# Hysteron

Only a very simple modification of the non-linearity "hysteron" is described below. See [a1] for the general definition and an identification theorem, that is, qualitative conditions under which a "black box" is a hysteron. Consider in the -plane the graphs of two continuous functions , satisfying the inequality , . Suppose that the set is partitioned into the disjoint union of the continuous family of graphs of continuous functions , where is a parameter. Each function is defined on its interval , , and , , that is, the end-points of the graphs of the functions belong to the union of the graphs of and (see Fig.a2.).

Figure: h110430a

Figure: h110430b

Hysteron: Prandtl non-linearity

A hysteron is a transducer with internal states from the segment and with the following input–output operators. The variable output () is defined by the formula

for monotone inputs , . The value of is determined by the initial state to satisfy and the corresponding variable internal state is defined by

For piecewise-monotone continuous inputs the output is constructed by the semi-group identity. The input–output operators can then be extended to the totality of all continuous inputs by continuity (see [a1]). The operators , are defined for each continuous input and each initial state. They are continuous as operators in the space of continuous functions with the uniform metric (cf. also Metric).

A hysteron is called a Prandtl non-linearity if , ; , , . This non-linearity describes the Prandtl model of ideal plasticity with Young modulus and elastic limits . The parallel connections of a finite numbers of such elements describe the Besseling model of elasto-plasticity and the continual counterpart describe the Ishlinskii model.

See also Hysteresis.

#### References

[a1] | M.A. Krasnosel'skii, A.V. Pokrovskii, "Systems with hysteresis" , Springer (1989) (In Russian) |

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Hysteron.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Hysteron&oldid=11383