# Romanenko O. Yu.

### Oleksandr Mykolaiovych Sharkovs’kyi (on his 80th birthday)

Fedorenko V. V., Ivanov А. F., Khusainov D. Ya., Kolyada S. F., Maistrenko Yu. L., Parasyuk I. O., Pelyukh G. P., Romanenko O. Yu., Samoilenko V. G., Shevchuk I. A., Sivak A. G., Tkachenko V. I., Trofimchuk S. I.

Ukr. Mat. Zh. - 2017. - 69, № 2. - pp. 257-260

### Self-stochasticity phenomenon in dynamical systems generated by difference equations with continuous argument

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 954–975

For dynamical systems generated by the difference equations *x*(*t*+1) = *f*(*x*(*t*)) with continuous time (*f* is a continuous mapping of an interval onto itself), we present a mathematical substantiation of the self-stochasticity phenomenon, according to which an attractor of a deterministic system contains random functions.

### 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.

### Aleksandr Nikolaevich Sharkovsky (on his 60th birthday)

Berezansky Yu. M., Fedorenko V. V., Kolyada S. F., Romanenko O. Yu., Sivak A. G., Vereikina M. B.

Ukr. Mat. Zh. - 1996. - 48, № 12. - pp. 1602-1603

### 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.