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Dynamical transitions of a low-dimensional model for Rayleigh-Bénard convection under a vertical magnetic field. (English) Zbl 1415.34073
Summary: 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.

##### MSC:
 34C23 Bifurcation theory for ordinary differential equations 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 35Q35 PDEs in connection with fluid mechanics 76E06 Convection in hydrodynamic stability
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