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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Numerical Bifurcation Analysis in 3D Kolmogorov Flow Problem

Journal of Applied Nonlinear Dynamics 8(4) (2019) 595--619 | DOI:10.5890/JAND.2019.12.007

N.M. Evstigneev, N.A. Magnitskii, O.I. Ryabkov

Federal Research Center "Computer Science and Control" of the Russian Academy of Sciences, 117312, pr. 60-letiya Oktyabrya, 9, Moscow, Russia

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Abstract

A three dimensional Kolmogorov problem with extended forcing term for Navier-Stokes equations is considered in the extended periodic domain. The Galerkin{Fourier method is applied and two problem congurations are considered: the symmetry preserving subset of solutions and the full set of solutions. The bifurcation patterns are revealed through the numerical analysis: the eigenvalues of the linearised perturbed system are analyzed (for stationary and periodic solutions) as well as the phase space trajectories that the system generates. Linear stability of the main solution is analyzed. For cubic domain we detected a fast transition to chaos through double Hopf bifurcation. Then the full bifurcation scenario is analyzed for rectangular parallelepiped with double side stretching. The initial stage of laminar-turbulent transition undergoes supercritical pitchfork bifurcation followed by subcritical Hopf bifurction. The system can either go through the series of cycles in Feigenbaum and Sharkovsky ordering or through the bifurcations on invariant tori up to the three and four dimensional tori. The transition to chaos goes either through the emergence of singular attractors of dimension greater than three or through the resonance of tori.

Acknowledgments

The work is supported by the Russian Foundation for Basic Research (grants RFBR 17-07-00116 and 18-29-10008mk).

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