Modeling and Analysis of Duhem Hysteresis Operators With Butterfly Loops

In this work, we study and analyze a class of Duhem hysteresis operators that can exhibit butterfly loops. We study first the consistency property of such operator, which corresponds to the existence of an attractive periodic solution when the operator is subject to a periodic input signal. Subsequently, we study the two defining functions of the Duhem operator such that the corresponding periodic solutions can admit a butterfly input-output phase plot. We present a number of examples where the Duhem butterfly hysteresis operators are constructed using two zero-level set curves that satisfy some mild conditions.