In the article, the possibility of using a bispectrum under the investigation of regular and chaotic behaviour of one-dimensional point mappings is discussed. The effectiveness of the transfer of this concept to nonlinear dynamics was demonstrated by an example of the Feigenbaum mapping. Also in the work, the application of the Kullback-Leibler entropy in the theory of point mappings is considered. It has been shown that this information-like value is able to describe the behaviour of statistical ensembles of one-dimensional mappings. In the framework of this theory some general properties of its behaviour were found out. Constructivity of the Kullback-Leibler entropy in the theory of point mappings was shown by means of its direct calculation for the ”saw tooth” mapping with linear initial probability density. Moreover, for this mapping the denumerable set of initial probability densities hitting into its stationary probability density after a finite number of steps was pointed out.

 

We consider a solution in a moving front form of the initial-boundary value problem for a singularly perturbed reaction-diffusion equation in a band with periodic conditions in one of the variables. Interest in solutions of the front type is associated with combustion problems or nonlinear acoustic waves. In the domain of the function which describes the moving front there is a subdomain where the function has a large gradient. This subdomain is called the internal transition layer. Boundary value problems with internal transition layers have a natural small parameter that is equal to the ratio of the transition layer width to the width of the region under consideration. The presence of a small parameter at the highest spatial derivative makes the problem singularly perturbed. The numerical solution of such problems meets certain difficulties connected with the choice of grids and initial conditions. To solve these problems the use of analytical methods is especially successful. Asymptotic analysis which uses Vasilieva’s algorithm was carried out in the paper. That made it possible to obtain an asymptotic approximation of the solution, which can be used as an initial condition for a numerical algorithm. We also determined the conditions for the existence of a front type solution. In addition, the analytical methods used in the paper make it possible to obtain in an explicit form the front motion equation approximation. This information can be used to develop mathematical models or numerical algorithms for solving boundary value problems for the reaction-diffusion-advection type equations.

 

We consider a moving front solution of a singularly perturbed FitzHugh–Nagumo type system of equations. The solution contains an internal transition layer, that is, a subdomain where a sharp change in the values of the functions describing the solution occurs. In initial-boundary value problems with moving front solutions, there naturally exists a small parameter that is equal to the ratio of the inner transition layer width to the width of the considered region. Taking into account this small parameter leads to the fact that the equations become singularly perturbed, thus the problems are classified as ”hard”, the numerical solution of which meets certain difficulties and does not always give a reliable result. In connection with this, the role of an analytical investigation of the existence of a solution with an internal transition layer increases. For these purposes the use of differential inequalities method is especially effective. The method consists in constructing continuous functions, which are called upper and lower solutions. An important role is played by the so-called ”quasimonotonicity condition” for functions which describe reactive terms. In this paper, we present an algorithm for constructing the upper and the lower solutions of a parabolic system with a single-scale internal transition layer. It should be mentioned that the quasimonotonicity condition in the present paper differs from the analogous condition in previous publications. The above algorithm can be further generalized to more complex systems with two-scale transition layers or to systems with discontinuous reactive terms. The study is of great practical importance for creating mathematically grounded models in biophysics.

 

Oscillations of an elastic beam with longitudinal compression are considered. The beam consists of two steel strips connected on free ends and fixed on opposite ones. Compression is achieved by a strained string. Excitation of oscillations is performed by exposure of alternating magnetic field on a magnet placed on the loose end. The law of motion with a change in the frequency of the harmonic action is registered. As a result of the full-scale experiment a large set of data is obtained. This set contains ordered periodic oscillations as well as disordered oscillations specific to dynamical systems with chaotic behaviour. To study the invariant numerical characteristics of the attractor of the corresponding dynamical system, a correlation integral and a correlation dimensionality as well as β-statentropy are calculated. A large numerical experiment showed that the calculation of β-statentropy is preferable to the calculation of the correlation index. Based on the developed algorithms the dependence of β- statentropy on the frequency of the external action is constructed. The constructed dependence can serve as an effective tool for measuring the adequacy of the mathematical model of forced oscillations of buckling beam driven oscillations.

 

We consider the problem of mathematical modeling of oxidation-reduction oscillatory chemical reactions based on the Belousov reaction mechanism. The process of the main components interaction in such a reaction can be interpreted by a “predator – prey” model phenomenologically similar to it. Thereby, we consider a parabolic boundary value problem consisting of three Volterratype equations, which is a mathematical model of this reaction. We carry out a local study of the neighborhood of the system non-trivial equilibrium state, define a critical parameter, at which the stability is lost in this neighborhood in an oscillatory manner. Using standard replacements, we construct the normal form of the considering system and the form of its coefficients defining the qualitative behaviour of the model and show the graphical representation of these coefficients depending on the main system parameters. On the basis of it, we prove a theorem on the existence of an orbitally asymptotically stable limit cycle, which bifurcates from the equilibrium state, and find its asymptotics. To identificate the limits of found asymptotics applicability, we compare the oscillation amplitudes of one periodic solution component obtained on the basis of asymptotic formulas and by numerical integration of the model system. Along with the main case of Andronov–Hopf bifurcation, we consider various combinations of normal form coefficients obtained by changing the parameters of the studied system, and the corresponding to them solutions behaviour near the equilibrium state. In the second part of the paper, we consider the problem of the diffusion loss of stability of a spatially homogeneous cycle obtained in the first part. We find a critical value of diffusion parameter, at which this cycle of distributed system loses the stability.

