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Vol 60, No 12 (2024)
ORDINARY DIFFERENTIAL EQUATIONS
CLASSIFICATION OF THE QUASILINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND ITS APPLICATION FOR NORMALIZATION OF SYSTEMS IN CRITICAL CASE OF BOGDANOV–TAKENS
Abstract
Two-dimensional autonomous system with a quasihomogenous polynomial of a first degree and weight (1, 2) in an unperturbed part is considered. The classification of an unperturbed parts is provided. According to it, the set of polynomials is constructively divided into eight equivalence classes with respect to quasihomogenuos zero degree substitutions. In each class the respresentatives, called the canonical forms, are determined. All structures of the generalized normal forms for the so far unstudied system with one of the canonical forms in its unperturbed part are obtained. Normalization in the system with unperturbed part (
Differencial'nye uravneniya. 2024;60(12):1587-1600
1587-1600
ON THE STABILITY BY THE NONLINEAR NON-STATIONARY HYBRID APPROXIMATION
Abstract
The paper investigates the effect of non-stationary perturbations on the stability of nonlinear nonautonomous systems with switching and impulsive effects. Sufficient conditions have been obtained to guarantee the asymptotic stability of a given equilibrium position of the initial system, and restrictions have been established under which the asymptotic stability is preserved under perturbations acting on the system. Note that the non-stationarities present both in the system itself and in perturbations can be described by unbounded functions with respect to time, as well as functions arbitrarily close to zero. It is assumed that the basic system is homogeneous in terms of the state vector. To find the required results, the second Lyapunov method is used in combination with the theory of differential inequalities.
Differencial'nye uravneniya. 2024;60(12):1640-1652
1640-1652
ON SENSITIVITY OF SOLUTIONS OF RICCATI EQUATIONS UNDER SMALL PARAMETER PERTURBATIONS AND OPTIMALITY IN LINEAR STOCHASTIC CONTROL SYSTEMS
Abstract
We investigate sensitivity of solutions of Riccati equations under asymptotically small perturbations of their coefficients. Upper bound on the difference between solutions of algebraic and differential Riccati equations is derived. The result is applied to study optimality in the stochastic linear-quadratic control problem over an infinite time-horizon for an asymptotically autonomous system. We also treat an issue related to performance of invariant control strategy.
Differencial'nye uravneniya. 2024;60(12):1623-1639
1623-1639
CONSTRUCTION OF SOLUTIONS WITH NEGATIVE EXPONENTS OF A DIFFERENTIAL SYSTEM IN THE TWO-DIMENSIONAL ANTI-PERRON EFFECT UNDER HIGHER-ORDER PERTURBATIONS
Abstract
The two-dimensional anti-Perron effect of changing all positive characteristic exponents of the linear approximation to four different negative exponents, respectively, of four non-trivial solutions of a differential system with a perturbation of higher order of smallness, has been realized.
Differencial'nye uravneniya. 2024;60(12):1616-1622
1616-1622
1601-1615
PARTIAL DERIVATIVE EQUATIONS
INTEGRATION OF THE NEGATIVE ORDER MODIFIED KORTEWEG–DE VRIES EQUATION WITH A LOADED TERM IN THE CLASS OF PERIODIC FUNCTIONS
Abstract
Spectral data of the Dirac operator with periodic potential are found. This operator is associated with the negative order modified Korteweg–de Vries equation with a loaded term. The obtained results make it possible to construct a solution to the negative order modified Korteweg–de Vries equation with a loaded term in the class of periodic functions using the inverse spectral problem method. The solvability of the Cauchy problem for an infinite system of Dubrovin–Trubovitz differential equations in the class of three times continuously differentiable periodic functions is proved.
Differencial'nye uravneniya. 2024;60(12):1703-1712
1703-1712
SOLVABILITY OF NONLINEAR BOUNDARY VALUE PROBLEMS FOR DIFFERENTIAL EQUILIBRIUM EQUATIONS OF NON-FLAT TIMOSHENKO TYPE SHELLS OF NON-ZERO GAUSSIAN CURVATURE IN ISOMETRIC COORDINATES
Abstract
We study the solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic non-flat inhomogeneous isotropic shells with loose edges in the framework of the Timoshenko shear model, referred to isometric coordinates.The boundary value problem is reduced to a nonlinear operator equation with respect to generalized displacements in Sobolev space, the solvability of which is established using the principle of compressed mappings.
Differencial'nye uravneniya. 2024;60(12):1685-1702
1685-1702
LOCALIZATION OF EIGENFUNCTIONS OF THE DIRICHLET PROBLEM NEAR A CONTOUR AT THE BOUNDARY OF A THIN DOMAIN
Abstract
We consider the spectral Dirichlet problem for the Laplace operator in a thin three-dimensional domain of a variable thickness which admits a maximum value at a smooth closed contour either inside the longitudinal cross-section, or at its boundary. We find out asymptotic expansions of the eigenvalues which involve eigenvalues of the harmonic oscillator at the axis or the half-axis as well as of a certain second order ordinary differential equation at the contour. The eigenfuctions are localized in the vicinity of the contour.
Differencial'nye uravneniya. 2024;60(12):1664-1684
1664-1684
1653-1663
BRIEF MESSAGES
BOUNDARY PROBLEM FOR THE LAPLACE EQUATION WITH MIXED BOUNDARY CONDITIONS IN A SEMIBAND
Abstract
Theorems on the existence and uniqueness of the solution to the Laplace equation with mixed boundary conditions in a semiband have been proven in the work. Additionally, integral representations for the partial derivatives of the solution have been obtained.
Differencial'nye uravneniya. 2024;60(12):1713-1718
1713-1718
Articles
AVTORSKIY UKAZATEL' TOMA 60, 2024 g.
Differencial'nye uravneniya. 2024;60(12):1719-1728
1719-1728