1. Representation of Hysteresis with Return Point Memory: Expanding the Operator Basis Gary Friedman Department of Electrical and Computer Engineering Drexel University

2. Hysteresis forms H M D int D ave Form most frequently associated with hysteresis: magnets Ratchets, swimming, molecular motors, etc.

3. Return Point (wiping-out) Memory The internal state variables return when the input returns to its previous extremum. Experimentally observed in: magnetic materials, superconductors, piezo-electric materials, shape memory alloys, absorption Also found in micro-models: Random Field Ising Models (with positive interactions), Sherrington - Kirkpatrick type models, models of domain motion in random potential,

4. How can we represent any hysteresis with wipe-out memory in general? Can we approximate any hysteresis with wipe-out memory? Preisach model represents some hysteresis with wipe-out memory because each bistable relay has wipe-out memory. It also has the property of Congruency which is an additional restriction

5. Congruency Any higher order reversal curve is congruent to the first order reversal curve. All loops bounded between the same input values are congruent. Higher order reversal curves could, in general, deviate from first order reversal curve. These deviations can not be accounted for in the Preisach model.

6. Examples of systems with Return Point Memory, but without Congruency Interacting networks of economic agents Mean-field models in physics Theorem: as long as interactions are positive, such systems have RPM (Jim Senthna, Karin Dahmen) Problem: Not clear if or when model unique model parameters can be identified using macroscopic observations

7. Mapping of history into output of the model Any hysteresis with wipe-out memory can be represented by a mapping of the interface function into the output (Martin Brokate)

8. How can an approximation be devised? Assume both, the given hysteresis transducer and the approximation we seek are sufficiently smooth mappings of history into the output

9. Nth order approximation

10. Building Nth order approximation Key point: as long as operators are functions of elementary rectangular loop operator, the system retains Return Point Memory Matryoshka threshold set

11. Higher order elementary operators Second order elementary operator example

12. Why use only Matryoshka threshold sets? Non-Matryoshka operators can be reduced to lower order Matryoshka operators

13. Nth order Preisach model Loops appear only after Nth order reversal. Reversal curves following that are congruent to Nth order reversal as long they have the same preceding set of first N reversals

14. Nth order approximation Due to first order Preisach model Due to second order Preisach model

15. Conclusion As long as the hysteretic system with RPM is a smooth mapping of history, it is possible to approximate it with arbitrary accuracy on the basis of higher order rectangular hysteresis operators. It is a sort of analog to Taylor series expansion of functions; Nth order approximation satisfies Nth order congruency property which is much less restrictive than the first order congruency property