Dynamical hysteresis is a phenomenon which arises in ferromagnetic systems below the critical temperature as a response to adiabatic variations of the external magnetic field. We study the problem in the context of the mean-field Ising model with Glauber dynamics, proving that for frequencies of the magnetic field oscillations of order $N^{-2/3}$, $N$ the size of the system, the “critical” hysteresis loop becomes random.

This paper deals with the asymptotic behavior as $t\to \infty$ of solutions $u$ to the forced Preisach oscillator equation $\ddot{w}\left(t\right)+u\left(t\right)=\psi \left(t\right)$, $w=u+𝒫\left[u\right]$, where $𝒫$ is a Preisach hysteresis operator, $\psi \in L^{\infty }(0,\infty )$ is a given function and $t\ge 0$ is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function $\psi$ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show...