Fig. 4: Phase diagram of the swarmalator model.
From: Tunable colloidal swarmalators with hydrodynamic coupling
K and J are the parameters governing the distance-dependent synchronization and synchronization-dependent attraction or repulsion, respectively (see equations (3) and (4)). a For K < 0, J > 0, the swarmalators arrange randomly without synchronization. b For K > 0, J > 0, highly synchronized clusters emerge. c For K < 0, J < 0, the swarmalators arrange in chain-like structures in which adjacent swarmalators have opposite phases. d For K > 0, J < 0, the swarmalators synchronize but repel each other. The colors denote the phases of the swarmalators as indicated in (d). For a quantitative description of the four phases, see SI Section 8. Additional snapshots for different parameter values are provided in Fig. S4, and an animated phase diagram is available in Supplementary video 3. For all J, K values, the noise was fixed with an amplitude Dθ = Dη = 0.001, where \(\langle {\xi }_{i,\theta }(t){\xi }_{j,\theta }(t^{\prime} )\rangle={D}_{\theta }\delta (t-t^{\prime} )\) and \(\langle {{{{\boldsymbol{\xi }}}}}_{i,\eta,}(t)\cdot {{{{\boldsymbol{\xi }}}}}_{j,\eta }(t^{\prime} )\rangle=d{D}_{\eta }\delta (t-t^{\prime} )\), where d is the dimensionality (here d = 2). We set ω0 = 1, integration time-step dt = 0.005/ω0, and ran the simulation for a total time of 20,000/ω0.
