# The Structure of State Transition Graphs in Systems with Return Point Memory: I. General Theory

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## Abstract

We consider the athermal quasi-static dynamics (AQS) of disordered systems driven by an external field. Our interest is in an automaton description (AQS-A) that represents the AQS dynamics via a graph of state transitions triggered by the field and in the presence of return-point-memory (RPM) property, a tendency for the system to return to the same microstate upon cycling driving. The existence of three conditions, (1) a partial order on the set of configuration; (2) a no-passing property; and (3) an adiabatic response to monotonously changing fields, implies RPM. When periodically driven, such systems settle into a cyclic response after a transient of at most one period. While sufficient, conditions (1)–(3) are not necessary. We show that the AQS dynamics provides a more selective partial order which, due to its explicit connection to hysteresis loops, is a natural choice for establishing the RPM property. This enables us to consider AQS-A exhibiting RPM without necessarily possessing the no-passing property. We call such automata \(\ell \)AQS-A and work out the structure of their state transition graphs. We find that the RPM property constrains the *intra-loop* structure of hysteresis loops, namely its hierarchical organization into sub-loops, but not the *inter-loop* structure. We prove that the topology of the intra-loop structure can be represented as an ordered tree and show that the corresponding state transition graph is planar. On the other hand, the RPM property does not significantly restrict the inter-loop transitions. A system exhibiting RPM and subject to periodic forcing can undergo a large number of transient cycles before settling into a periodic response. Such systems can even exhibit subharmonic response.

## Mathematics Subject Classification

Primary 82D30 Secondary 82C44## Notes

### Acknowledgements

The authors would like to thank M. Işeri and D.C. Kaspar for many fruitful exchanges. They also acknowledge discussions with B. Behringer, A. Bovier, K. Dahmen, N. Keim, J. Krug, C. Maloney, A. A. Middleton, S. Nagel, A. Rosso, S. Sastry, K. Sekimoto, J. Sethna, D. Vandembroucq, T. A. Witten, and A. Yılmaz. Many of these took place during the Memory Formation in Matter program of KITP, and the authors thank KITP for its kind hospitality.

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