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Mathematical models of gas dynamics and its computational industry, in our opinion, are far from perfect. We will look at this problem from the point of view of a clear probabilistic micro-model of a gas from hard spheres, relying on both the theory of random processes and the classical kinetic theory in terms of densities of distribution functions in phase space, namely, we will first construct a system of nonlinear stochastic differential equations (SDE), and then a generalized random and nonrandom integro-differential Boltzmann equation taking into account correlations and fluctuations. The key feature of the initial model is the random nature of the intensity of the jump measure and its dependence on the process itself.

Briefly recall the transition to increasingly coarse meso-macro approximations in accordance with a decrease in the dimensionalization parameter, the Knudsen number. We obtain stochastic and non-random equations, first in phase space (meso-model in terms of the Wiener — measure SDE and the Kolmogorov – Fokker – Planck equations), and then — in coordinate space (macro-equations that differ from the Navier – Stokes system of equations and quasi-gas dynamics systems). The main difference of this derivation is a more accurate averaging by velocity due to the analytical solution of stochastic differential equations with respect to the Wiener measure, in the form of which an intermediate meso-model in phase space is presented. This approach differs significantly from the traditional one, which uses not the random process itself, but its distribution function. The emphasis is placed on the transparency of assumptions during the transition from one level of detail to another, and not on numerical experiments, which contain additional approximation errors.

The theoretical power of the microscopic representation of macroscopic phenomena is also important as an ideological support for particle methods alternative to difference and finite element methods.

We propose a three-component discrete-time model of the phytoplankton-zooplankton community, in which toxic and non-toxic species of phytoplankton compete for resources. The use of the Holling functional response of type II allows us to describe an interaction between zooplankton and phytoplankton. With the Ricker competition model, we describe the restriction of phytoplankton biomass growth by the availability of external resources (mineral nutrition, oxygen, light, etc.). Many phytoplankton species, including diatom algae, are known not to release toxins if they are not damaged. Zooplankton pressure on phytoplankton decreases in the presence of toxic substances. For example, Copepods are selective in their food choices and avoid consuming toxin-producing phytoplankton. Therefore, in our model, zooplankton (predator) consumes only non-toxic phytoplankton species being prey, and toxic species phytoplankton only competes with non-toxic for resources.

We study analytically and numerically the proposed model. Dynamic mode maps allow us to investigate stability domains of fixed points, bifurcations, and the evolution of the community. Stability loss of fixed points is shown to occur only through a cascade of period-doubling bifurcations. The Neimark – Sacker scenario leading to the appearance of quasiperiodic oscillations is found to realize as well. Changes in intrapopulation parameters of phytoplankton or zooplankton can lead to abrupt transitions from regular to quasi-periodic dynamics (according to the Neimark – Sacker scenario) and further to cycles with a short period or even stationary dynamics. In the multistability areas, an initial condition variation with the unchanged values of all model parameters can shift the current dynamic mode or/and community composition.

The proposed discrete-time model of community is quite simple and reveals dynamics of interacting species that coincide with features of experimental dynamics. In particular, the system shows behavior like in prey-predator models without evolution: the predator fluctuations lag behind those of prey by about a quarter of the period. Considering the phytoplankton genetic heterogeneity, in the simplest case of two genetically different forms: toxic and non-toxic ones, allows the model to demonstrate both long-period antiphase oscillations of predator and prey and cryptic cycles. During the cryptic cycle, the prey density remains almost constant with fluctuating predators, which corresponds to the influence of rapid evolution masking the trophic interaction.

The paper formulates a mathematical model for hydroelastic oscillations of a plate resting on a nonlinear hardening elastic foundation and interacting with a pulsating fluid layer. The main feature of the proposed model, unlike the wellknown ones, is the joint consideration of the elastic properties of the plate, the nonlinearity of elastic foundation, as well as the dissipative properties of the fluid and the inertia of its motion. The model is represented by a system of equations for a twodimensional hydroelasticity problem including dynamics equation of Kirchhoff’s plate resting on the elastic foundation with hardening cubic nonlinearity, Navier – Stokes equations, and continuity equation. This system is supplemented by boundary conditions for plate deflections and fluid pressure at plate ends, as well as for fluid velocities at the bounding walls. The model was investigated by perturbation method with subsequent use of iteration method for the equations of thin layer of viscous fluid. As a result, the fluid pressure distribution at the plate surface was obtained and the transition to an integrodifferential equation describing bending hydroelastic oscillations of the plate is performed. This equation is solved by the Bubnov –Galerkin method using the harmonic balance method to determine the primary hydroelastic response of the plate and phase response due to the given harmonic law of fluid pressure pulsation at plate ends. It is shown that the original problem can be reduced to the study of the generalized Duffing equation, in which the coefficients at inertial, dissipative and stiffness terms are determined by the physical and mechanical parameters of the original system. The primary hydroelastic response and phases response for the plate are found. The numerical study of these responses is performed for the cases of considering the inertia of fluid motion and the creeping fluid motion for the nonlinear and linearly elastic foundation of the plate. The results of the calculations showed the need to jointly consider the viscosity and inertia of the fluid motion together with the elastic properties of the plate and its foundation, both for nonlinear and linear vibrations of the plate.

