||Hysteresis is a nonlinear phenomenon exhibited by systems stemming from various
science and engineering areas. To detect experimentally the presence of hysteresis in a system,
the graph output versus input of the system is plotted for different frequencies of the input. For
hysteresis systems, these graphs converge to a quasi-static limit set when the frequency goes
to zero. Moreover, the quasi-static graph approaches asymptotically a periodic orbit. Thus,
hysteresis is nonlinear phenomenon that can be detected only in the quasi-static regime, that is
when the frequency content of the input goes to zero. The relevance of hysteresis in applications
and the fact that it is essentially a quasi-static phenomenon makes it important to characterize
mathematically the quasi-static regime, which is the purpose of this paper. Although this work
is motivated by hysteresis systems, the tools that are presented are not limited to this class of
systems. For this reason, the systems that we consider are seen as operators that map an input
signal and initial condition to an output signal, all of them belonging to some specified sets.
The main result of this paper is a new criterion for the existence, uniqueness and mathematical
characterization of the quasi-static regime.