Vol 28, No 141 (2023)

In the article, linear systems of ordinary differential equations of fractional order  are investigated. In contrast to previously known results, the authors consider the case when the matrix before the fractional differentiation operation is degenerate. Problems in such a formulation are called differential-algebraic equations of fractional order. The fundamental differences of such systems from the classical problems of fractional differentiation and integration are emphasized, namely, the systems under consideration can have an infinite number of solutions, or a solution of the original problem depends on the high fractional derivative of the right-hand side. Corresponding examples are given. The authors pass to a different, equivalent formulation of the problem, namely, they rewrite it in the form of a system of linear integral equations of the Abel type (with a weak singularity). This technique allows one to use the apparatus of regular matrix bundles to investigate the existence and uniqueness of the original problem. Using this result, the authors give sufficient conditions for the existence of a unique solution to the class of problems under consideration. Further, an algorithm for the numerical solution of such equations is proposed. The method is based on the product integration method and the quadrature formula of right rectangles. Calculations and graphs of the errors of the proposed method for various fractional differentiation exponents and various indices of the initial matrix bundles are presented.

A model of a fully connected association of neurons with a synaptic electrical connection which is a system of $m$ differential equations with delay is considered. By a special substitution, this system is reduced to a system of impulsive ordinary differential equations. For the corresponding dynamical system in the case $m=3$, we study the existence, stability, and asymptotic representation of periodic solutions on the basis of a bifurcation analysis of a two-dimensional mapping, a shift operator along trajectories of a solution to a special system of two differential equations. Particular attention is paid to the number of coexisting stable regimes. We study the problem of finding parameters for which the number of such modes is maximum. In order to search the fixed points of the resulting two-dimensional mapping, a numerical study is used based on the following iterative procedure. Selected the starting point, the Runge-Kutta method with a given step calculates the solution values on the segment $[0,T]$. At the end point $T$ of this segment, the solution value is compared with the initial one and if the deviation exceeds the specified value, then the value at the end point is taken as the initial one and the calculation cycle by the Runge-Kutta method is repeated. The calculations terminate when the required small deviation is reached, i.e., a fixed point of the shift operator is found, and so is the corresponding stable periodic mode, or when the number of iterations reaches a given large number, and this indicates the absence of a fixed point. The paper presents the results of a numerical study that made it possible to demonstrate the main rearrangements occurring in the phase space of a two-dimensional mapping. The obtained fixed points allow us to find asymptotic stable solutions of the original problem.

The research subject of this work is at the junction of two sections of the qualitative theory of differential equations, namely: the theory of Lyapunov exponents and the theory of oscillation. In this paper, we study the spectra (i. e., sets of different values on nonzero solutions) of the exponents of oscillation of signs (strict and nonstrict), zeros, roots, and hyperroots of linear homogeneous differential systems with coefficients continuous on the positive semiaxis. For any $n\ge 2$, the existence of an -dimensional differential system with continuum spectra of the oscillation exponents is established. For even $n$, the spectra of all the oscillation exponents fill the same segment of the numerical axis with predetermined arbitrary positive incommensurable ends, and for odd $n$, zero is added to the indicated spectra. It turns out that for each solution of the constructed differential system, all the oscillation exponents coincide with each other. When proving the results of this work, the cases of even and odd $n$ are considered separately. The results obtained are theoretical in nature, they expand our understanding of the possible spectra of oscillation exponents of linear homogeneous differential systems.

The article proposes the development of a software module for modeling the kinematics and dynamics of a manipulator with five degrees of freedom. To solve the forward kinematics problem of the manipulator, the Denavit–Hartenberg method was used. To solve the inverse kinematics and dynamics problem of the manipulator, analytical methods (the Levenberg-Marquardt method, the Newton–Euler method) and a soft computing method (adaptive neurofuzzy inference system) were used. The software module for modeling the kinematics and dynamics of the manipulator was developed using the software package of the SolidWorks computer-aided design system and the MatLab program. The developed software module is able to simulate the kinematics and dynamics of the manipulator based on the described methods, visualize the simulation results, generate a trajectory for the target position and orientation of the end-effector of the manipulator, simulate the movement of the manipulator along a given trajectory.