Abstract
A two-dimensional Boussinesq fluid with nonlinear interaction between Rayleigh–Bénard convection and an external magnetic field is investigated numerically and analytically. A simplified model consisting of a fifth-order system of nonlinear ordinary differential equations with five parameters is introduced and integrated numerically in certain parameter regions. Various types of bifurcations from periodic solutions are found numerically: period-doubling bifurcation, heteroclinic bifurcation, intermittency, and saddle-node bifurcation. A normal form equation is also derived from the fifth-order system, and center manifold theory is applied to it. An expression for the renormalized Holmes–Melnikov boundary for the evaluation of the numerical results is given. It is shown from the normal form equation that each property of the two phase portraits described by the Duffing equation and the van der Pol equation emanates from one common attractor in the five-dimensional space of the fifth-order system.
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