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Reversible to irreversible (R-IR) transitions arise in numerous periodicallydriven collectively interacting systems that, after a certain number of drivingcycles, organize into a reversible state where the particle trajectoriesrepeat, or remain irreversible with chaotic motion. R-IR transitions were firstsystematically studied for periodically sheared dilute colloids, and appear ina wide variety of both soft and hard matter systems, including amorphoussolids, crystals, vortices in type-II superconductors, and magnetic textures.In some cases, the reversible transition is an absorbing phase transition witha critical divergence in the organization time scale. R-IR systems can storemultiple memories and exhibit return point memory. We give an overview of R-IRtransitions including recent advances in the field, and discuss how the generalframework of R-IR transitions could be applied to a much broader class ofperiodically driven nonequilibrium systems, including soft and hard condensedmatter systems, astrophysics, biological systems, and social systems. Somelikely candidate systems are commensurate-incommensurate states, systemsexhibiting hysteresis or avalanches, and nonequilibrium pattern forming states.Periodic driving could be applied to hard condensed matter systems to see ifR-IR transitions occur in metal-insulator transitions, semiconductors, electronglasses, electron nematics, cold atom systems, or Bose-Einstein condensates.R-IR transitions could also be examined in dynamical systems wheresynchronization or phase locking occurs. We discuss the use of complex periodicdriving such as changing drive directions or multiple frequencies as a methodto retain complex multiple memories. Finally, we describe features of classicaland quantum time crystals that could suggest the occurrence of R-IR transitionsin these systems.

Perspective on Reversible to Irreversible Transitions in Periodic Driven Many Body Systems and Future Directions For Classical and Quantum Systems