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Hysteresis (from the Greek  $  \upsilon \sigma \tau \varepsilon \rho \varepsilon \sigma \iota \zeta $: 
a coming short) is the collective name for a class of non-linear natural phenomenon which arise in mechanics, physics, etc., and have certain common features. The best known examples are plastic hysteresis in mechanics and ferro-magnetic hysteresis in physics, but many other types of hystereses are also important. A general mathematical theory which adequately describes phenomenological models of hysteresis and is convenient for the analysis of closed-loop systems with hysteresis non-linearities is presently (1996) under construction [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]].

In this theory, a hysteresis non-linearity is treated as a transducer, with an input, an output and an internal state. The variable output and internal state for  $  t \geq  t _ {0} $
are uniquely defined by the internal state at the moment  $  t _ {0} $
and the value of the input on  $  [ t _ {0} ,t ] $. 
The relationships  "input–internal state"  and  "input–output"  for a fixed internal state at the initial moment are operators in suitable function spaces. Usually, in hysteresis models the output depends not only on the value of the input but also on the direction of the input variation and the input–output relationships are, at least in the first approximation, independent from a time scale (the rate-independence property).

The main types of hysteresis non-linerities are:

1) Elementary hysteresis non-linearities: the [[Hysteron|hysteron]]; in particular, play and generalized play, stop or Prandtl non-linearity; Duhem non-linearity; [[Non-ideal relay|non-ideal relay]] or thermostate non-linearity; multi-dimensional play and stop; etc.

2) Hysteresis non-linearities treated as block-diagrams aggregated from elementary hysteresis non-linearities. Especially important are non-linearities admitting spectral decompositions into (often infinite) system of parallelly connected elementary non-linearities. These include the Preisach–Giltay model of magnetism and the Mayergoyz model of magnetism; the Besseling–Ishlinskii model in plasticity, and many others.

3) Differential models. These models include vibro-stable ordinary differential equations with delimiters ([[#References|[a1]]], [[#References|[a3]]]), the Bliman dry friction model and the Sorinedry friction model. These models are connected with other branches of mathematics, e.g. stochastic differential equations (cf. also [[Stochastic differential equation|Stochastic differential equation]]).

4) Non rate-independent hysteresis. The general investigation of such models is emerging (1996).

The operator properties of the non-linearities mentioned above can be investigated in detail. These operators are usually (with an exception of relay) continuous in certain function spaces, which reflects robustness to noises of the underlying physical phenomena; they have different monotonicity properties, etc. These properties allow one to describe the dynamics of closed systems by ordinary or partial differential equations with hysteresis operators and can be analyzed by topological methods. The initial value problem is mostly studied and various problems on forced and auto-oscillations, such as bifurcation at zero and at infinity (including the [[Hopf bifurcation|Hopf bifurcation]]), asymptotic and numerical methods (including an averaging principle), etc. have also been investigated.

See also [[Hysteron|Hysteron]]; [[Non-ideal relay|Non-ideal relay]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.A. Krasnosel'skii,   A.V. Pokrovskii,   "Systems with hysteresis" , Springer  (1989)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.D. Mayergoyz,   "Mathematical models of hysteresis" , Springer  (1991)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Visintin,   "Differential models of hysteresis" , ''Appl. Math. Sci.'' , '''111''' , Springer  (1994)</TD></TR></table>