The article studies the localized vibration mode of a system consisting of two pendulums connected elastically and placed in a magnetic field. The inertial properties of these pendulums are significantly different, which leads to the phenomenon of vibration localization. The model of the pendulum system takes into account the resistance of the medium and damping force created by the elastic element. The analytical investigation are carried out using the multiple scales method, then numerical modeling is used to study the effect of various parameters and initial conditions on the complex dynamics of the system, which can be both regular and chaotic. The stability of the vibration mode is assessed using a numerical-analytical approach that is related to the Lyapunov stability criterion. As a result, areas of parameters have been identified that ensure the existence of the pendulum system regular behavior when the system parameters are changed, in the form of the stable localized vibration mode.