# Volume 49, № 1, 1997

### On the 80th birthday of Academician Yu. A. Mitropol’skii

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 3-4

### On the contribution of Yu. A. Mitropol’skii to the development of asymptotic methods in nonlinear mechanics

Kolomiyets V. G., Samoilenko A. M.

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 5–10

We present a survey of the most important scientific results of Yu. A. Mitropol’skii in the fields of nonlinear differential equations, mathematical physics,and the theory of nonlinear oscil.

### Global solutions and invariant tori of difference-differential equations

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 11–24

We prove the existence of an *m*-parameter family of global solutions of a system of difference-differential equations. For difference-differential equations on a torus, we introduce the notion of rotation number. We also consider the problem of perturbation of an invariant torus of a system of difference-differential equations and study the problem of the existence of periodic and quasiperiodic solutions of second-order difference-differential equations.

### Singularly perturbed stochastic systems

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 25–34

Problems of singular perturbation of reducible invertible operators are classified and their applications to the analysis of stochastic Markov systems represented by random evolutions are considered. The phase merging, averaging, and diffusion approximation schemes are discussed for dynamical systems with rapid Markov switchings.

### Qualitative analysis of the influence of random perturbations of “white-noise” type applied along the vector of phase velocity on a harmonic oscillator with friction

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 35–46

We consider representations in the phase plane for the harmonic oscillator with friction under random perturbations applied along the vector of phase velocity. We investigate the behavior of the amplitude, phase, and total energy of the damped oscillator.

### Normalization and averaging on compact lie groups in nonlinear mechanics

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 47–67

We consider the method of normal forms, the Bogolyubov averaging method, and the method of asymptotic decomposition proposed by Yu. A. Mitropol’skii and the author of this paper. Under certain assumptions about group-theoretic properties of a system of zero approximation, the results obtained by the method of asymptotic decomposition coincide with the results obtained by the method of normal forms or the Bogolyubov averaging method. We develop a new algorithm of asymptotic decomposition by a part of the variables and its partial case — the algorithm of averaging on a compact Lie group. For the first time, it became possible to consider asymptotic expansions of solutions of differential equations on noncommutative compact groups.

### On properties of central manifolds of a stationary point

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 68–76

We present results concerning properties of central manifolds of a stationary point. The results are illustrated by examples.

### Problems with free boundaries and nonlocal problems for nonlinear parabolic equations

Berezovsky A. A., Mitropolskiy Yu. A.

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 84–97

We present statements of problems with free boundaries and nonlocal problems for nonlinear parabolic equations arising in metallurgy, medicine, and ecology. We consider some constructive methods for their solution.

### Stability of solutions of pulsed systems

Chernikova O. S., Perestyuk N. A.

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 98–111

We present the principal results in the theory of stability of pulse differential equations obtained by mathematicians of the Kiev scientific school of nonlinear mechanics. We also present some results of foreign authors.

### Existence of equilibrium states of systems of hard spheres in the Boltzmann-Enskog limit within the frame work of the grand canonical ensemble

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 112–121

We study equilibrium states of systems of hard spheres in the Boltzmann-Enskog limit (*d*→0, 1/*v*→∞ (*z*→∞), and *d* ^{3} (1/*v*)=const (*d* ^{3} *z*=const)). For this purpose, we use the Kirkwood-Salsburg equations. We prove that, in the Boltzmann-Enskog limit, solutions of these equations exist and the limit distribution functions are constant. By using the cluster and compatibility conditions, we prove that all distribution functions are equal to the product of one-particle distribution functions, which can be represented as power series in *z*=*d* ^{3} *z* with certain coefficients.

### Nilpotent flows of S^{1}-invariant Hamiltonian systems on 4-dimensional symplectic manifolds

Parasyuk I. O., Samoilenko A. M.

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 122–140

We investigate S^{1}-invariant Hamiltonian systems on compact 4-dimensional symplectic manifolds with free symplectic action of a circle. We show that, in a rather general case, such systems generate ergodic flows of types (quasiperiodic and nilpotent) on their isoenergetic surfaces. We solve the problem of straightening of these flows.

### On periodic solutions of linear differential equations with pulsed influence

Elgondyev K. K., Samoilenko V. G.

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 141–148

We study periodic solutions of ordinary linear second-order differential equations with publsed influence at fixed and nonfixed times.

### Pointwise estimates of potentials for higher-order capacities

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 149–163

In a domain *D*=Ω\*E*∈ *R* ^{ n }, we consider a nonlinear higher-order elliptic equation such that the corresponding energy space is *W* _{ p } ^{ m } *(D)*⩜*W* _{ q } ^{1} *(D)*, *q*>*mp*, and estimate a solution *u(x)* of this equation satisfying the condition *u(x)−kf(x)*∈*W* _{ p } ^{ m } *(D)*⩜*W* _{ q } ^{1} *(D)*, where *k*∈*R* ^{1}, *f(x)*∈ *C* _{0} ^{∞} (Ω), and *f(x)*=1 for *x*∈*F*. We establish a pointwise estimate for *u(x)* in terms of the higher-order capacity of the set *F* and the distance from the point *x* to the set *F*.

### Symmetry of equations of linear and nonlinear quantum mechanics

Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 164–176

We describe local and nonlocal symmetries of linear and nonlinear wave equations and present a classification of nonlinear multidimensional equations compatible with the Galilean principle of relativity. We propose new systems of nonlinear equations for the description of physical phenomena in classical and quantum mechanics.