Nonlinear Feedback Models of Hysteresis JINHYOUNG OH, BOJANA DRINCIC, and DENNIS S. BERNSTEIN BACKLASH, BIFURCATION, AND MULTISTABILITY

ysteresis arises in diverse applications, including structural mechanics, aerodynamics, and electromagnetics [1]–[4]. The word “hysteresis” connotes lag, although this nomenclature is misleading since delay per se is not the mechanism that gives rise to hysteresis. As discussed in [5], hysteresis is a quasi-static phenomenon in which a sequence of periodic inputs produces a nontrivial input-output loop as the period of the input increases without bound. In this limit, the input can be viewed as completing its cycle increasingly slowly. This limiting input-output loop is due to the existence, for a given constant input, of multiple equilibria that map to distinct output values (whether or not the output map is one-to-one). Intuitively speaking, as the input slowly increases, the state of the system is attracted to input-dependent equilibria that are different from the attracting equilibria when the input slowly decreases. The existence of multiple attracting equilibria is called multistability [5]–[10]. Multistability corresponds to the intuitive notion that, under asymptotically slowly changing inputs, the state of a hysteretic system converges to equilibria that belong to an equilibrium set that has a multivalued structure, that is, multiple state equlibria can exist for a given constant input. To the extent that this phenomenon is history dependent, it is appropriate to say that a hysteretic system has memory. For a single-input, single-output system, hysteresis is the persistence of a nondegenerate input-output closed curve as the frequency of excitation tends toward dc. Such systems are necessarily nonlinear since a linear system cannot have a nontrivial limiting closed input-output curve at asymptotically low frequencies. A hysteretic system whose periodic input-output map is the same for all frequencies is called rate independent; when the periodic input-output map is different for different frequencies, the hysteretic system is rate dependent. All of the hysteresis examples in this article exhibit rate-dependent hysteresis. The concept of ratedependent hysteresis is thus central to the study of hysteresis arising in nonlinear feedback models. Nonlinear Feedback Models of Hysteresis