# Sharkovsky O. M.

### Anatolii Mykhailovych Samoilenko (on his 80th birthday)

Antoniouk A. Vict., Berezansky Yu. M., Boichuk О. A., Gutlyanskii V. Ya., Khruslov E. Ya., Kochubei A. N., Korolyuk V. S., Kushnir R. M., Lukovsky I. O., Makarov V. L., Marchenko V. O., Nikitin A. G., Parasyuk I. O., Pastur L. A., Perestyuk N. A., Portenko N. I., Ronto M. I., Sharkovsky O. M., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 2018. - 70, № 1. - pp. 3-6

### On the 100th birthday of outstanding mathematician and mechanic Yurii Oleksiiovych Mytropol’s’kyi (03.01.1917 – 14.06.2008)

Berezansky Yu. M., Boichuk О. A., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Parasyuk I. O., Perestyuk N. A., Samoilenko A. M., Sharkovsky O. M.

Ukr. Mat. Zh. - 2017. - 69, № 1. - pp. 132-144

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Boichuk О. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

### Yurii Stephanovych Samoilenko (on his 70th birthday)

Berezansky Yu. M., Boichuk О. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Nizhnik L. P., Samoilenko A. M., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1408-1409

### Anatolii Mykhailovych Samoilenko (on his 75th birthday)

Berezansky Yu. M., Boichuk О. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Nikitin A. G., Perestyuk N. A., Portenko N. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M., Trohimchuk Yu. Yu

Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 3 - 6

### Fifty years devoted to science (on the 70th birthday of Anatolii Mykhailovych Samoilenko)

Berezansky Yu. M., Dorogovtsev A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Perestyuk N. A., Rebenko A. L., Ronto A. M., Ronto M. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 3–7

### 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

### 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.

### Asymptotic Discontinuity of Smooth Solutions of Nonlinear $q$-Difference Equations

Derfel' G. A., Romanenko Ye. Yu., Sharkovsky O. M.

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1615-1629

We investigate the asymptotic behavior of solutions of the simplest nonlinear *q*-difference equations having the form *x*(*qt*+ 1) = *f*(*x*(*t*)), *q*> 1, *t*∈ *R* ^{+}. The study is based on a comparison of these equations with the difference equations *x*(*t*+ 1) = *f*(*x*(*t*)), *t*∈ *R* ^{+}. It is shown that, for “not very large” *q*> 1, the solutions of the *q*-difference equation inherit the asymptotic properties of the solutions of the corresponding difference equation; in particular, we obtain an upper bound for the values of the parameter *q*for which smooth bounded solutions that possess the property \(\begin{array}{*{20}c} {\max } \\ {t \in [0,T]} \\ \end{array} \left| {x'(t)} \right| \to \infty \) as *T*→ ∞ and tend to discontinuous upper-semicontinuous functions in the Hausdorff metric for graphs are typical of the *q*-difference equation.

### Dynamics of solutions of the simplest nonlinear boundary-value problems

Romanenko O. Yu., Sharkovsky O. M.

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 810–826

We investigate two classes of essentially nonlinear boundary-value problems by using methods of the theory of dynamical systems and two special metrics. We prove that, for boundary-value problems of both these classes, all solutions tend (in the first metric) to upper semicontinuous functions and, under sufficiently general conditions, the asymptotic behavior of almost every solution can be described (by using the second metric) by a certain stochastic process.

### Tracing of pseudotrajectories of dynamical systems and stability of prolongations of orbits

Sharkovsky O. M., Vereikina M. B.

Ukr. Mat. Zh. - 1997. - 49, № 8. - pp. 1016–1024

We investigate properties of dynamical systems associated with the approximation of pseudotrajectories of a dynamical system by its trajectories. According to modern terminology, a property of this sort is called the “property of tracing pseudotrajectories” (also known in the English literature as the “shadowing property”). We prove that dynamical systems given by mappings of a compact set into itself and possessing this property are systems with stable prolongation of orbits. We construct examples of mappings of an interval into itself that prove that the inverse statement is not true, i.e., that dynamical systems with stable prolongation of orbits may not possess the property of tracing pseudotrajectories.

