Hysteresis in the Linearized Landau-Lifshitz Equation

The Landau-Lifshitz equation describes the behaviour of magnetization inside a ferromagnetic object. It is known that the Landau-Lifshitz equation has an infinite number of stable equilibrium points. The existence of multiple stable equilibria is closely related to hysteresis. This is a phenomenon that is often characterized by a looping behaviour; however, the existence of a loop is not sufficient to identify hysteretic systems, but is defined more precisely as the presence of looping as the frequency of the input goes to zero. We describe these two approaches to identification of hysteresis and demonstrate that both the linear and nonlinear Landau-Lifshitz equations exhibit hysteresis. The presence of hysteresis in the linear Landau-Lifshitz equation, as well as in a simpler system also described here, indicates that nonlinearity is not necessary for hysteresis to exist.