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The development of a viral infection in the organism is a complex process which depends on the competition race between virus replication in the host cells and the immune response. To study different regimes of infection progression, we analyze the general mathematical model of immune response to viral infection. The model consists of two ODEs for virus and immune cells non-dimensionalized concentrations. The proliferation rate of immune cells in the model is represented by a bell-shaped function of the virus concentration. This function increases for small virus concentrations describing the antigen-stimulated clonal expansion of immune cells, and decreases for sufficiently high virus concentrations describing down-regulation of immune cells proliferation by the infection. Depending on the virus virulence, strength of the immune response, and the initial viral load, the model predicts several scenarios: (a) infection can be completely eliminated, (b) it can remain at a low level while the concentration of immune cells is high; (c) immune cells can be essentially exhausted, or (d) completely exhausted, which is accompanied (c, d) by high virus concentration. The analysis of the model shows that virus concentration can oscillate as it gradually converges to its equilibrium value. We show that the considered model can be obtained by the reduction of a more general model with an additional equation for the total viral load provided that this equation is fast. In the case of slow kinetics of the total viral load, this more general model should be used.

This paper provides an overview of studies on nonlinear supratransmission and related phenomena. This effect consists in the transfer of energy at frequencies not supported by the systems under consideration. The supratransmission does not depend on the integrability of the system, it is resistant to damping and various classes of boundary conditions. In addition, a nonlinear discrete medium, under certain general conditions imposed on the structure, can create instability due to external periodic influence. This instability is the generative process underlying the nonlinear supratransmission. This is possible when the system supports nonlinear modes of various nature, in particular, discrete breathers. Then the energy penetrates into the system as soon as the amplitude of the external harmonic excitation exceeds the maximum amplitude of the static breather of the same frequency.

The effect of nonlinear supratransmission is an important property of many discrete structures. A necessary condition for its existence is the discreteness and nonlinearity of the medium. Its manifestation in systems of various nature speaks of its fundamentality and significance. This review considers the main works that touch upon the issue of nonlinear supratransmission in various systems, mainly model ones.

Many teams of authors are studying this effect. First of all, these are models described by discrete equations, including sin-Gordon and the discrete Schr¨odinger equation. At the same time, the effect is not exclusively model and manifests itself in full-scale experiments in electrical circuits, in nonlinear chains of oscillators, as well as in metastable modular metastructures. There is a gradual complication of models, which leads to a deeper understanding of the phenomenon of supratransmission, and the transition to disordered structures and those with elements of chaos structures allows us to talk about a more subtle manifestation of this effect. Numerical asymptotic approaches make it possible to study nonlinear supratransmission in complex nonintegrable systems. The complication of all kinds of oscillators, both physical and electrical, is relevant for various real devices based on such systems, in particular, in the field of nano-objects and energy transport in them through the considered effect. Such systems include molecular and crystalline clusters and nanodevices. In the conclusion of the paper, the main trends in the research of nonlinear supratransmission are given.

The second part presents numerical studies of the parameters of the lower ionosphere at altitudes of 40–90 km when heated by powerful high-frequency radio waves of various frequencies and powers. The problem statement is considered in the first part of the article. The main attention is paid to the interrelation between the energy and kinetic parameters of the disturbed $D$-region of the ionosphere in the processes that determine the absorption and transformation of the radio beam energy flux in space and time. The possibility of a significant difference in the behavior of the parameters of the disturbed region in the daytime and at nighttime, both in magnitude and in space-time distribution, is shown. In the absence of sufficiently reliable values of the rate constants for a number of important kinetic processes, numerical studies were carried out in stages with the gradual addition of individual processes and kinetic blocks corresponding at the same time to a certain physical content. It is shown that the energy thresholds for inelastic collisions of electrons with air molecules are the main ones. This approach made it possible to detect the effect of the emergence of a self-oscillating mode of changing parameters if the main channel for energy losses in inelastic processes is the most energy-intensive process — ionization. This effect may play a role in plasma studies using high-frequency inductive and capacitive discharges. The results of calculations of the ionization and optical parameters of the disturbed $D$-region for daytime conditions are presented. The electron temperature, density, emission coefficients in the visible and infrared ranges of the spectrum are obtained for various values of the power of the radio beam and its frequency in the lower ionosphere. The height-time distribution of the absorbed radiation power is calculated, which is necessary in studies of higher layers of the ionosphere. The influence on the electron temperature and on the general behavior of the parameters of energy losses by electrons on the excitation of vibrational and metastable states of molecules has been studied in detail. It is shown that under nighttime conditions, when the electron concentration begins at altitudes of about 80 km, and the concentration of heavy particles decreases by two orders of magnitude compared to the average $D$-region, large-scale gas-dynamic motion can develop with sufficient radio emission power The algorithm was developed based on the McCormack method and two-dimensional gas-dynamic calculations of the behavior of the parameters of the perturbed region were performed with some simplifications of the kinetics.