### From one-dimensional to infinite-dimensional dynamical systems: Ideal turbulence

Romanenko O. Yu., Sharkovsky O. M.

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1604-1627

There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical systems with the “most complicated” phase space containing random functions, as well. This statement is justified in this paper. By using “simple” examples of dynamical systems (one-dimensional and two-dimensional boundary-value problems), we consider notions that generally characterize the phenomenon of turbulence—first of all, the emergence of structures (including the cascade process of emergence of coherent structures of decreasing scales) and self-stochasticity.

### Asymptotic periodicity of solutions of difference equations with continuous time

Maistrenko Yu. L., Romanenko Ye. Yu., Sharkovsky O. M.

Ukr. Mat. Zh. - 1987. - 39, № 1. - pp. 123-129

Sharkovsky O. M., Shevelo V. N.

Ukr. Mat. Zh. - 1984. - 36, № 5. - pp. 597 – 601

### С^{ ∞ }-mappings of an interval with attracting cycles of arbitrarily large periods

Ukr. Mat. Zh. - 1983. - 35, № 4. - pp. 537—539

### The center of a coarse dynamical system

Ukr. Mat. Zh. - 1968. - 20, № 5. - pp. 717–719

### Attractive sets not containing cycles

Ukr. Mat. Zh. - 1968. - 20, № 1. - pp. 136–142

### On a classification of stationary points

Ukr. Mat. Zh. - 1965. - 17, № 5. - pp. 80-95

### On cycles and structure of continuous mapping

Ukr. Mat. Zh. - 1965. - 17, № 3. - pp. 104-111

### Coexistence of the cycles of a continuous mapping of the line into itself

Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 61-71

The basic result of this investigation may be formulated as follows. Consider a set of natural numbers in which the following relationship is introduced: $n_1$ precedes $n_2$ ($n_1 \preceq n_2$) if for any continuous mappings of
the real line into itself the existence of a cycle of order $n_2$ follows from the existence of a cycle of order The following theorem holds.

Theorem. The introduced relationship transforms the set of natural numbers into an ordered set, ordered in the following way:

$3 \prec 5 \prec 7 \prec 9 \prec 11 \prec ... \prec 3 \cdot 2 \prec 5 \cdot 2 \prec ... \prec 3 \cdot 2^2 \prec 5 \cdot 2^2 \prec ... \prec 2^3 \prec 2^2 \prec 2 \prec 1$

### Leter to Oditor. Correction

Ukr. Mat. Zh. - 1961. - 13, № 4. - pp. 116

### On the solution of a class of functional equations

Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 86-94

The author considers functional equations of form (1). It is asserted that the process of solving (1) is determined by the kernel $\varphi(x)$. If the kernel consists of stationary points only, the solution of equation (1) is reduced to the solution of a system of ordinary equations; if the kernel has no stationary points, the method of steps is employed. If the kernel has no more than an even number of stationary points, then (see [4]) the real axis is divided into sets $M-i,\; i = 0, 1, 2, ...$ so that $\varphi(M_i) \subseteq M_i$. The method of steps is applied to each set except $M_0$. The idea of dividing the region of definition of the function into invariants in respect to the function of sets is then used once more for determining the nature of the solution obtained.

### Rapidly converging iterative processes

Ukr. Mat. Zh. - 1961. - 13, № 2. - pp. 210-215

The author shows that every rapidly converging iterative process $F(x)$, constructed by a given iterative process $f(x)$, in a certain vicinity of a stationary point, where $F(x)$ and $f(x)$ are assumed to be sufficiently smooth, can be presented in the form (4).

The necessary and sufficient conditions of convergence of an iterative sequence generated by $F(x)$ are found. It is shown that any interval containing a stationary point of the iterative process $f(x)$, in which (8) occurs, is a region of attraction of this stationary point as a stationary point of the process $F(x)$.

### Necessary and Sufficient Conditions for Convergence of Monomerie Processes

Ukr. Mat. Zh. - 1960. - 12, № 4. - pp. 484 - 489