Vol 216 (2022)
Статьи
On the stability of the trivial solution to a periodic system of ordinary differential equations
Abstract
In this paper, we examine a normal system of ordinary differential equations whose right-hand side is periodic in the independent variable and locally smoothly depends on the small parameter and the phase variable. Using the properties of nonlinear approximations of the right and left monodromy operators, we prove conditions that guarantee the arbitrary smallness of perturbed solutions for sufficiently small initial values of the solutions and the parameter.
3-11
Lie algebras of projective motions of five-dimensional pseudo-riemannian spaces. V. Lie algebras of projective and affine motions of h-spaces H221 of type {221}
Abstract
This work is devoted to the problem of studying multidimensional pseudo-Riemannian manifolds that admit Lie algebras of infinitesimal projective (in particular, affine) transformations, wider than Lie algebras of infinitesimal homotheties. Such manifolds have numerous geometric and physical applications. This paper is the final part of the work. The first part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — 212. — P. 10–29. The second part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — xxx. — P. 10–37. The third part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — xxx. — P. 3–20. The fourth part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — xxx. — P. 18–31.
12-28
Bifurcations in a dynamic system modeling pedagogical impacts on a group of students with a negative informal leader
Abstract
We consider a system of ordinary differential equations, which describes a model of the pedagogical impact on a group of students. The impact is expressed as the sum of a constant and a control parameter. We find equilibrium states of the system and determine the types of their bifurcations that arise when the control parameter changes. Also, we obtain coefficient conditions for the emergence of stable equilibrium states and the corresponding bifurcation values of the parameter.
29-43
Cyclic spaces
Abstract
We consider group algebras whose base groups are cyclic groups, prove a theorem on the multiplicativity of circulants for cyclic and anticyclic numbers, and describe geometric structures on linear spaces of cyclic and anticyclic algebras.
44-49
Elliptic problems in domains with degenerate singularities
Abstract
We consider a model elliptic pseudodifferential equation in Sobolev– Slobodetsky spaces in a reflex angle on the plane. Using the wave factorization, in the case of a unique solution, we study the situation where the aperture of the explementary angle tends to zero. We prove that this limit exists only if the right-hand side satisfies a certain additional condition and obtain this condition using the properties of singular integral operators.
50-56
Some problems of convex analysis in the Lobachevsky space
Abstract
The shadow problem in the Euclidean space was posed by G. Khudaiberganov in 1982. Its solution for dimensions >2 and various generalizations were obtained by a group of Ukrainian mathematicians led by Yu. B. Zelinsky in 2015. In this paper, we consider some variations of such problems and their generalizations in the Lobachevsky space and a closed lighting problem for the Lobachevsky space. In the Euclidean space, this problem was posed by V. G. Boltyansky.
57-65
Invariant tori of the weakly dissipative version of the Ginzburg—Landau equation
Abstract
We consider a periodic boundary value-problem for a weakly dissipative variant of the complex Ginzburg– Landau equation in the case where the period (wavelength) is small. The possibility of the existence of finite-dimensional invariant tori is proved. For solutions that belong to such tori, asymptotic formulas are obtained. We prove that all invariant tori, except for tori of dimension one (i.e., limit cycles), are unstable.We used various methods of the theory of dynamical systems with an infinitedimensional space of initial conditions, for example, the method of integral (invariant) manifolds, the method of normal forms, and methods of perturbation theory.
66-75
Cycles of two competing macroeconomic systems within a certain version of the Goodwin model
Abstract
In this paper, we examine the problem of competitive interaction of two macroeconomic systems. As the basic model, the well-known Goodwin model is chosen. We obtain sufficient conditions under which stable limit cycles can appear in the system considered.
76-87
Hidden synchronization of phase-locked loops with nonlinear delay
Abstract
In this paper, we consider a mathematical model of a phase locked loop system taking into account nonlinearity in the delay in the case of a fractional rational second-order integrating filter. We obtain conditions for the existence of several quasi-synchronous modes of the system, which determine the phase synchronization modes, and analyze the influence of the nonlinear delay on the phase multistability. We develop numerical and analytical conditions for the existence of hidden synchronization of phase systems and construct an algorithm for determining the influence of nonlinear delays on synchronization modes.
88-96
97-105
On the existence and completeness of enumeration of three-dimensional RR -polyhedra
Abstract
An RR-polyhedron is a closed convex polyhedron in E3 whose set of faces can be divided into two nonempty disjoint class: the class of regular polygons of the same type and the class of faces that form stars of symmetric rhombic vertices. A theorem on the existence and completeness of enumeration of closed convex three-dimensional RR-polyhedra is proved.
106-115
Existence of a surface with prescribed geometric characteristics in the Galilean space
Abstract
In this paper, we prove the existence of a cyclic surface spanned by two given curved spaces, the existence of a complete cyclic surface with a given total curvature on the whole plane, and the existence of a surface with given coefficients of the first quadratic form and the curvature defect.
116-123
On the isometry groups of foliated manifolds
Abstract
In this paper, we study the isometry group IsoF (M) of a foliated manifold with an Fcompact-open topology. This topology depends on the foliation F and coincides with the compact-open topology if F is an n-dimensional foliation. If the codimension of the foliation is equal to n, then the convergence in this topology coincides with the pointwise convergence. Some properties of the group
IsoF (M) are proved.
124-132
Integrable homogeneous dynamical systems with dissipation on the tangent bundles of smooth finite-dimensional manifolds. III. Equations of motion on the tangent bundle of an n -dimensional manifold in a force field with variable dissipation
Abstract
This paper is the conclusion of the work on the integrability of general classes of homogeneous dynamical systems with variable dissipation on the tangent bundles of n-dimensional manifolds. The first part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of smooth finite-dimensional manifolds. I. Equations of geodesics on the tangent bundle of a smooth n-dimensional manifold// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory. — 2022. — V. xxx. — P. xx–xx. The second part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of smooth finite-dimensional manifolds. II. Equations of motion on the tangent bundle of an n-dimensional manifold in a potential force field// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory. — 2022. — V. xxx. — P. xx–xx.
133-152
Polynomial automorphisms, quantization, and Jacobian conjecture related problems. IV. Approximations by polynomial symplectomorphisms
Abstract
This paper is the fourth part of a review of results concerning the quantization approach to the some classical aspects of noncommutative algebras. The first part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 213 (2022), pp. 110–144. The second part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 214 (2022), pp. 107–126. The third part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 215 (2022), pp. 95–128. The final part of the survey will be published in the next issue.
153-171