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

We introduce an abstract automaton model (DAMA) that captures the state transitions in athermal discrete disordered systems that are induced by a varying external field. A broad class of models, including the random-field Ising model (RFIM) as well as various charge-density-wave type depinning models can be formulated as DAMAs. These systems exhibit return point memory (RPM), a tendency for the system to return to the same microstate upon cycling the external field. It is known that the existence of three conditions, (1) a partial order on a set of states; (2) a no-passing property; (3) an adiabatic response to monotonous driving fields, implies RPM. When periodically driven, such systems settle therefore into a cyclic response after a transient of at most one period. However conditions (1) - (3) are only sufficient, they are not necessary. The DAMA description allow us to consider models exhibiting the RPM property without necessarily having the no-passing property. We work out in detail the structure of their associated state transition graphs. The RPM property constrains the intra-loop structure of hysteresis loops, namely its hierarchical organization in sub loops. We prove that the topology of this 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 the inter-loop structure. A system exhibiting RPM and subject to periodic forcing can thus undergo a large number of transient cycles before settling into a periodic response. RPM alone, e.g. in the absence of no-passing, does not necessarily imply a short transient response to cyclic forcing.