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Hysteresis Operators

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

Abstract

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.

Keywords

Shape Function Input Function Input String Hysteresis Model Superposition Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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.
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  5. 16.
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  6. 18.
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  7. 19.
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  8. 23.
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  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.
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  18. 48.
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  19. 49.
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  20. 50.
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  26. 68.
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  27. 70.
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    This function has been introduced in Everett (1955).Google Scholar
  34. 90.
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  37. 95.
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  41. 104.
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  45. 111.
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  47. 114.
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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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