# Romanenko Ye. Yu.

### 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.

### Dynamics of neighborhoods of points under a continuous mapping of an interval

Ukr. Mat. Zh. - 2005. - 57, № 11. - pp. 1534–1547

Let $\{ I, f Z^{+} \}$ be a dynamical system induced by the continuous map $f$ of a closed bounded interval $I$ into itself. In order to describe the dynamics of neighborhoods of points unstable under $f$, we suggest a notion of $\varepsilon \omega - {\rm set} \omega_{f, \varepsilon}(x)$ of a point $x$ as the $\omega$-limit set of $\varepsilon$-neighborhood of $x$. We investigate the association between the $\varepsilon \omega - {\rm set}$ and the domain of influence of a point. We also show that the domain of influence of an unstable point is always a cycle of intervals. The results obtained can be directly applied in the theory of continuous time difference equations and similar equations.

### 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.

### Simulation of spatial-temporal chaos: The simplest mathematical patterns and computer graphics

Romanenko Ye. Yu., Vereikina M. B.

Ukr. Mat. Zh. - 1993. - 45, № 10. - pp. 1398–1410

The article presents three scenarios of the evolution of spatial-temporal chaos and specifies the corresponding types of chaotic solutions to a certain nonlinear boundary-value problem for PDE. Analytic assertions are illustrated by numerical analysis and computer graphics.

### Representation of the local general solution of a certain class of differential-functional equations

Ukr. Mat. Zh. - 1990. - 42, № 2. - pp. 206–210

### Asymptotic of the solution of a certain class of functional-differential equations

Ukr. Mat. Zh. - 1989. - 41, № 11. - pp. 1526–1532

### 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

### Representation of the solutions of quasilinear functional differential equations of neutral type in case of resonance

Ukr. Mat. Zh. - 1977. - 29, № 2. - pp. 280–283

### Representation of the solutions of quasilinear differential-functional equations of neutral type

Ukr. Mat. Zh. - 1974. - 26, № 6. - pp. 749–761