In this paper, we consider the problem of construction of real autonomous quadratic systems of three differential equations with the nonlocal existence of an infinite number of limit cycles. This means that an infinite number of limit cycles, emerging from the focus due to the Andronov – Hopf bifurcation, can exist in the phase space not only in the vicinity of the focus and not only for parameter values close to the bifurcation value. To solve this problem we use the method of determination of limit cycles as the curves of intersection of an invariant plane with a family of invariant elliptic paraboloids. Then the study of the limit cycles of the constructed system of the third order is reduced to the study of the corresponding system of the second order on each of the invariant elliptic paraboloids. The proof of the nonlocal existence of the limit cycle and the investigation of its stability for such a second-order system is carried out by constructing a topographic system of Poincaré functions or by transforming to polar coordinates.

In this paper, we consider a nonlinear differential equation of the first order of the Riccati hierarchy. The concept of a generalized solution for such an equation cannot be introduced within the framework of the classical theory of generalized functions because the product of generalized functions is not defined. To introduce the concept of a generalized solution, two approaches are considered. In the first approach, approximation by solutions of the Cauchy problem with complex initial conditions is used, and generalized solutions are defined as limits of approximating families in the sense of convergence in D′(¡). It is shown that there are two generalized solutions of the Cauchy problem. The type of solution depends on whether the poles of the approximating solution are located in the upper or lower half-plane. The second approach uses approximation with a system of equations. It is shown that there are many approximating systems, meanwhile, generalized solutions of the Cauchy problem depend on the choice of the approximating system.

In this paper, we study the computational complexity for a problem of partitioning the edge set of a bipartite graph into the minimal number of subgraphs isomorphic to those of a simple cycle of order 4 in special graph classes. This problem is NP-hard and finds application in organizing the distribution of network packets over communication channels in the process of transmission from one router to another. We develop an O(nlogn) algorithm for solving that problem in a class of n order trees. Intractable cases of the problem are identified.

It is known that the existence of many classical combinatorial structures in a graph, such as perfect matchings, Hamiltonian cycles, effective dominating sets, etc., can be characterized using (κ,τ)-regular sets, whose determination is equivalent to the determination of these classical combinatorial structures. On the other hand, the determination of (κ,τ)regular sets is closely related to the properties of the main spectrum of a graph. We use the previously obtained generalized properties of (κ,τ)-regular sets of graphs to develop a recognition algorithm of the traceability of a graph. We also obtained new sufficient conditions for the existence of a longest simple path in a graph in terms of the spectral radius of the adjacency matrix and the signless Laplacian of the graph.

In this paper, we consider a multicriteria integer linear programming problem with a parametric principle of optimality. Parameterization is realized by dividing the set of criteria into several disjoint groups (subsets) of criteria ordered by importance, with Pareto dominance within each group. The introduced parametric principle of optimality made it possible to connect such classical principles of optimality as lexicographic and Pareto ones. For the stability radius, which is the limiting level of perturbations of the parameters of the problem, not causing the appearance of new optimal solutions, the upper and lower estimations are obtained in the case of arbitrary Hölder’s norms in the criterion space and solution space. Some previously known results on the stability of the Boolean linear programming problem are formulated as corollaries.

In this paper, we consider the application of the matrix decomposition method to analyze Chua’s chaotic oscillator. It is shown that Chua’s system of equations describing the oscillator can be expanded into linear, quadratic, and cubic terms using the matrix decomposition method. Decomposition into a matrix series permits to study transition to chaos in Chua’s system from the point of view of Landau’s model of initial turbulence. The emerging new chaotic state in the system when a new stationary value of a state-space variable is chosen is explained using the Poincaré section method. For the system of equations that are obtained using the matrix decomposition method, the spectral and bifurcation analysis is conducted. Simulations using MATLAB and Simulink are carried out. A computational Simulink-model is the basis for building an information technology for recognizing the chaotic dynamics of Chua-type oscillators.

