Unified Ott-Antonsen framework for stability and bifurcations in ring swarmalators

Swarmalators, a prototypical model that integrates temporal rhythms with spatial swarming, have been widely used to explore the collective dynamics observed in active-matter systems. Although numerical simulations have revealed a variety of collective states, a general theoretical framework for the stability and bifurcations of the states remains unavailable. Here, by the model of one-dimensional ring-structured swarmalators with general distributions of natural frequencies and velocities, we conduct a systematic theoretical analysis of the stability and bifurcations of four representative collective states observed in simulations: the desynchronization state, the phase-wave state, the synchronization state, and the mixed state. We show that the macroscopic dynamics are governed by a low-dimensional invariant manifold, namely the generalized Ott-Antonsen manifold, and further establish that the same manifold underlies the collective dynamics of high-dimensional agent models, thereby unifying these seemingly different descriptions within a common reduction framework. Building on this connection, we derive stability conditions and bifurcation boundaries for the collective states and construct a bifurcation diagram in the two-parameter coupling plane, with theoretical predictions validated by large-scale numerical simulations. Our results clarify the dynamical origins of state selection in swarmalator systems and provide a theoretical foundation for analyzing nonequilibrium phase transitions in active-matter systems with coupled internal rhythms and spatial motions.