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Convergence of attractors for the simplified magnetic Bénard equations

Published online by Cambridge University Press:  26 September 2008

Hitoshi Imai ,
Naoyuki Ishimura  and
Masaaki Nakamura
Hitoshi Imai
Affiliation:
Faculty of Engineering, University of Tokushima, Tokushima, Tokushima 770, Japan
Naoyuki Ishimura
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Hongo, Tokyo 113, Japan
Masaaki Nakamura
Affiliation:
College of Science and Technology, Nihon University, Kanda-Surugadai, Tokyo 101, Japan
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Abstract

The simplified magnetic Bénard; equations (SMB) are derived from the two-dimensional magnetic Bénard system and extend the well-known Lorenz equations (L). (SMB) contains a parameter Q associated with the magnetic field. When Q = 0, the long time dynamics of (SMB) agrees with that of (L). In this paper, we investigate the long time behaviour of (SMB) as Q→0. We prove analytically that the global attractor of (SMB) converges to the one of (L) as Q→0 upper semicontinuously in the Hausdorff sense. However, numerical computation indicates that generically there is no continuity in the dimension of the attractor.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 1 , February 1996 , pp. 53 - 62
Copyright
Copyright © Cambridge University Press 1996

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