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.
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References
Carr, J.: Applications of Center Manifold Theory. Applied Mathematical Sciences. Springer-Verlag New York(1982)
Gelfgat, A.Y.: Different modes of Rayleigh-Bénard instability in two and three-dimensional rectangular enclosures. J. Comp. Phy. 156, 300–324(1999)
Guckenheimer, J. Holmes, P. J.: Non-linear oscillations, dynamical systems, and bifurcations of vector fields. Springer Science and Business Media(2013)
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)
Haragus, M. Iooss, G.: Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. Springer Science and Business Media(2010)
Henry, D.: Geometric theory of semi-linear parabolic equations. Springer-Verlag New York(1981)
Kelley, A.: Stability of the center-stable manifold. J. Math. Anal. Appl. 18, 336–344(1967)
Kelley, A.: The stable, center-stable, center, center-unstable, unstable manifolds. J. Differential Equations 3, 546–570(1967)
Knobloch, H. W. Aulbach, B.: The role of center manifolds in ordinary differential equations. Equadiff 5, 179–189(1982)
Perko, L.: Differential Equations and Dynamical systems. Springer Science and business media(2013)
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)
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)
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)
Sijbrand, J.: Properties of center manifolds. Trans. Amer. Math. Soc. 289, 431–469(1985)
Scarpellini, B.: Center manifolds of infinite dimensions: Main results and applications. Z. Angew. Math. Phys. 42, 1–32(1991)
Vanderbauwhede, A. Iooss, G.: Center manifold theory in infinite dimensions. Springer(1992)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical systems and chaos. Springer-Verlag NewYork(1990)
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Siddheshwar, P.G., Manjunath, S., Sushma, T.S. (2019). Solution of the Lorenz Model with Help from the Corresponding Ginzburg-Landau Model. In: Rushi Kumar, B., Sivaraj, R., Prasad, B., Nalliah, M., Reddy, A. (eds) Applied Mathematics and Scientific Computing. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01123-9_6
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DOI: https://doi.org/10.1007/978-3-030-01123-9_6
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