Why are some hysteresis loops shaped like a butterfly?

The contribution of this paper is a framework for relating butterfly-shaped hysteresis maps to simple (single-loop) hysteresis maps, which are typically easier to model and more amenable to control design. In particular, a unimodal mapping is used to transform simple loops to butterfly loops. For the practically important class of piecewise monotone hysteresis maps, we provide conditions for producing butterfly-shaped maps and examine the properties of the resulting butterflies. Conversely, we present conditions under which butterfly-shaped maps can be converted to simple piecewise monotone hysteresis maps to facilitate hysteresis compensation and control design. Examples of a preloaded two-bar linkage mechanism and a magnetostrictive actuator illustrate the theory and its utility for understanding, modeling, and controlling systems with butterfly-shaped hysteresis.