The structure of state transition graphs in hysteresis models with return point memory: I. General Theory

We consider the athermal quasi-static dynamics (AQS) of disordered systems driven by a slowly varying external field. Our interest is in an automaton description (AQS-A) that represents the AQS dynamics in terms of the graph of state transitions triggered by the driving field. A particular feature of these systems is return point memory (RPM), a tendency for the system to return to the same microstate upon cycling the field. It is known that 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 monotonous driving fields, implies RPM. When periodically driven, such systems settle into a cyclic response after a transient of at most one period. However conditions (1)-(3) are only sufficient but not necessary. In fact, we show that the AQS dynamics naturally provides a more selective partial order which, due to its connection to hysteresis loops, is a natural choice for establishing RPM. 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. Our central finding is that RPM constrains the {\em intra-loop} structure of hysteresis loops, namely its hierarchical organization into sub loops, but not the {\em inter-loop} structure. We prove that the topology of the intra-loop structure has a natural representation in terms of an ordered tree and that the corresponding state transition graph is planar. On the other hand, the RPM property does not significantly restrict inter-loop transitions. A system exhibiting RPM and subject to periodic forcing can thus undergo a large number of transient cycles before settling into a periodic response. Such systems can even exhibit subharmonic response.