Considered the application of physics-informed neural networks using multi layer perceptrons to solve Cauchy initial value problems in which the right-hand sides of the equation are continuous monotonically increasing, decreasing or oscillating functions. With the use of the computational experiments the influence of the construction of the approximate neural network solution, neural network structure, optimization algorithm and software implementation means on the learning process and the accuracy of the obtained solution is studied. The analysis of the efficiency of the most frequently used machine learning frameworks in software development with the programming languages Python and C# is carried out. It is shown that the use of C# language allows to reduce the time of neural networks training by 20–40%. The choice of different activation functions affects the learning process and the accuracy of the approximate solution. The most effective functions in the considered problems are sigmoid and hyperbolic tangent. The minimum of the loss function is achieved at the certain number of neurons of the hidden layer of a single-layer neural network for a fixed training time of the neural network model. It’s also mentioned that the complication of the network structure increasing the number of neurons does not improve the training results. At the same time, the size of the grid step between the points of the training sample, providing a minimum of the loss function, is almost the same for the considered Cauchy problems. Training single-layer neural networks, the Adam method and its modifications are the most effective to solve the optimization problems. Additionally, the application of twoand three-layer neural networks is considered. It is shown that in these cases it is reasonable to use the LBFGS algorithm, which, in comparison with the Adam method, in some cases requires much shorter training time achieving the same solution accuracy. The specificity of neural network training for Cauchy problems in which the solution is an oscillating function with monotonically decreasing amplitude is also investigated. For these problems, it is necessary to construct a neural network solution with variable weight coefficient rather than with constant one, which improves the solution in the grid cells located near by the end point of the solution interval.

We have investigated dynamics of synthetic genetic oscillators — repressilators — coupled through autoinducer diffusion. The model of the system with phase-repulsive coupling structure is under consideration. We have examined emergence of periodic regimes, stable inhomogeneous steady states depending on the main systems’ parameters: coupling strength and maximal transcription rate. It has been shown that autoinducer production module added to the isolated repressilator cause the limit cycle to disappear through infinite period bifurcation for sufficiently large transcription rate. We have found hysteresis of limit cycle and stable steady state the size of which is determined by ratio between mRNA and protein lifetimes. Two coupled oscillators system demonstrates stable anti-phase oscillations which can become a chaotic regime through invariant torus emergence or via Feigenbaum scenario.

In this study the calculation of 1/2-wave transducer of longitudinal ultrasonic oscillations and one wave transducer of longitudinal-torsion ultrasonic oscillations were carried out by finite element method with use of Abaqus. Dimensions of the concentrator of longitudinal-torsional oscillations and frequency-amplitude characteristics of the transducers of longitudinal and longitudinal-torsional oscillations were defined by calculation. Application of ultrasonic longitudinal-torsional oscillations at tool during surface-strengthening treatment of details made of titanium will allow to reduce adhesion portion of friction in the contact zone. A comparison of results of finite-element calculation of frequency-amplitude characteristics with experimental ones were made and calculation error does not exceed 2.5 %.

By numerical simulation methods the interaction processes of oscillating soliton (breather) with a 180-degree Neel domain wall in the framework of a (2 + 1)-dimensional supersymmetric O(3) nonlinear sigma model is studied. The purpose of this paper is to investigate nonlinear evolution and stability of a system of interacting localized dynamic and topological solutions. To construct the interaction models, were used a stationary breather and domain wall solutions, where obtained in the framework of the two-dimensional sine-Gordon equation by adding specially selected perturbations to the A3-field vector in the isotopic space of the Bloch sphere. In the absence of an external magnetic field, nonlinear sigma models have formal Lorentz invariance, which allows constructing, in particular, moving solutions and analyses the experimental data of the nonlinear dynamics of an interacting solitons system. In this paper, based on the obtained moving localized solutions, models for incident and head-on collisions of breathers with a domain wall are constructed, where, depending on the dynamic parameters of the system, are observed the collisions and reflections of solitons from each other, a long-range interactions and also the decay of an oscillating soliton into linear perturbation waves. In contrast to the breather solution that has the dynamics of the internal degree of freedom, the energy integral of a topologically stable soliton in the all experiments the preserved with high accuracy. For each type of interaction, the range of values of the velocity of the colliding dynamic and topological solitons is determined as a function of the rotation frequency of the A3-field vector in the isotopic space. Numerical models are constructed on the basis of methods of the theory of finite difference schemes, using the properties of stereographic projection, taking into account the group-theoretical features of constructions of the O(N) class of nonlinear sigma models of field theory. On the perimeter of the two-dimensional modeling area, specially developed boundary conditions are established that absorb linear perturbation waves radiated by interacting soliton fields. Thus, the simulation of the interaction processes of localized solutions in an infinite two-dimensional phase space is carried out. A software module has been developed that allows to carry out a complex analysis of the evolution of interacting solutions of nonlinear sigma models of field theory, taking into account it’s group properties in a two-dimensional pseudo-Euclidean space. The analysis of isospin dynamics, as well the energy density and energy integral of a system of interacting dynamic and topological solitons is carried out.

