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Reversible to irreversible (R-IR) transitions arise in numerous periodically driven collectively interacting systems that, after a certain number of driving cycles, organize into a reversible state where the particle trajectories repeat during every or every few cycles. On the irreversible side of the transition, the motion is chaotic. R-IR transitions were first systematically studied for periodically sheared dilute colloids, and have now been found in a wide variety of both soft and hard matter periodically driven systems, including amorphous solids, crystals, vortices in type-II superconductors, and magnetic textures. It has been shown that in several of these systems, the transition to a reversible state is an absorbing phase transition with a critical divergence in the organization timescale at the transition. The same systems are capable of storing multiple memories and may exhibit return point memory. We give an overview of R-IR transitions including recent advances in the field and discuss how the general framework of R-IR transitions could be applied to a much broader class of nonequilibrium systems in which periodic driving occurs, including not only soft and hard condensed matter systems, but also astrophysics, biological systems, and social systems. In particular, some likely candidate systems are commensurate-incommensurate states, systems exhibiting hysteresis or avalanches, nonequilibrium pattern forming states, and other systems with absorbing phase transitions. Periodic driving could be applied to hard condensed matter systems to see if organization into reversible states occurs for metal-insulator transitions, semiconductors, electron glasses, electron nematics, cold atom systems, or Bose-Einstein condensates. R-IR transitions could also be examined in dynamical systems where synchronization or phase locking occurs. We also discuss the possibility of using complex periodic driving, such as changing drive directions or using multiple frequencies, to determine whether these systems can still organize to reversible states or retain complex multiple memories. Finally, we describe features of classical and quantum time crystals that could suggest the occurrence of R-IR transitions in these systems.
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Chaotic heteroclinic networks as models of switching behavior in biological systems
Key features of biological activity can often be captured by transitions between a finite number of semi-stable states that correspond to behaviors or decisions. We present here a broad class of dynamical systems that are ideal for modeling such activity. The models we propose are chaotic heteroclinic networks with nontrivial intersections of stable and unstable manifolds. Due to the sensitive dependence on initial conditions, transitions between states are seemingly random. Dwell times, exit distributions, and other transition statistics can be built into the model through geometric design and can be controlled by tunable parameters. To test our model’s ability to simulate realistic biological phenomena, we turned to one of the most studied organisms, C. elegans, well known for its limited behavioral states. We reconstructed experimental data from two laboratories, demonstrating the model’s ability to quantitatively reproduce dwell times and transition statistics under a variety of conditions. Stochastic switching between dominant states in complex dynamical systems has been extensively studied and is often modeled as Markov chains. As an alternative, we propose here a new paradigm, namely, chaotic heteroclinic networks generated by deterministic rules (without the necessity for noise). Chaotic heteroclinic networks can be used to model systems with arbitrary architecture and size without a commensurate increase in phase dimension. They are highly flexible and able to capture a wide range of transition characteristics that can be adjusted through control parameters.
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- Award ID(s):
- 1901009
- PAR ID:
- 10635835
- Publisher / Repository:
- AIP Publishing
- Date Published:
- Journal Name:
- Chaos: An Interdisciplinary Journal of Nonlinear Science
- Volume:
- 32
- Issue:
- 12
- ISSN:
- 1054-1500
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0122184
