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The equilibrium configurations of charged electrons, confined in the hard disk potential, are analysed by means of the hybrid numerical algorithm. The algorithm is based on the interpolation formulas, that are obtained from the analysis of the equilibrium configurations, provided by the variational principle developed in the circular model. The solution of the nonlinear equations of the circular model yields the formation of the shell structure which is composed of the series of rings. Each ring contains a certain number of particles, which decreases as one moves from the boundary ring to the central one. The number of rings depends on the total number of electrons. The interpolation formulas provide the initial configurations for the molecular dynamics calculations. This approach makes it possible to significantly increase the speed at which an equilibrium configuration is reached for an arbitrarily chosen number of particles compared to the Metropolis annealing simulation algorithm and other algorithms based on global optimization methods.

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.

This article discusses the mechanical properties of carbon nanotube (CNT)-reinforced platinum under uniaxial tensile loading using the molecular dynamics method. A review of current computational and experimental studies on the use of carbon nanotube-reinforced composites from a structural point of view. However, quantitative and qualitative studies of CNTs to improve the properties of composites are still rare. Composite selection is a promising application for platinum alloys in many cases where they may be subjected to mechanical stress, including in biocompatibility sources. Pt-reinforced with CNTs may have additional possibilities for implantation of the implant and at the same time obtain the required mechanical characteristics.

The structure of the composite is composed of a Pt crystal with a face-centered cubic lattice with a constant of 3.92 Å and a carbon nanotube. The Pt matrix has the shape of a cube with dimensions of $43.1541 Å \times 43.1541 Å \times 43.1541 Å$. The hole size in the average platinum dimension is the radius of the carbon nanotube of the «zigzag» type (8,0), which is 2.6 Å. A carbon nanotube is placed in a hole with a radius of 4.2 Å. At such parameters, the maximum energy level was mutually observed. The model under consideration is contained in 320 atomic bombs and 5181 atomic platinum. The volume fraction of deaths in the Pt-C composite is 5.8%. At the first stage of the study, the strain rate was analyzed for stress-strain and energy change during uniaxial action on the Pt-C composite.

Analysis of the strain rate study showed that the consumption yield strength increases with high strain rate, and the elasticity has increased density with decreasing strain rate. This work also increased by 40% for Pt-C, the elasticity of the composite decreased by 42.3%. In general, fracture processes are considered in detail, including plastic deformation on an atomistic scale.

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.

Bistability is a fundamental property of nonlinear systems and is found in many applied and theoretical studies of biological systems (populations and communities). In the simplest case it is expressed in the coexistence of diametrically opposed alternative stable equilibrium states of the system, and which of them will be achieved depends on the initial conditions. Bistability in simple models can lead to quad-stability as models become more complex, for example, when adding genetic, age and spatial structure. This occurs in different models from completely different subject area and leads to very interesting, often counterintuitive conclusions. In this article, we review such situations. The paper deals with bifurcations leading to bi- and quad-stability in mathematical models of the following biological objects. The first one is the system of two populations coupled by migration and under the action of natural selection, in which all genetic diversity is associated with a single diallelic locus with a significant difference in fitness for homo- and heterozygotes. The second is the system of two limited populations described by the Bazykin model or the Ricker model and coupled by migration. The third is a population with two age stages and density-dependent regulation of birth rate which is determined either only by population density, or additionally depends on the genetic structure of adjacent generations. We found that all these models have similar scenarios for the birth of equilibrium states that correspond to the formation of spatiotemporal inhomogeneity or to the differentiation by phenotypes of individuals from different age stages. Such inhomogeneity is a consequence of local bistability and appears as a result of a combination of pitchfork bifurcation (period doubling) and saddle-node bifurcation.

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.

In this paper, a mathematical model of cellular tissue dynamics is considered. The first part gives the conclusion of the model, the main provisions and the formulation of the problem. In the second part, the final system is investigated numerically and the simulation results are presented. It is postulated that cellular tissue is a three-phase medium that consists of a solid skeleton (which is an extracellular matrix), cells and extracellular fluid. In addition, the presence of nutrients in the tissue is taken into account. The model is based on the equations of conservation of mass, taking into account mass exchange, the equations of conservation of momentum for each phase, as well as the diffusion equation for nutrients. The equation describing the cellular phase also takes into account the term describing the chemical effect on the tissue, which is called chemotaxis — the movement of cells caused by a gradient in the concentration of chemicals. The initial system of equations is reduced to a system of three equations for finding porosity, cell saturation and nutrient concentration. These equations are supplemented by initial and boundary conditions. In the one-dimensional case, the distribution of porosity, concentration of the cell phase and nutrients is set at the initial moment of time. A constant concentration of nutrients is set on the left border, which corresponds, for example, to the supply of oxygen from the vessel, as well as the flow of cell concentration on it is zero. Two types of conditions are considered at the right boundary: the first is the condition of impermeability of the right boundary, the second is the condition of constant concentration of the cell phase and zero flow of nutrient concentration. In both cases, the conditions for the matrix and extracellular fluid are the same, it is assumed that there is a source of nutrients (blood vessel) on the left border of the modeling area. As a result of modeling, it was revealed that chemotaxis has a significant effect on tissue growth. In the absence of chemotaxis, the compaction zone extends to the entire modeling area, but with an increase in the effect of chemotaxis on the tissue, a degradation area is formed in which the concentration of cells becomes lower than the initial one.

We examine a phenomenological algorithm for generating morphology of astrocytes, a major class of glial brain cells, based on morphometric data of rat brain protoplasmic astrocytes and observations of general cell development trends in vivo, based on current literature. We adapted the Space Colonization Algorithm (SCA) for procedural generation of astrocytic morphology from scratch. Attractor points used in generation were spatially distributed in the model volume according to the synapse distribution density in the rat hippocampus tissue during the first week of postnatal brain development. We analyzed and compared astrocytic morphology reconstructions at different brain development stages using morphometry estimation techniques such as Sholl analysis, number of bifurcations, number of terminals, total tree length, and maximum branching order. Using morphometric data from protoplasmic astrocytes of rats at different ages, we selected the necessary generation parameters to obtain the most realistic three-dimensional cell morphology models. We demonstrate that our proposed algorithm allows not only to obtain individual cell geometry but also recreate the phenomenon of tiling domain organization in the cell populations. In our algorithm tiling emerges due to the cell competition for territory and the assignment of unique attractor points to their processes, which then become unavailable to other cells and their processes. We further extend the original algorithm by splitting morphology generation in two phases, thereby simulating astrocyte tree structure development during the first and third-fourth weeks of rat postnatal brain development: rapid space exploration at the first stage and extensive branching at the second stage. To this end, we introduce two attractor types to separate two different growth strategies in time. We hypothesize that the extended algorithm with dynamic attractor generation can explain the formation process of fine astrocyte cell structures and maturation of astrocytic arborizations.