Dynamical Transitions of a Low-Dimensional Model for Rayleigh-Bénard Convection under a Vertical Magnetic Field


In this article, we study the dynamic transitions of a low-dimensional dynamical system for the Rayleigh-Bénard convection subject to a vertically applied magnetic field. Our analysis follows the dynamical phase transition theory for dissipative dynamical systems based on the principle of exchange of stability and the center manifold reduction. We find that, as the Rayleigh number increases, the system undergoes two successive transitions: the first one is a well-known pitchfork bifurcation, whereas the second one is structurally more complex and can be of different type depending on the system parameters. More precisely, for large magnetic field, the second transition is of continuous type and gives to a stable limit cycle; on the other hand, for low magnetic field or small height-to-width aspect ratio, a jump transition occurs where an unstable periodic orbit eventually collides with the stable steady state, leading to the loss of stability at the critical Rayleigh number. Finally, numerical results are presented to corroborate the analytic predictions.


Mathematics and Statistics

Research Center/Lab(s)

Center for High Performance Computing Research


The work of D. Han was supported in part by ONR grant N00014-15-1-2385, by the Research Fund of Indiana University and by a seed fund of Material Research Center at Missouri University of Science and Technology; M. Hernandez was supported in part by the National Science Foundation (NSF) grant DMS-1515024, and by the Office of Naval Research (ONR) grant N00014-15-1-2662; Q. Wang was supported by the NOAA HFIP funding (Award NA16NWS4680026). The authors wish to thank Dr. Shouhong Wang for helpful discussions.

Keywords and Phrases

Aspect ratio; Bifurcation (mathematics); Chaos theory; Magnetic fields; Natural convection; Center manifold reductions; Centre manifold reduction; Critical Rayleigh number; Dynamical phase transition; Dynamical transition; Low-dimensional dynamical systems; Principle of exchange of stabilities; Rayleigh; Dynamical systems; Dynamical transitions; Rayleigh-Benard convection

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Article - Journal

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© 2018 Elsevier, All rights reserved.

Publication Date

01 Sep 2018