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The mathematical model of the magnetic memory cell MRAM with the in-plane anisotropy axis parallel to the edge of a free ferromagnetic layer (longitudinal anisotropy) has been constructed using approximation of uniform magnetization. The model is based on the Landau–Lifshits–Gilbert equation with the injection-current term in the Sloncžewski–Berger form. The set of ordinary differential equations for magnetization dynamics in a three-layered Co/Cu/Cu valve under the control of external magnetic field and spin-polarized current has been derived in the normal coordinate form. It was shown that the set of equations has two main stationary points on the anisotropy axis at any values of field and current. The stationary analysis of them has been performed. The algebraic equations for determination of additional stationary points have been derived. It has been shown that, depending on the field and current magnitude, the set of equations can have altogether two, four, or six stationary points symmetric in pairs relatively the anisotropy axis. The bifurcation diagrams for all the points have been constructed. The classification of the corresponding phase portraits has been performed. The typical trajectories were calculated numerically using Runge–Kutta method. The regions, where stable and unstable limit cycles exist, have been determined. It was found that the unstable limit cycles exist around the main stable equilibrium point on the axis that coincides with the anisotropy one, whereas the stable cycles surround the unstable additional points of equilibrium. The area of their existence was determined numerically. The new types of dynamics, such as accidental switching and non-complete switching, have been found. The threshold values of switching current and field have been obtained analytically. The estimations of switching times have been performed numerically.

The work belongs to the direction of experimental mathematics, which investigates the properties of mathematical objects by the computing facilities of a computer. The base is an exponential map, its topological properties (Cantor's bouquets) differ from properties of polynomial and rational complex-valued functions. The subject of the study are the character and features of the Fatou and Julia sets, as well as the equilibrium points and orbits of the zero of three iterated complex-valued mappings: $f:z \to (1+ \mu) \exp (iz)$, $g : z \to \big(1+ \mu |z - z^*|\big) \exp (iz)$, $h : z \to \big(1+ \mu (z - z^* )\big) \exp (iz)$, with $z,\mu \in \mathbb{C}$, $z^* : \exp (iz^*) = z^*$. For a quasilinear map g having no analyticity characteristic, two bifurcation transitions were discovered: the creation of a new equilibrium point (for which the critical value of the linear parameter was found and the bifurcation consists of “fork” type and “saddle”-node transition) and the transition to the radical transformation of the Fatou set. A nontrivial character of convergence to a fixed point is revealed, which is associated with the appearance of “valleys” on the graph of convergence rates. For two other maps, the monoperiodicity of regimes is significant, the phenomenon of “period doubling” is noted (in one case along the path $39\to 3$, in the other along the path $17\to 2$), and the coincidence of the period multiplicity and the number of sleeves of the Julia spiral in a neighborhood of a fixed point is found. A rich illustrative material, numerical results of experiments and summary tables reflecting the parametric dependence of maps are given. Some questions are formulated in the paper for further research using traditional mathematics methods.

In this paper, using our bifurcation-geometric approach, we study global dynamics and solve the problem of the maximum number and distribution of limit cycles (self-oscillating regimes corresponding to states of dynamical equilibrium) in a planar polynomial mechanical system of the Euler–Lagrange–Liйnard type. Such systems are also used to model electrical, ecological, biomedical and other systems, which greatly facilitates the study of the corresponding real processes and systems with complex internal dynamics. They are used, in particular, in mechanical systems with damping and stiffness. There are a number of examples of technical systems that are described using quadratic damping in second-order dynamical models. In robotics, for example, quadratic damping appears in direct-coupled control and in nonlinear devices, such as variable impedance (resistance) actuators. Variable impedance actuators are of particular interest to collaborative robotics. To study the character and location of singular points in the phase plane of the Euler–Lagrange–Liйnard polynomial system, we use our method the meaning of which is to obtain the simplest (well-known) system by vanishing some parameters (usually, field rotation parameters) of the original system and then to enter sequentially these parameters studying the dynamics of singular points in the phase plane. To study the singular points of the system, we use the classical Poincarй index theorems, as well as our original geometric approach based on the application of the Erugin twoisocline method which is especially effective in the study of infinite singularities. Using the obtained information on the singular points and applying canonical systems with field rotation parameters, as well as using the geometric properties of the spirals filling the internal and external regions of the limit cycles and applying our geometric approach to qualitative analysis, we study limit cycle bifurcations of the system under consideration.

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.

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.

In this study we discuss methods to solve optimal control problems based on neural network techniques. We study hierarchical dynamical two-level system for surface water quality control. The system consists of a supervisor (government) and a few agents (enterprises). We consider this problem from the point of agents. In this case we solve optimal control problem with constraints. To solve this problem, we use Pontryagin’s maximum principle, with which we obtain optimality conditions. To solve emerging ODEs, we use feedforward neural network. We provide a review of existing techniques to study such problems and a review of neural network’s training methods. To estimate the error of numerical solution, we propose to use defect analysis method, adapted for neural networks. This allows one to get quantitative error estimations of numerical solution. We provide examples of our method’s usage for solving synthetic problem and a surface water quality control model. We compare the results of this examples with known solution (when provided) and the results of shooting method. In all cases the errors, estimated by our method are of the same order as the errors compared with known solution. Moreover, we study surface water quality control problem when no solutions is provided by other methods. This happens because of relatively large time interval and/or the case of several agents. In the latter case we seek Nash equilibrium between agents. Thus, in this study we show the ability of neural networks to solve various problems including optimal control problems and differential games and we show the ability of quantitative estimation of an error. From the numerical results we conclude that the presence of the supervisor is necessary for achieving the sustainable development.