The transformation of a conical-type light beam by a biaxial crystal during propagation along one of its optical axes is herein investigated. The beam is formed by an axicon from the circularly polarized Gaussian input field. Depending on the position of the axicon either a Bessel beam or a combination of Bessel and conical beams falls on the crystal. The conversion coefficient from a zero-order Bessel beam to a first-order beam with phase dislocation is calculated. We show that if the angle of the beam cone and its diameter are large enough, then it is transformed into a field that is a first-order BesselGaussian beam with high accuracy. At the same time the conversion coefficient is close to 1. The case of a small cone angle of an incident Bessel beam is also investigated. In this case the efficiency of transformation significantly depends on the type of the spatial spectrum. At a small cone angle, the shape of the spatial spectrum is determined by the diameter of the incident Gaussian beam. Namely, as the diameter decreases, the beam spectrum changes from annular to close to Gaussian, passing through an intermediate form in the form of a superposition of these two profiles. The influence of the spatial spectrum is the conversion coefficient decreasing with a decrease of the of the ring component contribution to the spectrum. In this case, the conversion coefficient is always higher than for a scheme without an axicon when a Gaussian beam falls on the crystal. Consequently, the introduction of even a small conicity into the beam makes it possible to increase the field transformation coefficient. It can be implemented, for example, using a scheme with two axicons having close angles of a ray deflection. The results obtained are also of practical interest, in particular, for the development of conical-type laser emitters with a small cone angle for long-range reconnaissance and optical communication in free space.

The problem of absorption of nonpolarized (natural) light by a two-dimensional ensemble (monolayer) of spherical particles under oblique illumination is considered. The solution is based on the statistical theory of multiple scatteing of waves. The written equations make it possible to find the optimal conditions for the absorption of directed light to increase the absorption efficiency of monolayers with a periodic and partially ordered spatial arrangement of particles. The results of the calculations are presented for the absorption coefficient of polarized and natural light by ensembles of silver particles in a nonabsorbing medium. The particle size and monolayer filling factor are chosen so as to illustrate the dependence of absorption on the angle of incidence under conditions of resonance effects caused by the spatial organization of particles. It is shown that in the region of the resonant absorption peak a normally illuminated monolayer with a triangular lattice of silver particles can absorb almost an order of magnitude more incident light than a partially ordered layer. The absorption coefficient of a monolayer under directional oblique illumination can be almost an order of magnitude larger than under normal illumination. The results can be used to optimize the design of opto-electronic devices based on particulate layers.

In this paper, we derived the relation for the phase time of electromagnetic radiation tunneling through an ideal plasma layer in a dielectric for frequencies ω below the plasma frequency ω_p in the limit of low transparency of the layer. Within the framework of the model under consideration, the tunneling time is found to be independent of the layer thickness and determined only by the ω and  ω_p values. For lower frequencies the tunneling time tends to the limit defined by the inverse plasma frequency which allows us to treat the tunneling process in this case as a ‘splash’ of a plasma layer as a whole entity to form the transmitted radiation. Since the transmittance of the plasma layer is very low, the result obtained does not allow us to speak about superluminal energy transfer.

The effect of gain saturation in quantum-cascade structures with 2–4 quantum wells per period is herein analyzed on the basis of a system of balance equations. It is shown that the nonlinearity parameter decreases with an increase in the relaxation rate of laser levels, but the total current through the structure also increases. The use of the proposed multiphoton designs leads to a decrease in the non-linearity parameter without increasing the operating current. For example, in a two-photon scheme of laser transitions with the same transition probabilities and differential gains, two times slower saturation of the gain with an increase in the photon density is achieved, which leads to a high generation efficiency than in single-photon schemes.

Представлены результаты исследования спектров излучения кристаллов и тонких пленок CuInSe₂ при непрерывном (2 Вт/см²) и наносекундном импульсном лазерном возбуждении в диапазоне плотности мощности возбуждения ~1–100 кВт/см² и температурах 10–160 К. Обнаружено, что в кристаллах CuInSe₂ стимулированное излучение возникает в спектральной области 1,033 эВ с минимальным уровнем пороговой накачки 9,8 кВт/см², а при уровнях накачки 36–76 кВт/см² наблюдается лазерное излучение. Установлено, что для тонких пленок CuInSe₂, сформированных на стеклянных подложках с предварительно осажденным на стекло слоем молибдена (структура CuInSe₂/Mo/стекло), характерно появление только стимулированного излучения в области энергий 1,014–1,097 эВ с минимальным уровнем пороговой накачки 30 кВт/см² при температуре 10 К. Обсуждаются механизмы возникновения стимулированного и лазерного излучения в соединении CuInSe₂.