 

We consider electro-optic oscillator model which is described by a system of the delay differential equations (DDE). The essential feature of this model is a small parameter in front of a derivative that allows us to draw a conclusion about the action of processes with different order velocities. We analyse the local dynamics of a singularly perturbed system in the vicinity of the zero steady state. The characteristic equation of the linearized problem has an asymptotically large number of roots with close to zero real parts while the parameters are close to critical values. To study the existent bifurcations in the system, we use the method of the behaviour constructing special normalized equations for slow amplitudes which describe of close to zero original problem solutions. The important feature of these equations is the fact that they do not depend on the small parameter. The root structure of characteristic equation and the supercriticality order define the kind of the normal form which can be represented as a partial differential equation (PDE). The role of the ”space” variable is performed by ”fast” time which satisfies periodicity conditions. We note fast response of dynamic features of normalized equations to small parameter fluctuation that is the sign of a possible unlimited process of direct and inverse bifurcations. Also, some obtained equations possess the multistability feature.

 

On the basis of the modified asymptotic method of boundary functions and the asymptotic method of differential inequalities, the question of the existence of Lyapunov-stable stationary solutions with internal layers of the nonlinear heat equation in the case of nonlinear dependence of the power of thermal sources from temperature is investigated. The main conditions of the existence of such solutions are discussed. We construct an asymptotic approximation of an arbitrary-order accuracy to such solutions and suggest an efficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed formal asymptotics, we use an asymptotic method of differential inequalities. The main complexity is related to the description of the transition surface in whose neighborhood the internal layer is localized. We use a more efficient method for localizing the transition surface, which permits one to develop an approach to a more complicated case of balanced nonlinearity. The results can be used to create a numerical algorithm which uses the asymptotic analyses to construct space-non-uniform meshes while describing internal layer behaviour of the solution. As an illustration, we consider a problem on the plane that allows us to visualize the numerical calculations. Numerical and asymptotic solutions of zero order are compared for different values of the small parameter.

 

A periodic boundary value problem is considered for one version of the KuramotoSivashinsky equation, which is widely known in mathematical physics. Local bifurcations in a neighborhood of the spatially homogeneous equilibrium points in the case when they change stability are studied. It is shown that the loss of stability of homogeneous equilibrium points leads to the appearance of a two-dimensional attractor on which all solutions are periodic functions of time, except one spatially inhomogeneous state. A spectrum of frequencies of the given family of periodic solutions fills the entire number line, and they are all unstable in a sense of Lyapunov definition in the metric of the phase space (space of initial conditions) of the corresponding initial boundary value problem. It is chosen the Sobolev space as the phase space. For the periodic solutions which fill the two-dimensional attractor, the asymptotic formulas are given. In order to analyze the bifurcation problem it was used analysis methods for infinite-dimensional dynamical systems: the integral (invariant) manifold method, the Poincare normal form theory, and asymptotic methods. The analysis of bifurcations for periodic boundary value problem was reduced to analysing the structure of the neighborhood of the zero solution of the homogeneous Dirichlet boundary value problem for the considered equation.

 

The dynamics of an association of three coupled oscillators is studied. The link between the oscillators is a broadcast connection, that is, one element unilaterally effects the other two, which in turn interact with each other. An important property of the relation among the oscillators is the presence of a delay that obviously can often be found in applications. The studied system simulates the situation of population dynamics when populations are weakly connected, for example, are divided geographically. In this case one population can affect the other two, which in turn can influence each other but not the first one. Each individual oscillator is represented by the logistic equation with a delay (Hutchinson’s equation). Local asymptotic analysis of this system is done in the case of proximity of oscillator parameters to the values at which the Andronov-Hopf bifurcation occur, also the coupling coefficient in the system are assumed to be small. The method of normal forms is used. The study of the dynamics of the system in some neighborhood of a single equilibrium state is reduced to a system of ordinary differential equations on a stable integral manifold. For the construction of a normal form were found elementary modes obtained by using the symmetry of the problem, and the conditions for their stability. Taking into account the obtained asymptotic formulas, the phase reorganizations occurring in the system are numerically analyzed. It is shown that the delay in the communication circuits of the oscillators significantly affects the qualitative behaviour of the system solutions.

