Dynamical Hysteresis without Static Hysteresis: Scaling Laws and Asymptotic Expansions

We study dynamical hysteresis in a simple class of nonlinear ordinary differential equations, namely first-order equations subject to sinusoidal forcing. The assumed nonlinearities are such that the area of the hysteresis loop vanishes as the forcing frequency tends to zero; in other words, there is no static hysteresis. Using regular and singular perturbation techniques, we derive the first term in the asymptotic expansion for the loop area as a function of the driving frequency, in the limit of both large and small frequency. Although the theory was originally motivated by experiments on bistable semiconductor lasers, it is applied here to explain (and in some cases, to correct) the scaling laws that were recently reported in numerical studies of mean-field kinetic Ising models of ferromagnets.