Establish the stability of nonidentical-state solutions in the 1D ring swarmalator model

Establish the linear or nonlinear stability of the synchronized, static phase wave, and mixed states in the one-dimensional ring swarmalator model defined by Eqs. (\eqref{1d-x})–(\eqref{1d-theta}) with nonidentical natural frequencies (e.g., Lorentzian distributions). Provide rigorous stability criteria beyond the asynchronous state, whose stability is already known.

Background

The solvable 1D ring model captures key space–phase couplings using sinusoidal interactions and allows analytic treatment of existence conditions for several collective states with heterogeneous (nonidentical) oscillators, including sync, phase wave, mixed, and async.

Although existence conditions and bifurcation structure have been derived, a rigorous stability analysis for all nonidentical states other than the asynchronous state remains to be developed. Completing this analysis would clarify which macrostates are realized and robust under parameter variations and heterogeneity.