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Conference publicationsAbstractsXXIII conferenceLaminar-turbulent transition in generalized A.N.Kolmogorov problem for incompressible fluid flow on 3D torusFRC IC, ISA RAS, lab. 11-3 ''Chaotic Dynamic systems'', senior staff scientist, 117312, Moscow, pr-t 60-let Oktyabrya, 9. 8(495)998-7683, evstigneevnm@yandex.ru 1 pp. (accepted)The work is dedicated to the qualitative analysis of laminar-turbulent transition. The generalized A.N.Kolmogorov problem is posed on 3D torus for Navier Stokes equations. Projector-Operator is constructed and the Galerkin system is considered for trigonometric polynomials and for S-splines by using Galerkin-Petrov method: \begin{equation} A \frac{d \mathbf{\hat{u} }}{d t}+\mathcal{P}[\mathcal{B}(\mathbf{\hat{u}}, \mathbf{\hat{u}})] =\frac{1}{R}C\mathbf{\hat{u}}+\mathbf{\hat f}. \end{equation} Here: $\mathbf{\hat u}$ - coefficients for vector-function of velocity, $\mathbf{\hat f}$ - coefficients for external force, $A$ - mass matrix, $\mathcal{B}$ - tensor with dimension 3, $\mathcal{P}$ - projector, $C$ - Laplace matrix, $R$ - Reynolds number. Let Galerkin system on trigonometric polynomials with dimension $k: 0 0$: $\mathbf{\hat u} \in Im^3$.}\par Hence, the solution is symmetric about $(0,0,0)$ of the domain. The following bifurcations are found with different Reynolds numbers for systems (1) and (2): \begin{equation} PF \to C \to T_2 \to T_3 \to T_3 \times 2 \to Ch. \end{equation} Here: $PF$ - pitchfork, $C$ - Cycle (Andronov-Hopf bifurcation), $T_n$ - torus of dimension $n$; $\times 2$ - period doubling bifurcation; $Ch$ - Chaos. Further investigation is on the way.
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