• ### The Bifurcation and Stability of the Solutions of the Boussinesq Equations

Mohammad Yahia Ibrahim Awajan, Ruwaidiah Idris

School of Informatics and Applied Mathematics, Universiti Malaysia Terengganu (UMT), 21030 Kuala Nerus, Malaysia Terengganu.

Abstract: The study determined the bifurcation and stability of the solutions of the Boussinesq equations as well as the onset of the Rayleigh-Benard convection. The article established the nonlinear theory for this problem using a new notion of bifurcation known as attractor bifurcation. This article considered the theory that comprises the following three perspectives. We initially deal with the problem that bifurcates from the trivial solution an attractor  while the Rayleigh number  intersects the ﬁrst critical Rayleigh number  for all physically boundary conditions, despite the multiplicity of the eigenvalue  for the linear problem. Hence, secondly, the study considered the bifurcated attractor  as asymptotically stable. Lastly, the bifurcated solutions are also structurally stable when the spatial dimension is two, and are classiﬁed as a bifurcated solution as well. Furthermore, the technical method explained here provides a means, which can be adopted for many different problems in bifurcation and other pattern formation that are related.

Keywords: Rayleigh-Benard Convection, Dynamic bifurcation, Boussinesq equation, Rayleigh number, Attractor bifurcation.