A solvable normal form for coupled swarmalators

Swarmalators are mobile generalizations of phase oscillators. Introduced to model systems in which sync and self-assembly interact, they remain poorly understood theoretically. Unlike the Kuramoto model for coupled oscillators, existing swarmalator models lack a normal-form foundation, and their basic stabilities and bifurcations remain largely unsolved. Here we address both problems. Building on Tanaka's reduction of chemotactic oscillators, we show that the canonical one-dimensional swarmalator model -- previously introduced as an ad hoc toy model -- is recovered in the first-harmonic, zero-lag limit, implying its behavior is generic. We then derive the stability boundaries organizing its four collective states, show they meet at a single cusp, correct a previously published order-parameter formula, and uncover a non-monotonic sync response absent in the Kuramoto model.