Abstract
The theory of difference equations is rich in applications in many branches of the natural sciences. Difference equations with discrete and continuous argument are playing a fundamental role in our understanding of nonlinear phenomena and in processes occurring in various, drastically different systems. To a certain extent, the growing interest in difference equations may be also attributed to their simplicity. Although only quite simple computational and graphical representation tools are necessary to study the behavior of the solutions of difference equations and their bifurcations for changing parameters, it is possible to appreciate the complicated and surprisingly diverse dynamics of difference equations.
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However, it should be noted that despite great analytical difficulties, rapid progress in computer science generates great hopes of success in solving the Navier-Stokes equations numerically (Babenko and Petrovich (1983).
The concept of “dry turbulence” was introduced by Sharkovsky (1983) by analogy with the concept of “dry water” (see Feynman, Leighton, and Sands (1969)).
The system (0.13) may be derived from the Navier-Stokes equations under various simplifying assumptions; however, in the framework of the hydrodynamic statement of the problem, the effect of nonlinearity in the boundary conditions (0.14) remains unclear. Note that with the nonlinearity being carried from the system of equations itself to the boundary conditions, the problem can be handled analytically.
This attractor apparently turns into a finite-dimensional one, if viscosity is taken into account in (0.13)-(0.14). This fact is in good agreement with the results of Babin and Vishik (1982), Ilyashenko (1981), Ladyzhenskaya (1972, 1982), and Lorenz (1963) who established the finite-dimensionality and gave the estimates of the dimensionality of attractors arising in various boundary value problems for the two-dimensional Navier-Stokes equations.
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© 1993 Springer Science+Business Media Dordrecht
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Sharkovsky, A.N., Maistrenko, Y.L., Romanenko, E.Y. (1993). Introduction. In: Difference Equations and Their Applications. Mathematics and Its Applications, vol 250. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1763-0_1
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DOI: https://doi.org/10.1007/978-94-011-1763-0_1
Publisher Name: Springer, Dordrecht
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