  In this work we study and analyze a class of Duhem hysteresis operators that
can exhibit butterfly loops. We study firstly 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.
