Exploring Basins of Attraction in the Kuramoto Ring through Sign-Flip Dynamics

The Kuramoto model is a fundamental framework to describe the synchronization of coupled oscillators. Though studied extensively, its dynamics for finite connectivity is not fully understood even in simple topologies, and the shape and structure of the basins of attraction remain an open problem. In this paper, we introduce link flipping—the switching of an interaction from excitatory to inhibitory, or vice versa—as a hybrid dynamical process on a ring of identical oscillators. Used as a deterministic probe on systems at equilibrium, link flips reveal symmetric patterns that consistently move the system toward the synchronized state, suggesting a nested organization of the basins of attraction and connecting distant points of state space to the same attractor, in agreement with the tentacle like basin geometry proposed in recent literature. We then introduce stochastic flip dynamics, in which random sign changes turn the ring into a genuinely out of equilibrium system. We show that an invariant region, previously known for the fully excitatory ring, extends to signed couplings, and that the competition between the phase relaxation time and the flipping rate organizes the long time behavior: when flips are well separated the system is driven toward synchronization, whereas frequent flips expose trajectories to non synchronized states.