Hysteresis phenomena are common in numerous physical, mechanical, ecological and biological systems. The use of hysteresis operators (hysterons) offers an approach to macroscopic modelling of the dynamics of phase transitions and rheological systems.

1 Static Hysteron.- 1. Short-memory transducer.- 2. Generalized play.- 3. Hysteron.- 4. Canonical representation of hysteron and proof of Theorem 3.2.- 5. Distances.- 6. Various input spaces.- 2 Identification Theorem.- 7. Identification problem.- 8. Proof of Theorem 7.1.- 9. ? - identiflability.- 10. Approximate construction of hysteron.- 3 Vibro-Correct Differential Equations and Variable Hysterons.- 11. Necessary condition of vibro-correctness.- 12. Sufficient condition of vibro-correctness.- 13. Vibro-solutions.- 14. Equations with constraints.- 15. Variable hysteron.- 4 Multidimensional Hysterons.- 16. Multidimensional play and stop defined on smooth inputs.- 17. Strictly convex characteristics.- 18. Polyhedral characteristics.- 19. Arbitrary convex characteristics.- 20. Inputs with summable derivatives.- 21. Vibro-correct equations with vector input.- 22. Equations with vector inputs and smooth constraints.- 5 Discontinuous Nonlinearities.- 23. Static elements.- 24. Elements with monotone characteristics.- 25. Elements with multi-valued characteristics.- 26. Closures of static element.- 27. Weak closures and convexification procedure.- 28. Relay.- 6 Self-Magnetization Phenomenon.- 29. Madelung’s hysterons.- 30. Proofs of Theorems 29.1 and 29.2.- 31. Response to small perturbations of the input.- 32. Closure modulo sets of Wiener measure zero.- 7 Complex Hysteresis Nonlinearities.- 33. Parallel connections and bundles of hysterons.- 34. Sequential connections of hysterons.- 35. Ishlinskii’s material.- 36. Properties of Ishlinskii’s transducer.- 37. Finite systems of relays.- 38. Continual systems of relays.- 39. Rheological models.- Bibliographic comments.- References.

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