation. The question of the existence and stability of second-order equilibrium states has been studied in two different ways. In the first of these, the Galerkin method was used. The second approach is based on using strictly grounded methods of the theory of dynamic systems with infinite-dimensional phase space: the method of integral manifolds, the theory of normal forms, asymptotic methods.

In the work, in general, the approach from the well-known work of D.Armbruster, D.Guckenheimer, F.Holmes is repeated, where the approach based on the application of the Galerkin method is used. The results of this analysis are substantially supplemented and developed. Using the capabilities of modern computers has helped significantly complement the analysis of this task. In particular, to find all the solutions in the fourand five-term Galerkin approximations, which for the studied boundary-value problem should be interpreted as equilibrium states of the second kind. An analysis of their stability in the sense of A. M. Lyapunov’s definition is also given.

In this paper, we compare the results obtained using the Galerkin method with the results of a bifurcation analysis of a boundary value problem based on the use of qualitative analysis methods for infinite-dimensional dynamic systems. Comparison of two variants of results showed some limited possibilities of using the Galerkin method.

An algorithm for solving the conjugation of aerodynamic and ballistic problems, which is based on the method of modeling with the help of a grid system, has been complemented by a numerical mechanism that allows to take into account the relative movement and rotation of bodies relative to their centers of mass. For a given configuration of the bodies a problem of flow is solved by relaxation method. After that the state of the system is recalculated after a short amount of time. With the use of iteration it is possible to trace the dynamics of the system over a large period of time. The algorithm is implemented for research of flight of systems of bodies taking into account their relative position and rotation. The algorithm was tested on the problem of flow around a body with segmental-conical form. A good correlation of the results with experimental studies was shown. The algorithm is used to calculate the problem of the supersonic fight of a rotating body. For bodies of rectangular shape, imitating elongated fragments of a meteoroid, it is shown that for elongated bodies the aerodynamically more stable position is flight with a larger area across the direction of flight. This de facto leads to flight of bodies with the greatest possible aerodynamic resistance due to the maximum midship area. The algorithm is used to calculate the flight apart of two identical bodies of a rectangular shape, taking into account their rotation. Rotation leads to the fact that the bodies fly apart not only under the action of the pushing aerodynamic force but also the additional lateral force due to the acquisition of the angle of attack. The velocity of flight apart of two fragments with elongated shape of a meteoric body increases to three times with the account of rotation in comparison with the case, when it is assumed that the bodies do not rotate. The study was carried out in order to evaluate the influence of various factors on the velocity of fragmentation of the meteoric body after destruction in order to construct possible trajectories of fallen on earth meteorites. A developed algorithm for solving the conjugation of aerodynamic and ballistic problems, taking into account the relative movement and rotation of the bodies, can be used to solve technical problems, for example, to study the dynamics of separation of aircraft stages.

This work is dedicated to application of bicompact schemes to numerical solution of evolutionary hyperbolic equations. The main advantage of this class of schemes lies in combination of two beneficial properties: the first one is spatial approximation of high even order on a stencil that always occupies only one mesh cell; the second one is spectral resolution which is better in comparison to classic compact finite-difference schemes of the same order of spatial approximation. One feature of bicompact schemes is considered: their spatial approximation is rigidly tied to Cartesian meshes (with parallelepiped-shaped cells in three-dimensional case). This feature makes rather challenging any application of bicompact schemes to problems with complex computational domains as treated in the framework of unstructured meshes. This problem is proposed to be solved using well-known methods for treating complex-shaped boundaries and their corresponding boundary conditions on Cartesian meshes. The generalization of bicompact schemes on problems in geometrically complex domains is made in case of gas dynamics problems and Euler equations. The free boundary method is chosen as a particular tool to introduce the influence of arbitrary-shaped solid boundaries on gas flows on Cartesian meshes. A brief description of this method is given, its governing equations are written down. Bicompact schemes of fourth order of approximation in space with locally one-dimensional splitting are constructed for equations of the free boundary method. Its compensation flux is discretized with second order of accuracy. Time stepping in the obtained schemes is done with the implicit Euler method and the third order accurate $L$-stable stiffly accurate three-stage singly diagonally implicit Runge–Kutta method. The designed bicompact schemes are tested on three two-dimensional problems: stationary supersonic flows with Mach number three past one circular cylinder and past three circular cylinders; the non-stationary interaction of planar shock wave with a circular cylinder in a channel with planar parallel walls. The obtained results are in a good agreement with other works: influence of solid bodies on gas flows is physically correct, pressure in control points on solid surfaces is calculated with the accuracy appropriate to the chosen mesh resolution and level of numerical dissipation.

The using of determining the measure of interaction between channels when choosing the configuration structure of a control system for complex dynamic objects is considered in the work. The main methods for determining the measure of interaction between subsystems of complex control systems based on the methods RGA (Relative Gain Array), Dynamic RGA, HIIA (Hankel Interaction Index Array), PM (Participation matrix) are presented. When choosing a control configuration, simple configurations are preferable, as they are simple in design, maintenance and more resistant to failures. However, complex configurations provide higher performance control systems. Processes in large dynamic objects are characterized by a high degree of interaction between process variables. For the design of the control structure interaction measures are used, namely, the selection of the control structure and the decision on the configuration of the controller. The choice of control structure is to determine which dynamic connections should be used to design the controller. When a structure is selected, connections can be used to configure the controller. For large systems, it is proposed to pre-group the components of the vectors of input and output signals of the actuators and sensitive elements into sets in which the number of variables decreases significantly in order to select a control structure. A quantitative estimation of the decentralization of the control system based on minimizing the sum of the off-diagonal elements of the PM matrix is given. An example of estimation the measure of interaction between components of strong coupled subsystems and the measure of interaction between components of weak coupled subsystems is given. A quantitative estimation is given of neglecting the interaction of components of weak coupled subsystems. The construction of a weighted graph for visualizing the interaction of the subsystems of a complex system is considered. A method for the formation of the controllability gramian on the vector of output signals that is invariant to state vector transformations is proposed in the paper. An example of the decomposition of the stabilization system of the components of the flying vehicle angular velocity vector is given. The estimation of measures of the mutual influence of processes in the channels of control systems makes it possible to increase the reliability of the systems when accounting for the use of analytical redundancy of information from various devices, which reduces the mass and energy consumption. Methods for assessing measures of the interaction of processes in subsystems of control systems can be used in the design of complex systems, for example, motion control systems, orientation and stabilization systems of vehicles.