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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.

We derive a new model of circadian oscillations in Neurospora crassa, which is suitable to analyze both temporal and spatial dynamics of proteins responsible for mechanism of rythms. The model is based on the non-linear interplay between proteins FRQ and WCC which are products of transcription of frequency and white collar genes forming a feedback loop comprised both positive and negative elements. The main component of oscillations mechanism is supposed to be time-delay in biochemical reactions of transcription. We show that the model accounts for various features observed in Neurospora’s experiments such as entrainment by light cycles, phase shift under light pulse, robustness to action of fluctuations and so on. Wave patterns excited during spatial development of the system are studied. It is shown that the wave of synchronization of biorythms arises under basal transcription factors.

The adequate complexity three-dimensional finite element model of biomechanical system with space, shell and beam-type elements was built. The model includes the Ilizarov fixator and tibial bone’s simulator with the regenerating tissue at the fracture location. The proposed model allows us to specify the orthotropic elastic properties of tibial bone model in cortical and trabecular zones. It is also possible to change the basic geometrical and mechanical characteristics of biomechanical system, change the finite element mash density and define the different external loads, such as pressure on the bone and compression or distraction between the repositioned rings of Ilizarov device.

By using special APDL ANSYS program macros the mode of deformation was calculated in the fracture zone for various static loads on the simulator bone, for compression or distraction between the repositioned rings and for various mechanical properties during different stages of the bone regenerate formation (gelatinous, cartilaginous, trabecular and cortical bone remodeling). The obtained results allow us to estimate the permissible values of the external pressure on the bone and of the displacements of the Ilizarov fixator rings for different stages of the bone regeneration, based on the admittance criterion for the maximum of the stresses in the callus. The presented data can be used in a clinical condition for planning, realization and monitoring of the power modes for transosseous osteosynthesis with the external Ilizarov fixator.

The development of structured molecular systems based on a nucleic acid framework takes into account the ability of single-stranded DNA to form a stable double-stranded structure due to stacking interactions and hydrogen bonds of complementary pairs of nucleotides. To increase the stability of the DNA double helix and to expand the temperature range in the hybridization protocols, it was proposed to use more stable metal-mediated complexes of nucleotide pairs as an alternative to Watson-Crick hydrogen bonds. One of the most frequently considered options is the use of silver ions to stabilize a pair of cytosines from opposite DNA strands. Silver ions specifically bind to N3 cytosines along the helix axis to form, as is believed, a strong N3–Ag⁺–N3 bond, relative to which, two rotational isomers, the cis- and trans-configurations of C–Ag⁺–C can be formed. In present work, a theoretical study and a comparative analysis of the free energy profile of the dissociation of two С–Ag⁺–C isomers were carried out using the combined method of molecular mechanics and quantum chemistry (QM/MM). As a result, it was shown that the cis-configuration is more favorable in energy than the trans- for a single pair of cytosines, and the geometry of the global minimum at free energy profile for both isomers differs from the equilibrium geometries obtained previously by quantum chemistry methods. Apparently, the silver ion stabilization model of the DNA duplex should take into account not only the direct binding of silver ions to cytosines, but also the presence of related factors, such as stacking interaction in extended DNA, interplanar hydrogen bonds, and metallophilic interaction of neighboring silver ions.

Investigation of logical deterministic cellular automata models of population dynamics allows to reveal detailed individual-based mechanisms. The search for such mechanisms is important in connection with ecological problems caused by overexploitation of natural resources, environmental pollution and climate change. Classical models of population dynamics have the phenomenological nature, as they are “black boxes”. Phenomenological models fundamentally complicate research of detailed mechanisms of ecosystem functioning. We have investigated the role of fecundity and duration of resources regeneration in mechanisms of population growth using four models of ecosystem with one species. These models are logical deterministic cellular automata and are based on physical axiomatics of excitable medium with regeneration. We have modeled catastrophic death of population arising from increasing of resources regeneration duration. It has been shown that greater fecundity accelerates population extinction. The investigated mechanisms are important for understanding mechanisms of sustainability of ecosystems and biodiversity conservation. Prospects of the presented modeling approach as a method of transparent multilevel modeling of complex systems are discussed.

The paper proposes a mathematical model for the therapy of glioma, taking into account the blood-brain barrier, radiotherapy and antibody therapy. The parameters were estimated from experimental data and the evaluation of the effect of parameter values on the effectiveness of treatment and the prognosis of the disease were obtained. The possible variants of sequential use of radiotherapy and the effect of antibodies have been explored. The combined use of radiotherapy with intravenous administration of $mab$ $Cx43$ leads to a potentiation of the therapeutic effect in glioma.

Radiotherapy must precede chemotherapy, as radio exposure reduces the barrier function of endothelial cells. Endothelial cells of the brain vessels fit tightly to each other. Between their walls are formed so-called tight contacts, whose role in the provision of BBB is that they prevent the penetration into the brain tissue of various undesirable substances from the bloodstream. Dense contacts between endothelial cells block the intercellular passive transport.

The mathematical model consists of a continuous part and a discrete one. Experimental data on the volume of glioma show the following interesting dynamics: after cessation of radio exposure, tumor growth does not resume immediately, but there is some time interval during which glioma does not grow. Glioma cells are divided into two groups. The first group is living cells that divide as fast as possible. The second group is cells affected by radiation. As a measure of the health of the blood-brain barrier system, the ratios of the number of BBB cells at the current moment to the number of cells at rest, that is, on average healthy state, are chosen.

