Analytical resolution and characterization of mixed states in heterogeneous swarmalators

Develop an analytical framework to resolve and characterize the mixed states observed in the two-dimensional swarmalator model with nonidentical swarmalators (velocities and frequencies drawn from Lorentzian distributions). Derive invariant solutions for the Ott–Antonsen amplitude variables and determine whether these mixed states are genuine attractors, bistable coexisting solutions, or long transients.

Background

In the heterogeneous case, besides async, thick/thin phase waves, and sync, the authors observe mixed states that blend phase-wave and sync characteristics and appear as intermediates at small higher-order coupling. These states are unusual, not seen in standard Kuramoto-type models or in identical swarmalators.

The authors note these mixed states are not associated with fixed points in the Ott–Antonsen manifold for the Fourier amplitudes, making them analytically elusive. Understanding their nature—attractor vs. transient vs. bistable—requires new analytical tools.