Coupling-split clusters in a swarmalator model with uniform coupling disorder

We study the one-dimensional swarmalator model in which the phase coupling Kᵢ' is drawn from a uniform distribution. Our main result is a static coupling-split cluster, in which the population partitions across the threshold $K'=0$ that separates positively coupled (Kᵢ'>0) from negatively coupled (Kᵢ'<0) swarmalators, with smaller order parameter $s=μ/γ$ set by the positive-coupling excess. The familiar async, phase-wave, and sync states persist, but each stability boundary feels a different part of the distribution: async the mean same-coordinate response, sync the most negatively coupled particle, and the phase wave the full density through a logarithmic characteristic equation. At a cusp where its Hopf and real-eigenvalue branches meet, the phase-wave dispersion has a double zero -- the spectral signature of a Bogdanov--Takens point -- and simulations nearby show a small-amplitude breathing limit cycle. For supports containing strongly negatively coupled particles the order parameters instead oscillate persistently.