# Volume 59, № 2, 2007

### On the 90th birthday of Yurii Alekseevich Mitropol’skii

Berezansky Yu. M., Gorbachuk M. L., Korolyuk V. S., Koshlyakov V. N., Lukovsky I. O., Makarov V. L., Perestyuk N. A., Samoilenko A. M., Samoilenko Yu. I., Sharko V. V., Sharkovsky O. M., Stepanets O. I., Tamrazov P. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 147–151

### General method for the solution of some problems of stabilization and destabilization of motion

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 152–161

We demonstrate a complete mathematical analogy between the description of motion of an electron in a periodic field and the phenomenon of parametric resonance. A band approach to the analysis of the phenomenon of parametric resonance is formulated. For an oscillator under the action of an external force described by the Weierstrass function, we calculate the increments of increase in oscillations and formulate a condition for parametric resonance. For the known problem of a pendulum with vibrating point of suspension, we find exact conditions for the stabilization of the pendulum in the upper (unstable) equilibrium position by using the Lamé equation.

### Analytic study of some differential equations related to problems of space dynamics

Chichurin A. V., Grebenikov E. A.

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 162–171

We present a survey of results of the study of differential equations whose solutions have singularities of a certain type, in particular movable singular points with fairly simple topology. New statements on the form of partial and general solutions of these equations are obtained.

### On the countability of the number of solutions of a two-dimensional linear Pfaff system with different characteristic sets

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 172–189

We prove that a two-dimensional, completely integrable, linear Pfaff system has at most countably many solutions with all different characteristic sets.

### Resonance and nonlinearity: A survey

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 190–205

This paper surveys recent results about nonresonant and resonant periodically forced nonlinear oscillators. This includes the existence of periodic, unbounded or bounded solutions for bounded nonlinear perturbations of linear and piecewise-linear oscillators, as well as of some classes of planar Hamiltonian systems.

### Van der Pol oscillator under parametric and forced excitations

Nguyen Van Dao, Nguyen Van Dinh, Tran Kim Chi

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 206–216

We study a van der Pol oscillator under parametric and forced excitations. The case where a system contains a small parameter and is quasilinear and the general case (without the assumption of the smallness of nonlinear terms and perturbations) are studied. In the first case, equations of the first approximation are obtained by the Krylov-Bogolyubov-Mitropol’skii technique, their averaging is performed, frequency-amplitude and resonance curves are studied, and the stability of the given system is considered. In the second case, the possibility of chaotic behavior in a deterministic system of oscillator type is shown.

### Dynamical systems and simulation of turbulence

Romanenko Ye. Yu., Sharkovsky O. M.

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 217–230

We propose an approach to the analysis of turbulent oscillations described by nonlinear boundary-value problems for partial differential equations. This approach is based on passing to a dynamical system of shifts along solutions and uses the notion of ideal turbulence (a mathematical phenomenon in which an attractor of an infinite-dimensional dynamical system is contained not in the phase space of the system but in a wider functional space and there are fractal or random functions among the attractor “points”). A scenario for ideal turbulence in systems with regular dynamics on an attractor is described; in this case, the space-time chaotization of a system (in particular, intermixing, self-stochasticity, and the cascade process of formation of structures) is due to the very complicated internal organization of attractor “points” (elements of a certain wider functional space). Such a scenario is realized in some idealized models of distributed systems of electrodynamics, acoustics, and radiophysics.

### Some results of the local theory of smooth functions

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 231–267

We present results of the investigation of the local behavior of smooth functions in neighborhoods of their regular and critical points and prove theorems on the mean values of the functions considered similar to the Lagrange finite-increments theorem. We also study the symmetry of the derivative of an analytic function in the neighborhood of its multiple zero, prove new statements of the Weierstrass preparation theorem related to the critical point of a smooth function with finite smoothness, determine a nongradient vector field of a function in the neighborhood of its critical point, and consider one critical case of stability of an equilibrium position of a nonlinear system.

### A general class of evolutionary equations

Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 268–288

Using observable quantities and state variable of a dynamical process, a general evolutionary equation is defined which unifies classical ordinary differential equations, partial differential equations, and hereditary systems of retarded and neutral type. Specific illustrations are given using transmission lines nearest-neighbor coupled at the boundary and the theory of heat transfer in solids. Some spectral theory for linearization of the equations is also discussed.