2017
Том 69
№ 9

All Issues

Dynamical systems and simulation of turbulence

Romanenko Ye. Yu., Sharkovsky O. M.

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Abstract

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.

English version (Springer): Ukrainian Mathematical Journal 59 (2007), no. 2, pp 229-242.

Citation Example: Romanenko Ye. Yu., Sharkovsky O. M. Dynamical systems and simulation of turbulence // Ukr. Mat. Zh. - 2007. - 59, № 2. - pp. 217–230.

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