Recent Papers

Application of generalized Ljapunov functions to the investigation of non-homogeneous systems

Bounded solutions of differential equations of second order

Refinement of a theorem of V. V. Morozov

A class of dynamical systems with stable structures

Analytic form of solutions of linear systems of differential equations with quasi-periodic coefficients

Differential equations and inequalities with discontinuous right member. I

On a singular integro-differential equation

A. V. Bitsadze. Boundary value problems for second-order elliptic equations

Topological classification of singular points and generalized Ljapunov functions

Periodic solutions of second-order nonlinear equations

The paper presents the conditions for the existence and provides an algorithm for constructing periodic solutions of the equation $$\frac{d^2x}{dt^2}=f\biggl(t,x,\frac{dx}{dt}\biggr),$$ the right-hand side of which is periodic in $t$. Bibliography: 6 items.

The number of periodic solutions of the second kind to the differential equation $\ddot{\varphi}-f_1(\varphi)\dot{\varphi}-f_0(\varphi)=0$

The equation \begin{equation} \ddot{\varphi}-f_1(\varphi)\dot{\varphi}-f_0(\varphi)=0\tag{1} \label{1} \end{equation} is considered in the case where $f_0(\varphi)$ and $f_1(\varphi)$ are trigonometric polynomials of degree no higher than $n$, where $n$ is a fixed natural number. Equation (1) is equ…

Dynamical systems close to Hamiltonian ones

In problems related to determining the number of limit cycles bifurcating from a singular point of the second group, it is sometimes (RZhMat, 1965, 7B199) essential to establish the fact that for a system close to a Hamiltonian one, depending on a parameter $\mu$, under certain additional conditions…

Smoothness of thermal potentials. IV. Application of the theory of thermal potentials to the solution of a problem of biophysics on the distribution of concentrations in a living cell

The first boundary value problem for quasilinear parabolic equations with retarded argument

A question on the uniqueness of solutions of linear boundary value problems for ordinary differential equations

On the spectrum of a non-selfadjoint Sturm–Liouville operator

Two-point and multi-point Taylor formulas

A formula is considered by which the value of the function $f(x)\in C^n$ on a given interval is determined through its values and the values of its derivatives at $m$ points ($m\ge2$) of the given interval. The considered examples demonstrate the possibility of effectively using the obtained general…

The theory of characteristic vectors and its application to the study of the asymptotic behavior of solutions of differential systems. I

On the existence of an upper and a lower solution of a boundary value problem for a system of ordinary differential equations

Under certain conditions imposed on the right-hand sides of the system of equations, the existence of upper and lower solutions to the boundary value problem for finite and countable systems of ordinary differential equations on an interval of a specific length is proven. Bibliography 2.

The structure of a solution of the equation $u'=U_0(\nu)u+\sum_{n=1}^\infty(\nu)u^{1+\alpha_n}\equiv U(u,\nu)$ in a small neighborhood of the origin

The article constructs a solution to the equation \begin{equation} u'=U_0(\nu)u+\sum_{n=1}^\infty(\nu)u^{1+\alpha_n}\equiv U(u,\nu)\tag{1} \label{1} \end{equation} where $u$ and $\nu$ are polar coordinates. At the same time, there is one case where the solution to equation \eqref{1} cannot be obtain…

On the definition of the capture regions in the case of a system of linear differential equations with periodic coefficients

Regularity of a class of linear systems with almost periodic coefficients

The paper considers a class of systems of linear differential equations with almost-periodic coefficients. It is proved that the systems of the selected class are regular. Bibliography: 6 items.

Viktor Vladimirovič Nemyckiǐ

The field of a horizontal magnetic dipole in the presence of an ellipsoid

The problem of electromagnetic field diffraction by a spheroid is solved. The source of the primary field is a magnetic dipole located on the polar axis of the spheroid. The dipole moment is directed at a right angle to the axis; the spheroid is perfectly conducting. The solution to the problem is r…

The classification of trajectories of a dynamical system with cylindrical phase space

A system of differential equations containing angular coordinates is considered. The phase space of such a system is cylindrical. Based on the behavior of trajectories on covering spaces, a classification of the trajectories of the system under consideration is introduced. Conditions for the absence…

The stability of a certain third-order system with variable structure

The paper considers an automatic control system with a variable structure, described by a system of third-order differential equations, the parameters of which vary within wide limits. A cone located in the phase space of variables $x_1$, $x_2$, $x_3$ is taken as the switching surface. The switching…