 

The work is aimed to study front solutions of a nonlinear system of parabolic equations in a two-dimensional region. The system can be considered as a mathematical model describing an abrupt change in physical characteristics of spatially heterogeneous media. We consider a system with small parameters raised to the different powers at a differential operator, that represents the difference of typical processes speeds for the system components. The study of the system is conducted by using the contrast structures theory methods, which allowed us to obtain conditions for the existence of front solutions contained in the neighborhood of a closed curve, to determine the front velocity depending on time and coordinate along the front curve, and to obtain the zero-order and the first-order terms of the asymptotic approximation to the solution. The scope of the system includes the description of autowave solutions in the field of ecology, biophysics, combustion physics and chemical kinetics. The approximate solution allows us to choose the model parameters so that the result corresponds to the processes observed, to explain and describe the characteristics of the solutions with sharp gradients, to create models with stable solutions and thereby to simplify the numerical analysis. Note that the numerical experiment for the two-dimensional spatial models requires a considerable amount of processing power and the use of parallel computing techniques and does not allow to effectively analyze and modify the model. In this paper, we obtain the asymptotic approximation that is to be justified, which can be done by the method of differential inequalities.

 

In the paper, we study a singularly perturbed periodic in time problem for the parabolic reaction-advection-diffusion equation with a weak linear advection. The case of the reactive term in the form of a cubic nonlinearity is considered. On the basis of already known results, a more general formulation of the problem is investigated, with weaker sufficient conditions for the existence of a solution with an internal transition layer to be provided than in previous studies. For convenience, the known results are given, which ensure the fulfillment of the existence theorem of the contrast structure. The justification for the existence of a solution with an internal transition layer is based on the use of an asymptotic method of differential inequalities based on the modification of the terms of the constructed asymptotic expansion. Further, sufficient conditions are established to fulfill these requirements, and they have simple and concise formulations in the form of the algebraic equation w(x₀,t) = 0 and the condition w_x(x₀,t) < 0, which is essentially a condition of simplicity of the root x₀(t) and ensuring the stability of the solution found. The function w is a function of the known functions appearing in the reactive and advective terms of the original problem. The equation w(x₀,t) = 0 is a problem for finding the zero approximation x₀(t) to determine the localization region of the inner transition layer. In addition, the asymptotic Lyapunov stability of the found periodic solution is investigated, based on the application of the so-called compressible barrier method. The main result of the paper is formulated as a theorem.

 

The multi-component extension problem of the (2+1)D-gauge topological Jackiw–Pi model describing the nonlinear quantum dynamics of charged particles in multi-layer Hall systems is considered. By applying the dimensional reduction (2 + 1)D → (1 + 1)D to Lagrangians with the Chern–Simons topologic fields , multi-component nonlinear Schrodinger equations for particles are constructed with allowance for their interaction. With Hirota‘s method, an exact two-soliton solution is obtained, which is of interest in quantum information transmission systems due to the stability of their propagation. An asymptotic analysis t →±∞ of soliton-soliton interactions shows that there is no backscattering processes. We identify these solutions with the edge (topological protected) states – chiral solitons – in the multi-layer quantum Hall systems. By applying the Hirota bilinear operator algebra and a current theorem, it is shown that, in contrast to the usual vector solitons, the dynamics of new solutions (chiral vector solitons) has exclusively unidirectional motion. The article is published in the author’s wording.

 

Let  \(n\in{\mathbb N}\) and let \(Q_n\) be the unit cube \([0,1]^n\). For a nondegenerate simplex \(S\subset{\mathbb R}^n\), by \(\sigma S\) denote the homothetic copy of \(S\) with center of homothety in the center of gravity of \(S\) and ratio of homothety \(\sigma.\) Put \(\xi(S)=\min \{\sigma\geq 1: Q_n\subset \sigma S\}.\) We call \(\xi(S)\)  an absorption index of simplex \(S\). In the present paper, we give new estimates for the minimal absorption index of the simplex contained in \(Q_n\), i.\,e., for the number \(\xi_n=\min \{ \xi(S): , S\subset Q_n \}.\) In particular, this value and its analogues have applications in estimates for the norms of interpolation projectors. Previously the first author proved some general estimates of \(\xi_n\). Always \(n\leq\xi_n< n+1\). If there exists an Hadamard matrix of order \(n+1\), then \(\xi_n=n\). The best known general upper estimate has the form \(\xi_n\leq \frac{n^2-3}{n-1}\)  \((n>2)\). There exists a constant \(c>0\) not depending on \(n\) such that, for any simplex \(S\subset Q_n\) of maximum volume, inequalities \(c\xi(S)\leq \xi_n\leq \xi(S)\) take place. It motivates the use of maximum volume simplices in upper estimates of \(\xi_n\). The set of vertices of such a simplex can be consructed with  application of maximum \(0/1\)-determinant of order \(n\) or maximum \(-1/1\)-determinant of order \(n+1\). In the paper, we compute absorption indices of maximum volume simplices in \(Q_n\) constructed from known maximum \(-1/1\)-determinants via a special procedure. For some \(n\), this approach makes it possible to lower theoretical upper bounds  of \(\xi_n\). Also we give best known upper estimates of \(\xi_n\) for \(n\leq 118\)