The instability and attractor bifurcation for the magnetic fluid between two concentric rotating cylinders

In this paper, we present the magnetohydrodynamic (MHD) equations for the flow between two concentric rotating cylinders. The primary objective is to investigate the instability and attractor bifurcation of the simplified governing equations. The analysis is based on recently developed theories, including the bifurcation theory for nonlinear dynamical systems and differential operators on 3-D Riemann manifolds. We begin by formulating the MHD equations coupled with temperature in cylindrical coordinates, defining the control parameters, which are related with the magnetic field, temperature difference, and the angular speeds of the inner and outer cylinders. It is shown that the system always undergoes a dynamical transition as control parameter equation crosses the first critical value $\lambda _{C}$ . The type of transition is determined by the multiplicity of the critical eigenvalue. Then, we provide approximate expressions for the bifurcation and the structure of the attractor, along with necessary explanations for the obtained results. Finally, we present numerical computations to interpret the results of the theorem and provide a visual analysis of the velocity field and temperature distribution for the four stable nodes of the attractor through graphical representation.