The continuous part of the model includes a description of the division of both types of glioma cells, the recovery of BBB cells, and the dynamics of the drug. Reducing the number of well-functioning BBB cells facilitates the penetration of the drug to brain cells, that is, enhances the action of the drug. At the same time, the rate of division of glioma cells does not increase, since it is limited not by the deficiency of nutrients available to cells, but by the internal mechanisms of the cell. The discrete part of the mathematical model includes the operator of radio interaction, which is applied to the indicator of BBB and to glial cells.

Within the framework of the mathematical model of treatment of a cancer tumor (glioma), the problem of optimal control with phase constraints is solved. The patient’s condition is described by two variables: the volume of the tumor and the condition of the BBB. The phase constraints delineate a certain area in the space of these indicators, which we call the survival area. Our task is to find such treatment strategies that minimize the time of treatment, maximize the patient’s rest time, and at the same time allow state indicators not to exceed the permitted limits. Since the task of survival is to maximize the patient’s lifespan, it is precisely such treatment strategies that return the indicators to their original position (and we see periodic trajectories on the graphs). Periodic trajectories indicate that the deadly disease is translated into a chronic one.

The paper analyzes the methods of studying the process of interaction of soil environments with the tillage tools of soil processing machines. The mathematical methods of numerical modeling are considered in detail, which make it possible to overcome the disadvantages of analytical and empirical approaches. A classification and overview of the possibilities the continuous (FEM — finite element method, CFD — computational fluid dynamics) and discrete (DEM — discrete element method, SPH — hydrodynamics of smoothed particles) numerical methods is presented. Based on the discrete element method, a mathematical model has been developed that represents the soil in the form of a set of interacting small spherical elements. The working surfaces of the tillage tool are presented in the framework of the finite element approximation in the form of a combination of many elementary triangles. The model calculates the movement of soil elements under the action of contact forces of soil elements with each other and with the working surfaces of the tillage tool (elastic forces, dry and viscous friction forces). This makes it possible to assess the influence of the geometric parameters of the tillage tools, technological parameters of the process and soil parameters on the geometric indicators of soil displacement, indicators of the self-installation of tools, power loads, quality indicators of loosening and spatial distribution of indicators. A total of 22 indicators were investigated (or the distribution of the indicator in space). This makes it possible to reproduce changes in the state of the system of elements of the soil (soil cultivation process) and determine the total mechanical effect of the elements on the moving tillage tools of the implement. A demonstration of the capabilities of the mathematical model is given by the example of a study of soil cultivation with a disk cultivator battery. In the computer experiment, a virtual soil channel of 5×1.4 m in size and a 3D model of a disk cultivator battery were used. The radius of the soil particles was taken to be 18 mm, the speed of the tillage tool was 1 m/s, the total simulation time was 5 s. The processing depth was 10 cm at angles of attack of 10, 15, 20, 25 and 30°. The verification of the reliability of the simulation results was carried out on a laboratory stand for volumetric dynamometry by examining a full-scale sample, made in full accordance with the investigated 3D-model. The control was carried out according to three components of the traction resistance vector: $F_x$, $F_y$ and $F_z$. Comparison of the data obtained experimentally with the simulation data showed that the discrepancy is not more than 22.2%, while in all cases the maximum discrepancy was observed at angles of attack of the disk battery of 30°. Good consistency of data on three key power parameters confirms the reliability of the whole complex of studied indicators.

In the last few years, significant progress has been observed in the field of rotating detonation engines for aircrafts. Scientific laboratories around the world conduct both fundamental researches related, for example, to the issues of effective mixing of fuel and oxidizer with the separate supply, and applied development of existing prototypes. The paper provides a brief overview of the main results of the most significant recent computational work on the study of propagation of a onedimensional pulsating gaseous detonation wave in a non-uniform medium. The general trends observed by the authors of these works are noted. In these works, it is shown that the presence of parameter perturbations in front of the wave front can lead to regularization and to resonant amplification of pulsations behind the detonation wave front. Thus, there is an appealing opportunity from a practical point of view to influence the stability of the detonation wave and control it. The aim of the present work is to create an instrument to study the gas-dynamic mechanisms of these effects.

The mathematical model is based on one-dimensional Euler equations supplemented by a one-stage model of the kinetics of chemical reactions. The defining system of equations is written in the shock-attached frame that leads to the need to add a shock-change equations. A method for integrating this equation is proposed, taking into account the change in the density of the medium in front of the wave front. So, the numerical algorithm for the simulation of detonation wave propagation in a non-uniform medium is proposed.

Using the developed algorithm, a numerical study of the propagation of stable detonation in a medium with variable density as carried out. A mode with a relatively small oscillation amplitude is investigated, in which the fluctuations of the parameters behind the detonation wave front occur with the frequency of fluctuations in the density of the medium. It is shown the relationship of the oscillation period with the passage time of the characteristics C₊ and C₀ over the region, which can be conditionally considered an induction zone. The phase shift between the oscillations of the velocity of the detonation wave and the density of the gas before the wave is estimated as the maximum time of passage of the characteristic C₊ through the induction zone.

Cognitive and regulatory processes in organism are ensured by the functioning of several different network systems — neural, endocrine, immune, and gene ones. These systems are, however, closely related and form a single integrated neurogenohumoral cognitive-regulatory dynamic system of organism. A review of publications is given which shows that it is possible to associate with this dynamic system a corresponding Lyapunov function (energy function, potential function) and that analyzing this function allows, due to its geometrical insight, to easily discover a set of general properties of cognitive and regulatory functioning of organism.