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The response, pathways and memory effects of cyclically driven complex mediacan be captured by hysteretic elements called hysterons. Here we demonstratethe profound impact of hysteron interactions on pathways and memory.Specifically, while the Preisach model of independent hysterons features arestricted class of pathways which always satisfy return point memory, we showthat three interacting hysterons generate more than 15,000 transition graphs,with most violating return point memory and having features completely distinctfrom the Preisach model. Exploring these opens a route to designer pathways andinformation processing in complex matter.

Profusion of Transition Pathways for Interacting Hysterons