We present a novel model for the dynamical trap of the stimulus – response type that mimics human control over dynamic systems when the bounded capacity of human cognition is a crucial factor. Our focus lies on scenarios where the subject modulates a control variable in response to a certain stimulus. In this context, the bounded capacity of human cognition manifests in the uncertainty of stimulus perception and the subsequent actions of the subject. The model suggests that when the stimulus intensity falls below the (blurred) threshold of stimulus perception, the subject suspends the control and maintains the control variable near zero with accuracy determined by the control uncertainty. As the stimulus intensity grows above the perception uncertainty and becomes accessible to human cognition, the subject activates control. Consequently, the system dynamics can be conceptualized as an alternating sequence of passive and active modes of control with probabilistic transitions between them. Moreover, these transitions are expected to display hysteresis due to decision-making inertia.

Generally, the passive and active modes of human control are governed by different mechanisms, posing challenges in developing efficient algorithms for their description and numerical simulation. The proposed model overcomes this problem by introducing the dynamical trap of the stimulus-response type, which has a complex structure. The dynamical trap region includes two subregions: the stagnation region and the hysteresis region. The model is based on the formalism of stochastic differential equations, capturing both probabilistic transitions between control suspension and activation as well as the internal dynamics of these modes within a unified framework. It reproduces the expected properties in control suspension and activation, probabilistic transitions between them, and hysteresis near the perception threshold. Additionally, in a limiting case, the model demonstrates the capability of mimicking a similar subject’s behavior when (1) the active mode represents an open-loop implementation of locally planned actions and (2) the control activation occurs only when the stimulus intensity grows substantially and the risk of the subject losing the control over the system dynamics becomes essential.

The study of the structural characteristics of composites and nanostructures is of fundamental importance in materials science. Theoretical and numerical modeling and simulation of the mechanical properties of nanostructures is the main tool that allows for complex studies that are difficult to conduct only experimentally. One example of nanostructures considered in this work are carbon nanotubes (CNTs), which have good thermal and electrical properties, as well as low density and high Young’s modulus, making them the most suitable reinforcement element for composites, for potential applications in aerospace, automotive, metallurgical and biomedical industries. In this review, we reviewed the modeling methods, mechanical properties, and applications of CNT-reinforced metal matrix composites. Some modeling methods applicable in the study of composites with polymer and metal matrices are also considered. Methods such as the gradient descent method, the Monte Carlo method, methods of molecular statics and molecular dynamics are considered. Molecular dynamics simulations have been shown to be excellent for creating various composite material systems and studying the properties of metal matrix composites reinforced with carbon nanomaterials under various conditions. This paper briefly presents the most commonly used potentials that describe the interactions of composite modeling systems. The correct choice of interaction potentials between parts of composites directly affects the description of the phenomenon being studied. The dependence of the mechanical properties of composites on the volume fraction of the diameter, orientation, and number of CNTs is detailed and discussed. It has been shown that the volume fraction of carbon nanotubes has a significant effect on the tensile strength and Young’s modulus. The CNT diameter has a greater impact on the tensile strength than on the elastic modulus. An example of works is also given in which the effect of CNT length on the mechanical properties of composites is studied. In conclusion, we offer perspectives on the direction of development of molecular dynamics modeling in relation to metal matrix composites reinforced with carbon nanomaterials.

This work gives results of numerical simulation of a superconducting magnetic focusing system. While modeling this system, special care was taken to achieve approximation accuracy over the condition u(∞)=0 by using Richardson method. The work presents the results of comparison of the magnetic field calculated distribution with measurements of the field performed on a modified magnet SP-40 of “MARUSYA” physical installation. This work also presents some results of numeric analysis of magnetic systems of “MARUSYA” physical installation with the purpose to study an opportunity of designing magnetic systems with predetermined characteristics of the magnetic field.

Charge transfer in DNA is simulated by a discrete Holstein model «quantum particle + classical site chain + interaction». Thermostat temperature is taken into account as stochastic force, which acts on classical sites (Langevin equation). Thus dynamics of charge migration along the chain is described by ODE system with stochastic right-hand side. To integrate the system numerically, algorithms of order 1 or 2 are usually applied. We developed «mixed» algorithm having 4th order of accuracy for fast «quantum» variables (note that in quantum subsystem the condition «sum of probabilities of charge being on site is time-constant» must be held), and 2nd order for slow classical variables, which are affecting by stochastic force. The algorithm allows us to calculate trajectories on longer time intervals as compared to standard algorithms. Model calculations of polaron disruption in homogeneous chain caused by temperature fluctuations are given as an example.

The difference scheme for numerical solution of a time-dependant system of two Schrödinger equations with the operator of a spin-orbit interaction for a two-component spinor wave function is offered on the basis of a split method for a time-dependant Schrödinger equations. The computer simulation of the external neutrons’ wave functions evolution with different values of the full moment projection upon internuclear axis and probabilities of their transfer are executed for head-on collisions of ¹⁸O and ⁵⁸Ni nuclei.

Isothermal electroconvection in a dielectric liquid arising in a plane-parallel electrode system due to unipolar injection of charges from the cathode is considered. Spatially periodic rolls structures stability is investigated.

The paper describes the results of computer simulation of time-dependent temperature fields arising in polar dielectrics irradiated by focused electron bunches with average electron energy when analyzing with electron microscopy techniques. The mathematical model was based on solving several-dimensional nonstationary heat conduction equation with use of numerical finite element method. The approximation of thermal source was performed taking into account the estimation of initial electron distribution determined by Monte-Carlo simulation of electron trajectories. The simulation program was designed in Matlab. The geometrical modeling and calculation results demonstrated the main features of model sample heating by electron beam were presented at the given experimental parameters as well as source approximation.