Vol 24, No 2 (2022)

n this paper, authors examine flows with a finite hyperbolic chain-recurrent set without heteroclinic intersections on arbitrary closed n-manifolds. For such flows, the existence of a dual attractor and a repeller is proved. These points are separated by a (n-1)-dimensional sphere, which is secant for wandering trajectories in a complement to attractor and repeller. The study of the flow dynamics makes it possible to obtain a topological invariant, called a spherical flow scheme, consisting of multi-dimensional spheres that are the intersections of a secant sphere with invariant saddle manifolds. It is worth known that for some classes of flows spherical scheme is complete invariant. Thus, it follows from G. Fleitas results that for polar flows (with a single sink and a single source) on the surface, it is the spherical scheme that is complete equivalence invariant. 

This article addresses systems of linear ordinary differential equations with an identically degenerate matrix in the main part. Such formulations of problems in literature are usually called differential-algebraic equations. In this work, attention is paid to the problems of the second order. Basing on the theory of matrix pencils and polynomials, sufficient conditions for existence and uniqueness of the equations’ solution are given. To solve them numerically, authors investigate a multistep method and its version based on a reformulated notation of the original problem. This representation makes it possible to construct methods whose coefficient matrices can be calculated at previous points. This approach has delivered good results in numerical solution of first-order differential-algebraic equations that contain stiff and rapidly oscillating components and have singular matrix pencil. The stability of proposed numerical algorithm is investigated for the well-known test equation. It is shown that this difference scheme has the first order of convergence. Numerical calculations of the model problem are presented.

The paper examines the stability problem of biological, economic and other processes modeled by the Lotka-Volterra equations with delay. The difference between studied equations and the known ones is that the adaptability functions and the coefficients of the relative change of the interacting subjects or objects are non-linear and take into account variable delay in the action of factors affecting the number of subjects or objects. Moreover, these functions admit the existence of equilibrium positions’ set that is finite in a bounded domain. The stability study of three types of equilibrium positions is carried out using direct analysis of perturbed equations and construction of Lyapunov functionals that satisfy conditions of well-known theorems. Corresponding sufficient conditions for asymptotic stability including global stability are derived, as well as instability and attraction conditions of these positions.

The average force acting on the system of polarizing particles from the electric field in a non-uniformly heated dielectric liquid is determined. The case of pair interactions in the system is examined. To find the force acting on the particles, the interaction of two particles in a liquid is modelled in the presence of a given temperature gradient and the electric field strength far from the particles. The dependence of the particle permittivity on temperature is taken into account. The resulting expression for the force acting on two particles has such a power-law dependence on the distance between the particles, that allows to carry out the direct averaging procedure for a system of particles located in an infinite volume of liquid. When determining the average force, the probability density function of a continuous random variable is used, and the vector connecting the centers of particles plays the role of this variable. The differential equation for finding the probability density function is derived from two conditions. First, the pairs of particles are preserved in the space of all their possible configurations. Second, each pair of particles moves like a point with a speed equal to the speed of their relative motion. The resulting equation in the case under consideration has a set of solutions. Basing on the physical analysis of the problem, the choice of the probability density function is proposed, which allows one to determine the average electro-thermophoretic force acting in such a system with an accuracy up to the second degree of the volume concentration of particles.

The photoconductivity kinetics of a resistor with homogeneous generation of electrons and holes in thickness is investigated. Calculations are carried out for an $n$-type semiconductor. The cases of linear and quadratic volumetric recombination are considered. The mathematical model of the process includes a non-linear parabolic partial differential equation. The cause of its non-linearity is quadratic recombination. Boundary conditions of the 3rd kind are used, thus allowing to examine the surface recombination of nonequilibrium charge carriers. This latter phenomenon makes it necessary to take into account the diffusion term when writing kinetic equations describing the distribution of electrons and holes. The model neglects the volumetric charge. In described circumstances it is possible to use the integration of the photocurrent flowing through the resistor to obtain the dependence of the light intensity on time for small optical pulse durations: $T < \max{(\tau_n, \tau_p)}$. Here $T$ is the pulse duration, $\tau_n$ and $\tau_p$ are the lifetimes of electrons and holes, respectively. Nonlinear distortions in this case are mainly associated with the appearance of the second and the third harmonics of the Fourier series expansion of the function that determines the photocurrent dependence on time. To "restore" the optical pulse, the operation of differentiating the photocurrent can be used. Nonlinear and phase distortions are small when the condition $T < \max{(\tau_n, \tau_p)}$ is met. Proposed methods make it possible to expand the range of optical pulse durations ($T$) in which its "recovery" is possible. In the vicinity of the region defined by the equality $T\approx \max{(\tau_n, \tau_p)}$, nonlinear and phase distortions are significant.