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Bifurcation Analysis Software and Chaotic Dynamics for Some Problems in Fluid Dynamics Laminar–Turbulent Transition

Author

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  • Nikolay M. Evstigneev

    (Federal Research Center ”Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia)

  • Nikolai A. Magnitskii

    (Federal Research Center ”Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia)

Abstract

The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which include system spectral and algebraic problems, specific nonlinear numerical methods, and computational implementation on parallel architectures. The software structure that is required to perform numerical bifurcation analysis for large-scale systems was considered in the paper. The software structure, specific features that are used for successful bifurcation analysis, globalization strategies, stabilization, and high-precision implementations are discussed. We considered the bifurcation analysis in the initial boundary value problem for a system of partial differential equations that describes the dynamics of incompressible ABC flow (3D Navier–Stokes equations). The initial stationary solution is characterized by the stability and connectivity to the main solutions branches. Periodic solutions were considered in view of instability transition problems. Finally, some questions of higher dimensional attractors and chaotic regimes are discussed.

Suggested Citation

  • Nikolay M. Evstigneev & Nikolai A. Magnitskii, 2023. "Bifurcation Analysis Software and Chaotic Dynamics for Some Problems in Fluid Dynamics Laminar–Turbulent Transition," Mathematics, MDPI, vol. 11(18), pages 1-25, September.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:18:p:3875-:d:1237569
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    References listed on IDEAS

    as
    1. Nikolai A. Magnitskii, 2023. "Universal Bifurcation Chaos Theory and Its New Applications," Mathematics, MDPI, vol. 11(11), pages 1-20, May.
    2. Elhajji, S. & Errachid, M., 2002. "Analysis of a bifurcation problem," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 58(3), pages 231-245.
    3. Rehman, Khalil Ur & Malik, M.Y., 2019. "On Lie symmetry mechanics for Navier–Stokes equations unified with non-Newtonian fluid model: A classical directory," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).
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    Cited by:

    1. Nikolay M. Evstigneev & Taisia V. Karamysheva & Nikolai A. Magnitskii & Oleg I. Ryabkov, 2024. "Disconnected Stationary Solutions in 3D Kolmogorov Flow and Their Relation to Chaotic Dynamics," Mathematics, MDPI, vol. 12(21), pages 1-16, October.

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