The function space$W_p^m,\alpha_1,\dots,\alpha_s(T,\Omega)$, and completely continuous operators in weight spaces

The existence and uniqueness theorem for the Cauchy problem of a hyperbolic equation with self-regulating delay

For a hyperbolic equation of the form $$U_{tx}=f(t,x,U(t,x),U(t-\tau,x),U_t(t,x),U_t(t-\tau,x),U_x(t,x)U_x(t-\tau,x))$$ with an initial function $\varphi(t,x)$ defined for $(t,x)\in[t_0-\tau_0,t_0]\times\Omega$ and with a delay $\tau=\tau(t,x,U,U_t,U_x)$ that depends not only on the independent vari…

The asymptotic representation of the solution of a boundary value problem for a system of ordinary differential equations with a complex parameter

In connection with the study of one-dimensional mixed problems for second-order parabolic systems containing time derivatives in the boundary conditions, this paper provides an asymptotic representation of the solution to the spectral problem (1)–(2) for a system of ordinary differential equations o…

Некоторые теоремы о предельных циклах

The equation $\ddot{x}=f(x)-R(x,\dot{x})$ with a discontinuous right-hand side, which can be interpreted as a generalized pendulum equation, is considered. This work is devoted to the issues of the existence and relative positioning of global limit cycles. Estimates of the distance between two globa…

On the structure of the coefficients of the succession function

The group of symmetric transformations of a generalized Gyldén problem

A generalized Gylden problem is considered, i.e., the problem of the motion of a point of variable mass in a nonstationary central force field whose reduced force law (the ratio of force to mass) is expressed by an arbitrary function $f_{\text{pr}}(t,r)$ of time $t$ and the distance $r$ of the point…

Stability of the trivial solution of a non-autonomous system of two differential equations

Stability of generalized processes. I, II

Solution by the method of lines of certain boundary value problems for equations of elliptic type

This paper considers the solution of certain boundary value problems for a two-dimensional elliptic equation with coefficients depending on two variables using the method of lines. The general solution to the system of differential equations of the method of lines is sought using the method proposed…

On the theory of operators of the form$\frac{d^m}{dt^m}-A$

A remark on limit cycles of an equation

Conditions of validity of Čaplygin's theorem for elliptic difference equations

The paper considers a necessary and sufficient condition for the validity of a theorem analogous to S. A. Chaplygin's theorem on differential inequalities for an elliptic difference equation. Effective estimates of the domain of applicability of the mentioned theorem to difference and differential e…

The basic existence theorems of space elasticity theory

A problem on a pairing of equations of parabolic and hyperbolic types when time derivatives occur in the boundary conditions

We consider the problem of finding a solution to the following system of equations: $$\frac{\partial u^{(1)}}{\partial t}=\sum_{l=0}^2C_{0l}^{(1)}(x)\frac{\partial^lu^{(1)}}{\partial x^l}+f^{(1)}(x,t),\quad x\in(a_1,b_1),\quad t\in(0,T),$$ $$\frac{\partial^2 u^{(2)}}{\partial t^2}=\sum_{\substack{{k…

The resolvent of a weighted integro-differential equation

The completeness of certain function systems

Linearly independent systems \begin{equation} {\ln r(x_i,y)},\quad\biggl{\frac{\partial}{\partial n_y}\ln r(x_i,y)\biggr}\quad(i=1,2,\dots), \label{1} \end{equation} are considered, where $x_i$ are uniformly distributed on the circle $S_1$, and $y \in S$; here, $S$ and $S_1$ are concentric circles. …

A nonlinear analog of the Ljapunov transformation

Uniqueness theorems for a system of multidimensional Volterra integral equations

Expansion formulas for a boundary value problem with a complex parameter

Application of the asymptotic method to the integration of autonomous systems with lag

A quasilinear autonomous system with many degrees of freedom and delays is considered. Under the assumption that the generating system possesses a multi-frequency periodic regime, the solution of the system is constructed using the asymptotic method of N. M. Krylov and N. N. Bogolyubov. The possibil…

Stability of a singular cycle in the critical case

Asymptotic representation of the solution of a mixed problem for a differential equation of hyperbolic type with a small parameter

On the smoothness of thermal potentials. IV

The paper considers an application of the theory of Panya’s special heat potential of a simple layer to a boundary value problem for a system of parabolic equations with discontinuous coefficients, arising in the study of the distribution of concentrations of substances involved in the vital process…

Solvability of an integro-differential equation