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Dynamical systems and simulation of turbulence

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Ukrainian Mathematical Journal Aims and scope

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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  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    E. Yu. Romanenko & A. N. Sharkovskii

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 2, pp. 217–230, February, 2007.

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Romanenko, E.Y., Sharkovskii, A.N. Dynamical systems and simulation of turbulence. Ukr Math J 59, 229–242 (2007). https://doi.org/10.1007/s11253-007-0018-z

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  • DOI: https://doi.org/10.1007/s11253-007-0018-z

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