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.

Hysteresis phenomena are common in numerous physical, mechanical, ecological and biological systems. They reflect memory effects and process irreversibility. The use of hysteresis operators (hysterons) offers an approach to macroscopic modelling of the dynamics of phase transitions and rheological systems. The applications cover processes in electromagnetism, elastoplasticity and population dynamics in particular. Hysterons are also typical elements of control systems where they represent thermostats and other discontinuous controllers with memory. The book offers the first systematic mathematical treatment of hysteresis nonlinearities. Construction procedures are set up for hysterons in various function spaces, in continuous and discontinuous cases. A general theory of variable hysterons is developed, including identification and stability questions. Both deterministic and non-deterministic hysterons are considered, with applications to the study of feedback systems. Many of the results presented - mostly obtained by the authors and their scientific group - have not been published before. The book is essentially self contained and is addressed both to researchers and advanced students.