Hysteresis Operators

  • Martin Brokate
  • Jürgen Sprekels
Part of the Applied Mathematical Sciences book series (AMS, volume 121)


Our first approach to the phenomenon of hysteresis is a direct one. We consider the hysteresis diagrams and loops as they present themselves, without trying to understand how their respective forms might follow from general physical principles such as, for example, the universal balance laws for mass, linear momentum and internal energy1; instead, we assume the form of the hysteresis loops as given, and we study them from a purely mathematical point of view. The central notion will be that of a hysteresis operator, usually denoted by W. Defined in accordance with the rules and functions that accompany a given hysteresis model, a hysteresis operator W maps input functions v = v(t) into output functions w = w(t) (we denote the independent variable by t since it will always represent a time variable). Using this formulation, we will examine the structures and the resulting memory effects of various kinds of hysteresis models. In addition, we establish the relevant connections between the different types of hysteresis operators and basic notions of calculus such as continuity and function spaces.


Shape Function Input Function Input String Hysteresis Model Superposition Operator 
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  1. 2.
    We also refer to the recent collection of lecture notes edited by Visintin (1994b).Google Scholar
  2. 4.
    In the literature, this operator is often referred to as the Ishlinskii operator or the Ishlinshii transducer, see Krasnoselskii-Pokrovskii (1989). We prefer to name it after Prandtl whose paper appeared more than a decade before Ishlinskii’s who studied this operator in the early forties.Google Scholar
  3. 5.
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  4. 14.
    See Krasnoselskii-Pokrovskii (1989), Section 34.2.Google Scholar
  5. 16.
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  6. 18.
    See Krasnoselskii-Pokrovskii (1989), p. 158, in their discussion of vector hysteresis models. Actually, this has only recently been used in the analysis of scalar hysteresis operators.Google Scholar
  7. 19.
    The estimate (3.21) constitutes the first result published by the group around Krasnoselskii on hysteresis operators, see Krasnoselskii et al. (1970).Google Scholar
  8. 23.
    See Krasnoselskii-Pokrovskii (1989), p. 365.Google Scholar
  9. 27.
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  10. 29.
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  11. 30.
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  12. 31.
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  17. 47.
    The generalized play operator serves as the basic hysteresis model in Krasnoselskii-Pokrovskii (1989), where an extensive discussion under minimal assumptions can be found. The connection with the Preisach model has been noted in Krasnoselskii-Pokrovskii (1976).Google Scholar
  18. 48.
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  19. 49.
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  20. 50.
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  21. 52.
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  26. 68.
    After Wöhler (1870).Google Scholar
  27. 70.
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  28. 71.
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    This function has been introduced in Everett (1955).Google Scholar
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    This feature is often referred to as the congruency property of the Preisach model, see Mayergoyz (1985).Google Scholar
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    See Remark 2.4.15.Google Scholar
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    For treatments of the aspects of numerical approximation and identification, we refer the interested reader to the works of Verdi-Visintin (1985,1989), Hoffmann-Sprekels-Visintin (1988), Hoffmann-Meyer (1989), Brokate (1990c), Mayergoyz (1991), Hütter (1991) and Verdi (1994).Google Scholar
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    See Proposition 2.2.16.Google Scholar
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    This result is originally due to Krejčí (1986a).Google Scholar
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    The results of this section are due to Brokate-Dreßler-Krejčí (1995b).Google Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Martin Brokate
    • 1
  • Jürgen Sprekels
    • 2
  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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