The article presents the results of numerical modeling of the vortex spatial non-stationary motion of the medium arising near the lateral and bottom surfaces of the descent module during its movement in the atmosphere of Mars. The numerical study was performed for the high-speed streamline regime at various angles of attack. Mathematical modeling was carried out on the basis of the Navier – Stokes model and the model of equilibrium chemical reactions for the Martian atmosphere gas. The simulation results showed that under the considered conditions of the descent module motion, a non-stationary flow with a pronounced vortex character is realized near its lateral and bottom surfaces. Numerical calculations indicate that, depending on the angle of attack, the nonstationarity and vortex nature of the flow can manifest itself both on the entire lateral and bottom surfaces of the module, and, partially, on their leeward side. For various angles of attack, pictures of the vortex structure of the flow near the surface of the descent vehicle and in its near wake are presented, as well as pictures of the gas-dynamic parameters fields. The non-stationary nature of the flow is confirmed by the presented time dependences of the gas-dynamic parameters of the flow at various points on the module surface. The carried out parametric calculations made it possible to determine the dependence of the aerodynamic characteristics of the descent module on the angle of attack. Mathematical modeling is carried out on the basis of the conservative numerical method of fluxes, which is a finitevolume method based on a finite-difference writing of the conservation laws of additive characteristics of the medium using «upwind» approximations of stream variables. To simulate the complex vortex structure of the flow over descent module, the nonuniform computational grids are used, including up to 30 million finite volumes with exponential thickening to the surface, which made it possible to reveal small-scale vortex formations. Numerical investigations were carried out on the basis of the developed software package based on parallel algorithms of the used numerical method and implemented on modern multiprocessor computer systems. The results of numerical simulation presented in the article were obtained using up to two thousand computing cores of a multiprocessor complex.

The work investigates the search for inefficient edges in the model of stable dynamics by Nestrov – de Palma (2003). For this purpose, we prove several general theorems about equilibrium properties, including the condition of equal costs for all used routes that can be extended to all paths involving edges from equilibrium routes. The study demonstrates that the standard problem formulation of finding edges whose removal reduces the cost of travel for all participants has no practical significance because the same edge can be both efficient and inefficient depending on the network’s load. In the work, we introduce the concept of an inefficient edge based on the sensitivity of total driver costs to the costs on the edge. The paper provides an algorithm for finding inefficient edges and presents the results of numerical experiments for the transportation network of the city of Anaheim.

The paper is devoted to the analysis of the proximity of the population system to dangerous boundaries. An intersection of these boundaries results in the collapse of the stable coexistence of interacting populations. As a reason of such destruction one can consider random perturbations inevitably presented in any living system. This study is carried out on the example of the well-known model of interaction between predator and prey populations, taking into account both a stabilizing factor of the competition of predators for another than prey resources, and also a destabilizing saturation factor for predators. To describe the saturation of predators, we use the second type Holling trophic function. The dynamics of the system is studied as a function of the predator saturation, and the coefficient of predator competition for resources other than prey. The paper presents a parametric description of the possible dynamic regimes of the deterministic model. Here, local and global bifurcations are studied, and areas of sustainable coexistence of populations in equilibrium and the oscillation modes are described. An interesting feature of this mathematical model, firstly considered by Bazykin, is a global bifurcation of the birth of limit cycle from the separatrix loop. We study the effects of noise on the equilibrium and oscillatory regimes of coexistence of predator and prey populations. It is shown that an increase of the intensity of random disturbances can lead to significant deformations of these regimes right up to their destruction. The aim of this work is to develop a constructive probabilistic criterion for the proximity of the population stochastic system to the dangerous boundaries. The proposed approach is based on the mathematical technique of stochastic sensitivity functions, and the method of confidence domains. In the case of a stable equilibrium, this confidence domain is an ellipse. For the stable cycle, this domain is a confidence band. The size of the confidence domain is proportional to the intensity of the noise and stochastic sensitivity of the initial deterministic attractor. A geometric criterion of the exit of the population system from sustainable coexistence mode is the intersection of the confidence domain and the corresponding separatrix of the unforced deterministic model. An effectiveness of this analytical approach is confirmed by the good agreement of theoretical estimates and results of direct numerical simulations.

The present article sets out the scientific approach of Dmitry Sergeevich Chernavskii to the modelling of economic processes. It recounts the history of works of Dmitry Sergeyevich on the economic front, its milestones and achievements. One of the most important advances in the economic analysis was the prediction by a team of scientists headed by D. S. Chernavskii, the major crises that have occurred in our country over the last 20 years, namely, the default of 1998, the crisis of industrial production in the second half of the 2000s, the 2008 crisis and the ensuing recession. As an example, the dynamic analysis of the global macroeconomic processes shows the model of functioning of the dollar as the world currency. On this particular example shows the possibility of seigniorage due to the issue of the dollar and the calculated “window of opportunity” that allows you to issue dollars as the global currency, without prejudice to its own economy.

A model for the development of a closed society (without external economic relations) in the one-product approach is considered as an example of dynamic analysis of the economy of a separate state. The model is based on the principles of market economy, i.e. the dynamics of prices is determined by the balance of supply and demand. It is shown that in the general case, the state of market equilibrium is not unique. Several steady states with different levels of production and consumption are possible. Effect of addressed emission of money in underproductive state is considered. It is shown that, depending on its size it can lead to the transition to a highly productive condition, and just cause inflation without transition. The relationship of these results with the “Keynesian” and “monetarist” approaches is discussed.