The paper reviews possibilities to predict buckling of thin cylindrical shells with non-destructive techniques during operation. It studies shallow shells made of high strength materials. Such structures are known for surface displacements exceeding the thickness of the elements. In the explored shells relaxation oscillations of significant amplitude can be generated even under relatively low internal stresses. The problem of the cylindrical shell oscillation is mechanically and mathematically modeled in a simplified form by conversion into an ordinary differential equation. To create the model, the researches of many authors were used who studied the geometry of the surface formed after buckling (postbuckling behavior). The nonlinear ordinary differential equation for the oscillating shell matches the well-known Duffing equation. It is important that there is a small parameter before the second time derivative in the Duffing equation. The latter circumstance enables making a detailed analysis of the obtained equation and describing the physical phenomena — relaxation oscillations — that are unique to thin high-strength shells.

It is shown that harmonic oscillations of the shell around the equilibrium position and stable relaxation oscillations are defined by the bifurcation point of the solutions to the Duffing equation. This is the first point in the Feigenbaum sequence to convert the stable periodic motions into dynamic chaos. The amplitude and the period of relaxation oscillations are calculated based on the physical properties and the level of internal stresses within the shell. Two cases of loading are reviewed: compression along generating elements and external pressure.

It is highlighted that if external forces vary in time according to the harmonic law, the periodic oscillation of the shell (nonlinear resonance) is a combination of slow and stick-slip movements. Since the amplitude and the frequency of the oscillations are known, this fact enables proposing an experimental facility for prediction of the shell buckling with non-destructive techniques. The following requirement is set as a safety factor: maximum load combinations must not cause displacements exceeding specified limits. Based on the results of the experimental measurements a formula is obtained to estimate safety against buckling (safety factor) of the structure.

The repressilator is the first genetic regulatory network in synthetic biology, which was artificially constructed in 2000. It is a closed network of three genetic elements $lacI$, $\lambda cI$ and $tetR$, which have a natural origin, but are not found in nature in such a combination. The promoter of each of the three genes controls the next cistron via the negative feedback, suppressing the expression of the neighboring gene. In our previous paper [Bratsun et al., 2018], we proposed a mathematical model of a delayed repressillator and studied its properties within the framework of a deterministic description. We assume that delay can be both natural, i.e. arises during the transcription / translation of genes due to the multistage nature of these processes, and artificial, i.e. specially to be introduced into the work of the regulatory network using gene engineering technologies. In this work, we apply the stochastic description of dynamic processes in a delayed repressilator, which is an important addition to deterministic analysis due to the small number of molecules involved in gene regulation. The stochastic study is carried out numerically using the Gillespie algorithm, which is modified for time delay systems. We present the description of the algorithm, its software implementation, and the results of benchmark simulations for a onegene delayed autorepressor. When studying the behavior of a repressilator, we show that a stochastic description in a number of cases gives new information about the behavior of a system, which does not reduce to deterministic dynamics even when averaged over a large number of realizations. We show that in the subcritical range of parameters, where deterministic analysis predicts the absolute stability of the system, quasi-regular oscillations may be excited due to the nonlinear interaction of noise and delay. Earlier, we have discovered within the framework of the deterministic description, that there exists a long-lived transient regime, which is represented in the phase space by a slow manifold. This mode reflects the process of long-term synchronization of protein pulsations in the work of the repressilator genes. In this work, we show that the transition to the cooperative mode of gene operation occurs a two order of magnitude faster, when the effect of the intrinsic noise is taken into account. We have obtained the probability distribution of moment when the phase trajectory leaves the slow manifold and have determined the most probable time for such a transition. The influence of the intrinsic noise of chemical reactions on the dynamic properties of the repressilator is discussed.

The paper presents an analysis of the nonlinear equation of spin wave transport by the method of characteristics. The conclusion of a new mathematical model of spin wave propagation is presented for the solution of which the characteristic is applied. The behavior analysis of the behavior of the real and imaginary parts of the wave and its amplitude is performed. The phase portraits demonstrate the dependence of the desired function on the nonlinearity coefficient. It is established that the real and imaginary parts of the wave oscillate by studying the nature of the evolution of the initial wave profile by the phase plane method. The transition of trajectories from an unstable focus to a limiting cycle, which corresponds to the oscillation of the real and imaginary parts, is shown. For the amplitude of the wave, such a transition is characterized by its amplification or attenuation (depending on the nonlinearity coefficient and the chosen initial conditions) up to a certain threshold value. It is shown that the time of the transition process from amplification (attenuation) to stabilization of the amplitude also depends on the nonlinearity parameter. It was found out that at the interval of amplification of the amplitude of the spin wave, the time of the transition process decreases, and lower amplitude values correspond to higher parameters of nonlinearity.