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Solution of the Lorenz Model with Help from the Corresponding Ginzburg-Landau Model

  • P. G. SiddheshwarEmail author
  • S. Manjunath
  • T. S. Sushma
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

Centre manifold theory, a useful tool in the study of dynamical systems, plays a crucial role in analysing the stability of the system. In the paper the three-dimensional manifold arising in the study of Rayleigh-Bénard-Brinkman convection in enclosures is reduced to a unidimensional manifold using a transformation dictated by the centre manifold theorem. Such a reduction is possible since the Lorenz model is autonomous. The advantage in this procedure is that the intractable Lorenz model gets reduced to a tractable Ginzburg-Landau equation and hence facilitates an analytical study of heat transport.

Keywords

Rayleigh-Bénard-Brinkman convection Center manifold Enclosure 

References

  1. 1.
    Carr, J.: Applications of Center Manifold Theory. Applied Mathematical Sciences. Springer-Verlag New York(1982)Google Scholar
  2. 2.
    Gelfgat, A.Y.: Different modes of Rayleigh-Bénard instability in two and three-dimensional rectangular enclosures. J. Comp. Phy. 156, 300–324(1999)CrossRefGoogle Scholar
  3. 3.
    Guckenheimer, J. Holmes, P. J.: Non-linear oscillations, dynamical systems, and bifurcations of vector fields. Springer Science and Business Media(2013)Google Scholar
  4. 4.
    Guillet, C. Mare, T. Nguyen, C. T.: Application of a non-linear local analysis method for the problem of mixed convection instability. Int. J. Non Linear Mech. 42, 981–988(2007)CrossRefGoogle Scholar
  5. 5.
    Haragus, M. Iooss, G.: Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. Springer Science and Business Media(2010)Google Scholar
  6. 6.
    Henry, D.: Geometric theory of semi-linear parabolic equations. Springer-Verlag New York(1981)CrossRefGoogle Scholar
  7. 7.
    Kelley, A.: Stability of the center-stable manifold. J. Math. Anal. Appl. 18, 336–344(1967)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Kelley, A.: The stable, center-stable, center, center-unstable, unstable manifolds. J. Differential Equations 3, 546–570(1967)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Knobloch, H. W. Aulbach, B.: The role of center manifolds in ordinary differential equations. Equadiff 5, 179–189(1982)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Perko, L.: Differential Equations and Dynamical systems. Springer Science and business media(2013)Google Scholar
  11. 11.
    Platten, J. K. Marcoux, M. Mojtabi, A.: The Rayleigh-Bénard problem in extremely confined geometries with and without the Soret effect. Comptes Rendus Mecanique 335, 638–654(2007)CrossRefGoogle Scholar
  12. 12.
    Pliss, V. A.: A reduction principle in the theory of stability of motion. Izv. Akad. Nauk S.S.S.R. Mat. Ser. 6, 1297–1324(1964)Google Scholar
  13. 13.
    Siddheshwar, P. G. Meenakshi, N.: Amplitude equation and heat transport for Rayleigh-Bénard convection in Newtonian liquids with nanoparticles. Int. J. Appl. and Comp. Math. 2, 1–22(2015)zbMATHGoogle Scholar
  14. 14.
    Sijbrand, J.: Properties of center manifolds. Trans. Amer. Math. Soc. 289, 431–469(1985)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Scarpellini, B.: Center manifolds of infinite dimensions: Main results and applications. Z. Angew. Math. Phys. 42, 1–32(1991)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Vanderbauwhede, A. Iooss, G.: Center manifold theory in infinite dimensions. Springer(1992)Google Scholar
  17. 17.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical systems and chaos. Springer-Verlag NewYork(1990)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • P. G. Siddheshwar
    • 1
    Email author
  • S. Manjunath
    • 2
  • T. S. Sushma
    • 2
  1. 1.Department of MathematicsBangalore UniversityBangaloreIndia
  2. 2.Department of MathematicsB. N. M. Institute of TechnologyBangaloreIndia

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