Path dependent versus energy dependent elements of hysteresis

Hysterons and their interactions are often used as a simple model to study memory formation in cyclically driven materials. The default hysteron model is based on a double-well potential with two adjacent energy minima. As the system is driven periodically, it lands in one of the two available ground states solely based on its energy and the evolution of the landscape as one of the minima turns into a saddle point and disappears when the driving amplitude goes above a critical value. However, in many mechanical systems with buckling modes such as spring networks and beams, the evolution of the double-well cyclic driving is qualitatively different and the ground state in which the system finds itself depends on its pathway rather than its energy. Here we introduce a generalized hysteron model that consists of a particle connected to two harmonic, slightly compressed springs and coupled to the environment via a weak oscillatory force. We show that there is a crossover frequency below which the hysteron is path-dependent and above which the hysteron becomes energy-dependent.