In this work, a cylindrically symmetric gradient-refractivity two-dimensional electromagnetic system is constructed by using the magnetic metamaterials consisting of an array of ferrite rods. With the change of the bias magnetic field, the different gradient-refractivity systems can be obtained, based on which a tunable flexible beam is demonstrated. Based on the effective-medium theory, the effective electric permittivity and the effective magnetic permeability can be retrieved and thus the effective refractive index is obtained straightforwardly. It is shown that with the variation of the ferrite rod radius, an effective refractivity profile with particular gradient can be realized, which exhibits the electromagnetic “black-hole-like” effect. Especially, the gradient refractivity profile is also designed by introducing the gradient bias magnetic field, which, in principle, results in the refractivity profile with many different gradients. Finally, the propagation of a Gaussian beam in the gradient-refractivity system is simulated by using the multiple scattering theory. A few different phenomena are observed such as the “black-hole” effect, the interior beam deflection, the exterior beam deflection, and the beam splitting. Furthermore, the functionalities can be switched between each other by controlling the bias magnetic field and adding an additional degree of freedom for beam propagation.