From one-dimensional to infinite-dimensional dynamical systems: Ideal turbulence. (English. Ukrainian original) Zbl 0936.37021
Ukr. Math. J. 48, No. 12, 1817-1842 (1996); translation from Ukr. Mat. Zh. 48, No. 12, 1604-1627 (1996).
Summary: There is a very short chain that joins dynamical systems with the simplest phase space (real line) and dynamical systems with the “most complicated” phase space containing random functions, as well. This statement is justified in this paper. By using “simple” examples of dynamical systems (one-dimensional and two-dimensional boundary-value problems), we consider notions that generally characterize the phenomenon of turbulence-first of all, the emergence of structures (including the cascade process of emergence of coherent structures of decreasing scales) and self-stochasticity.
MSC:
| 37H10 | Generation, random and stochastic difference and differential equations |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |
| 39A10 | Additive difference equations |
Keywords:
random functions; turbulence-first; cascade process; coherent structures; self-stochasticityReferences:
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