On reversibility of macroscopic and microscopic dynamics in the Kuramoto model

We address the question of reversibility of collective motions in the Kuramoto model with identical oscillators and global coupling. In such a system, the dynamics is constrained on a low-dimensional invariant submanifold. This submanifold is determined by the initial state of the system. The question can be roughly stated as follows: Supposed that we know the governing equations and the state of the system at a certain moment T1, can we recover an initial state? The answers to this question are exposed through simulations of the dynamics on two invariant submanifolds and comparison between them. We show that common noise induces non-reversible dynamics, however, depending on the initial state, this non-reversibility is sometimes invisible on the macroscopic level. In such a setup, reversibility appears as a geometric concept that is related to rotational symmetries in the system. •We analyze the impact of the global phase on dynamics in the Kuramoto model.•We introduce two concepts of reversibility of the collective dynamics.•The conclusions are illustrated in two examples of invariant submanifolds.•We point out analogies with SU(1,1) coherent states in Quantum Theory.•For coherent states non-reversibility is not visible on the macroscopic level.