# Volume 50, № 1, 1998

### Anatolii Mikhailovich Samoilenko (on his 60th birthday)

Berezansky Yu. M., Boichuk О. A., Korneichuk N. P., Korolyuk V. S., Koshlyakov V. N., Kulik V. L., Luchka A. Y., Mitropolskiy Yu. A., Pelyukh G. P., Perestyuk N. A., Skorokhod A. V., Skrypnik I. V., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 3–4

### A note on global attractivity in models of hematopoiesis

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 5–12

We consider the delay differential equations \(P'(t) = \frac{{\beta _0 \theta ^n [P(t - \tau )]^j }}{{\theta ^n + [P(t - \tau )]^n }} - \delta P(t),{\rm{ }}j = 0,1,\) which were proposed by Mackey and Glass as a model of blood cell production. We suggest new conditions sufficient for the positive equilibrium of the equation considered to be a global attractor. In contrast to the Lasota-Wazewska model, we establish the existence of the number δ_{j} = δ_{j}(*n*, τ) > 0 such that the equilibrium of the equation under consideration is a global attractor for all δ ε (0, δ_{j}] independently of β_{0} and θ.

### Invariant tori of differential equations in a Banach space

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 13–21

We establish conditions for the existence, uniqueness, and smoothness of the toroidal manifolds of differential equations with unbounded operator coefficients.

### Parametric bufferness in systems of parabolic and hyperbolic equations with small diffusion

Kolesov A. Yu., Mishchenko E. F., Rozov N. Kh.

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 22–35

We investigate the problem of parametric excitation of oscillations in systems of parabolic and hyperbolic equations with small coefficient of diffusion. We establish the phenomenon of parametric bufferness, i.e., the existence of an arbitrary fixed number of stable periodic solutions for a proper choice of the parameters of equations.

### Stability of stochastic systems in the diffusion-approximation scheme

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 36–47

By using a solution of a singular perturbation problem, we obtain sufficient conditions for the stability of a dynamical system with rapid Markov switchings under the condition of exponential stability of the averaged diffusion process.

### On the problem of estimation of the number of cycles in two-dimensional quadratic systems from the viewpoint of nonlinear mechanics

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 48–57

Two-dimensional quadratic systems are considered as a Liénard equation with certain special nonlinearities. Theorems on the existence or absence of cycles are given.

### On the construction of an asymptotic solution of a perturbed Bretherton equation

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 58–71

We consider the application of the asymptotic method of nonlinear mechanics to the construction of the first and second approximations of a solution of the Bremerton equation.

### Perturbations of degenerate coisotropic invariant tori of Hamiltonian systems

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 72–86

We consider a Hamiltonian system with a one-parameter family of degenerate coisotropic invariant tori. We prove a theorem on the preservation of the majority of tori under small perturbations of the Hamiltonian.

### On integral manifolds of oscillating systems with slowly varying frequencies

Lakusta L. M., Petryshyn R. I.

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 87–93

By using averages of functions, we construct the integral manifold of an oscillating system that passes through resonances in the course of its evolution. We investigate the smoothness of the integral manifold and obtain estimates for its partial derivatives.

### Forced frequency locking of rotating waves

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 94–101

We describe the frequency locking of an asymptotically orbitally stable rotating wave solution of an autonomous S_{1}-equivariant differential equation under the forcing of a rotating wave.

### The theory of the numerical-analytic method: Achievements and new trends of development. I

Ronto M. I., Samoilenko A. M., Trofimchuk S. I.

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 102–117

We describe the history of the development of the numerical-analytic method suggested by Samoilenko in 1965 and analyze the relation of this method to other investigations.

### Principle of additivity in averaging of degenerate nonlinear Dirichlet problems

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 118–135

We study the problem of averaging of Dirichlet problems for degenerate nonlinear elliptic equations of the second order in domains with fine-grained boundary under the condition that the weight function belongs to a certain Muckenhoupt class. We prove a pointwise estimate for solutions of the model degenerate nonlinear problem. The averaged boundary-value problem is constructed under new structural conditions for a perforated domain. In particular, we do not assume that the diameters of cavities are small as compared with the distances between them.

### On the exponential dichotomy of linear almost periodic pulse systems

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 136–142

For a linear almost periodic pulse system, we prove that the exponential dichotomy on a semiaxis implies the exponential dichotomy on the entire axis.

### On the irreversibility of time for an electromagnetic hereditary linear system

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 143–147

We give conditions for an electromagnetic linear system of hereditary type under which the time-reversal hypothesis does not hold if the relaxation functions of the electromagnetic field have different behavior at the extremes of the interval of definition. Under the same conditions, it is also possible to prove that these functions are constant if they have the same behavior at the extremes of the interval of definition.

### Quasidifferential equations in semilinear metric spaces

Ukr. Mat. Zh. - 1998. - 50, № 1. - pp. 148–153

For quasidifferential equations in semilinear metric spaces, we consider the problem of existence, uniqueness, and continuity of solutions and the problem of justification of the averaging method.