Hysteresis is an exciting and mathematically challenging phenomenon
that oc curs in rather different situations: jt, can be a byproduct
offundamental physical mechanisms (such as phase transitions) or
the consequence of a degradation or imperfection (like the play in
a mechanical system), or it is built deliberately into a system in
order to monitor its behaviour, as in the case of the heat control
via thermostats. The delicate interplay between memory effects and
the occurrence of hys teresis loops has the effect that hysteresis
is a genuinely nonlinear phenomenon which is usually non-smooth and
thus not easy to treat mathematically. Hence it was only in the
early seventies that the group of Russian scientists around M. A.
Krasnoselskii initiated a systematic mathematical investigation of
the phenomenon of hysteresis which culminated in the fundamental
monograph Krasnoselskii-Pokrovskii (1983). In the meantime, many
mathematicians have contributed to the mathematical theory, and the
important monographs of 1. Mayergoyz (1991) and A. Visintin (1994a)
have appeared. We came into contact with the notion of hysteresis
around the year 1980."
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