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=                             Hysteresis                             =
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                             Introduction
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Hysteresis is the dependence of the state of a system on its history.
For example, a magnet may have more than one possible magnetic moment
in a given magnetic field, depending on how the field changed in the
past. Plots of a single component of the moment often form a loop or
hysteresis curve, where there are different values of one variable
depending on the direction of change of another variable. This history
dependence is the basis of memory in a hard disk drive and the
remanence that retains a record of the Earth's magnetic field
magnitude in the past. Hysteresis occurs in ferromagnetic and
ferroelectric materials, as well as in the deformation of rubber bands
and shape-memory alloys and many other natural phenomena. In natural
systems it is often associated with irreversible thermodynamic change
such as phase transitions and with internal friction; and dissipation
is a common side effect.

Hysteresis can be found in physics, chemistry, engineering, biology,
and economics. It is incorporated in many artificial systems: for
example, in thermostats and Schmitt triggers, it prevents unwanted
frequent switching.

Hysteresis can be a dynamic lag between an input and an output that
disappears if the input is varied more slowly; this is known as
'rate-dependent' hysteresis. However, phenomena such as the magnetic
hysteresis loops are mainly 'rate-independent', which makes a durable
memory possible.

Systems with hysteresis are nonlinear, and can be mathematically
challenging to model. Some models such as the Preisach model
(originally applied to ferromagnetism) and the  Bouc-Wen model attempt
to capture general features of hysteresis; and there are also
phenomenological models for particular phenomena such as the
Jiles-Atherton model for ferromagnetism. See also Hysteretic model.


                        Etymology and history
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The term "hysteresis" is derived from , an Ancient Greek word meaning
"deficiency" or "lagging behind". It was coined around 1890 by Sir
James Alfred Ewing to describe the behaviour of magnetic materials.

Some early work on describing hysteresis in mechanical systems was
performed by James Clerk Maxwell. Subsequently, hysteretic models have
received significant attention in the works of Ferenc Preisach
(Preisach model of hysteresis), Louis Néel and Douglas Hugh Everett in
connection with magnetism and absorption. A more formal mathematical
theory of systems with hysteresis was developed in the 1970s by a
group of Russian mathematicians led by Mark Krasnosel'skii.


 Rate-dependent
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One type of hysteresis is a lag between input and output. An example
is a sinusoidal input  that results in a sinusoidal output , but with
a phase lag :
: \begin{align}
X(t) &= X_0 \sin \omega t \\ Y(t) &= Y_0 \sin\left(\omega
t-\varphi\right).
\end{align}
Such behavior can occur in linear systems, and a more general form of
response is
: Y(t) = \chi_\text{i} X(t) + \int_0^{\infty} \Phi_\text{d} (\tau)
X(t-\tau) \, \mathrm{d}\tau,
where \chi_\text{i} is the instantaneous response and \Phi_d(\tau) is
the impulse response to an impulse that occurred \tau time units in
the past.  In the frequency domain, input and output are related by a
complex 'generalized susceptibility' that can be computed from \Phi_d;
it is mathematically equivalent to a transfer function in linear
filter theory and analogue signal processing.

This kind of hysteresis is often referred to as 'rate-dependent
hysteresis'. If the input is reduced to zero, the output continues to
respond for a finite time. This constitutes a memory of the past, but
a limited one because it disappears as the output decays to zero. The
phase lag depends on the frequency of the input, and goes to zero as
the frequency decreases.

When rate-dependent hysteresis is due to dissipative effects like
friction, it is associated with power loss.


 Rate-independent
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Systems with 'rate-independent hysteresis' have a 'persistent' memory
of the past that remains after the transients have died out. The
future development of such a system depends on the history of states
visited, but does not fade as the events recede into the past. If an
input variable  cycles from    to  and back again, the output   may be
initially but a different value   upon return. The values of   depend
on the path of values that   passes through but not on the speed at
which it traverses the path. Many authors restrict the term hysteresis
to mean only rate-independent hysteresis. Hysteresis effects can be
characterized using the Preisach model and the generalized