Dynamical systems and simulation of turbulence

  • Ye. Yu. Romanenko
  • O. M. Sharkovsky

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.
Published
25.02.2007
How to Cite
Romanenko, Y. Y., and O. M. Sharkovsky. “Dynamical Systems and Simulation of Turbulence”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, no. 2, Feb. 2007, pp. 217–230, http://umj.imath.kiev.ua/index.php/umj/article/view/3305.
Section
Research articles