Interacting hysterons with asymptotically small or large spans

Models of interacting hysteretic elements, called hysterons, capture the sequential response and complex memory effects in a wide range of complex systems and can guide the design of intelligent metamaterials. However, even simple models with few hysterons feature a bewildering number and variety of behaviors. Here we study the hysteron model in two physically relevant limits, where the response of a hysteron system is easier to understand. First, when the hysteron span-the gap between its two hysteretic transitions-dominates all other scales, the range of pathways encoded in transition graphs becomes limited because many avalanches are absent. Second, when the hysteron span becomes vanishingly small, hysterons behave as interacting binary spins, which require avalanches in order to exhibit nontrivial pathways. Finally we show that hysterons can be mimicked by pairs of strongly interacting spins, such that collections of n interacting hysterons can be mapped to 2n interacting spins, albeit via highly specific interactions. Altogether, our work provides a deeper understanding of the role of the hysteron parameters on their collective behavior, and points to connections and differences between spin- and hysteron-based models of complex matter.