Dynamical systems and simulation of turbulence |
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Authors: | E Yu Romanenko A N Sharkovskii |
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Institution: | (1) Institute of Mathematics, Ukrainian Academy of Sciences, Kiev |
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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.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 217–230, February, 2007. |
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