## Mathematical Models of HysteresisThe purpose of this book is to describe in sufficient detail the mathematical models of hysteresis nonlinearities with "nonlocal memories. " The distinct feature of these nonlinearities is that their future states depend on past histories of input variations. It turns out that memories of hysteresis nonlinearities are quite selective. Indeed, experiments show that only some past input extrema (not the entire input variations) leave their marks upon future states of hysteresis nonlinearities. Thus, special mathematical tools are needed to describe nonlocal selective memories of hysteresis nonlinearities. The origin of such tools can be traced back to the landmark paper of Preisach. The book is primarily concerned with Preisach-type models of hysteresis. All these models have a common generic feature; they are constructed as superposi tions of simplest hysteresis nonlinearities-rectangular loops. The discussion is by and large centered around the following topics: various generalizations and extensions of the classical Preisach model (with special emphasis on vector generalizations), finding of necessary and sufficient conditions for the represen tation of actual hysteresis nonlinearities by various Preisach-type models, solution of identification problems for these models, and numerical implementa tion and experimental testing of Preisach-type models. Although the study of Preisach-type models constitutes the main subject of the book, some effort is also made to establish some interesting connections between these models and such topics as the critical state model for superconducting hysteresis, the classi cal Stoner-Wohlfarth model for vector magnetic hysteresis, thermal activation type models for viscosity, magnetostrictive hysteresis and neural networks. |

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### Contents

The Classical Preisach Model of Hysteresis | 1 |

Generalized Scalar Preisach Models of Hysteresis | 64 |

202 | 185 |

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According applications ascending assume B)dadß branch chapter classical Preisach model clear computed conclude condition Consequently Consider constant corresponding decreased defined definition depend derive described determined diagram differential direction discussion energy losses equal equation experimental experimental data expression extrema fact final first-order transition curves formula function given hysteresis nonlinearities hysteretic identification problem input variations instant integral integral equation limiting magnetic field matching mathematical means memories minor loops models of hysteresis monotonically increased natural negative numerical obtain occurs operators output output value particles particular past positive possible problem proof prove reaches reason reduced relaxation representation represented respectively restricted result reversal curves saturation scalar second-order shown in Fig similar solution substituting theorem tion transducer triangle vary vector hysteresis vector Preisach models vertical wiping-out property